From 7fa585a17db8e3030b0df68ce577e86631777bda Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Fri, 16 Aug 2024 00:34:14 -0400 Subject: [PATCH 01/51] add skeleton of tensor network tutorial --- .../tutorial_tn_basics.metadata.json | 55 +++++++++++++++++++ demonstrations/tutorial_tn_basics.py | 41 ++++++++++++++ 2 files changed, 96 insertions(+) create mode 100644 demonstrations/tutorial_tn_basics.metadata.json create mode 100644 demonstrations/tutorial_tn_basics.py diff --git a/demonstrations/tutorial_tn_basics.metadata.json b/demonstrations/tutorial_tn_basics.metadata.json new file mode 100644 index 0000000000..b2c5db8670 --- /dev/null +++ b/demonstrations/tutorial_tn_basics.metadata.json @@ -0,0 +1,55 @@ +{ + "title": "Tutorial: ABC of tensor networks", + "authors": [ + { + "id": "emiliano_godinez" + }, + { + "id": "ahmed_darwish" + } + ], + "dateOfPublication": "2022-03-29T00:00:00+00:00", + "dateOfLastModification": "2024-08-06T00:00:00+00:00", + "categories": [ + "Tensor Networks" + ], + "tags": [], + "previewImages": [ + { + "type": "thumbnail", + "uri": "/_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_tensor_network_quantum_circuits.png" + } + ], + "seoDescription": "An introductory tutorial to tensor networks.", + "doi": "", + "canonicalURL": "/qml/demos/tutorial_tn_circuits", + "references": [ + { + "id": "huggins", + "type": "article", + "title": "Towards quantum machine learning with tensor networks", + "authors": "W. Huggins, P. Patil, B. Mitchell, K. B. Whaley, and E. M. Stoudenmire", + "year": "2019", + "journal": "Quantum Science and Technology", + "url": "http://dx.doi.org/10.1088/2058-9565/aaea94" + }, + { + "id": "orus", + "type": "article", + "title": "A practical introduction to tensor networks: Matrix product states and projected entangled pair states", + "authors": "R. Orús", + "year": "2014", + "journal": "Annals of Physics", + "url": "https://www.sciencedirect.com/science/article/pii/S0003491614001596" + } + ], + "basedOnPapers": ["10.1088/2058-9565/aaea94"], + "referencedByPapers": [], + "relatedContent": [ + { + "type": "demonstration", + "id": "tutorial_variational_classifier", + "weight": 1.0 + } + ] +} \ No newline at end of file diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py new file mode 100644 index 0000000000..bfee3281a7 --- /dev/null +++ b/demonstrations/tutorial_tn_basics.py @@ -0,0 +1,41 @@ +r"""Tutorial: ABC of tensor networks +============================================================= + +Intro: +- introduce where are tensor networks used (physics, math, machine learning), why are they so widely spread, and mention how they are not just a data structure but also a diagrammatic language for formal reasoning and even proofs! +- Define the scope of this tutorial: basics of tensor networks, their relation to quantum circuits, and the importance of choosing the contraction paths. + +From matrices to tensors +^^^^^^^^^^^^^^^^^^^^^^^^^^^ +- Definition of a tensor network as an n-dimensional array. Show notation with n indices, stating that it belongs to C^{d1,...,dn}. Define rank, index, dimension (mention how these terms are sometimes used (wrongly?) interchangeably in literature). +- Graphical notation. Mention that there exist certain representations in the literature that allow to represent properties of the tensors (e.g. symmetry, orthogonality). In our case, we can adhere to a general circle. +- Specific examples used in a day-to-day: scalars, vectors, matrices. Mention that for quantum states, we can adopt the convention that the legs in one direction mean that the state belongs to one Hilbert space, and the legs to the other side to the dual space. +- CODE: include code using numpy creating a >2 dimensional array. + +From matrix multiplication to tensor contractions +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +- Show the matrix multiplication in terms of summation over indices, then using the diagrammatic representation. This results in another rank 2 tensor (matrix) +- Analagously, we can represent matrix-vector multiplication resulting in a rank 1 tensor (vector). Just as we expected! +- We can generalize this concept to tensors. This is done by summing over repeated indices (just as in einstein convention - external link for it) resulting in another tensor made up of the open legs of all the tensors together. + In diagrammatic notation, this is simply sticking together legs with same indices! (show nice diagram with >3 tensors). We have just formed a network of tensors, i.e. a Tensor Network! +- CODE: Talking about einstein convetion, we can perform this contraction of tensors using `np.einsum`. + +The cost of contracting a network (bubbling): +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +- Mention that the resulting tensor network doesn't change but the way we arrive to the final tensor affects how expensive it is to get there. +- Show how can we can calculate the complexity of a contraction by means of a simple example using 2 matrices (rank 2 tensors): dimension_contracted x (dimensions_open). + Intuition behind: we perform one operation (contraction) and repeat many times to "populate" the resulting tensor (dimension_open1 x dimension_open2). Show the equation with indices. +- Show an example with at least three tensors where they all have different dimensions. Walk through it showing that choosing to contract two indices (the ones with lower dimensions) results in a worst computational complexity than contracting other ones (the ones with higher dimensions). +- For this reason there exist heuristics for optimizing contraction path complexity. NP problem -> no perfect solution but great heuristics (https://arxiv.org/pdf/2002.01935). + (optional) mention the idea behind some of them + Link to quimb examples. +- CODE: show this using np.einsum, timeit, and very large dimensions expecting to see a difference. + +From tensor networks to quantum circuits: +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +- Quantum circuits are a restricted subclass of tensor networks +- show examples on https://arxiv.org/pdf/1912.10049 page 8 and 9 showing a quantum circuit for a bell state, defining each component as a tensor and show their contraction. +- What else? +""" + From cc0634b23f6fcbd4cf3ef9b8a3e9c11370af7efb Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Wed, 21 Aug 2024 16:57:02 -0400 Subject: [PATCH 02/51] add text to from matrices to tensors section --- demonstrations/tutorial_tn_basics.py | 80 +++++++++++++++++++++++++++- 1 file changed, 79 insertions(+), 1 deletion(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index bfee3281a7..6973175723 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -7,11 +7,89 @@ From matrices to tensors ^^^^^^^^^^^^^^^^^^^^^^^^^^^ -- Definition of a tensor network as an n-dimensional array. Show notation with n indices, stating that it belongs to C^{d1,...,dn}. Define rank, index, dimension (mention how these terms are sometimes used (wrongly?) interchangeably in literature). + +A common and intuitive way of thinking about tensors is as generalizations of vectors and matrices. That is, we can think of them as multi dimensional arrays - i.e., multi dimensional maps that are linear with respect to every parameter. +A tensor of dimensions :math:`d_1 \times d_2 \times \ldots \times \d_r` can be expressed as + +.. math:: + T_{i_1, i_2, \ldots, i_r} \in \mathbb{C}^{d_1 x d_2 x \ldots x \d_r}. + +Where each :math:`i_n` is an *index* of dimension :math:`d_n` - it can take values ranging from :math:`1` to :math:`d_n`-, and the number of indices :math:`r` is known as the *rank* of the tensor. We say :math:`T` is a rank-r tensor. + +.. tip:: + Some authors refer to the indices of the tensors as their dimensions. In this tutorial, these two concepts will have have different meanings, although related. + +For example, a scalar :math:`S` is a rank-0 tensor, a vector :math:`V_i` is a rank-1 tensor and a matrix :math:`G_{ij}` is a rank-2 tensor. + +A beautiful and powerful tool accompanying tensors (networks) is their graphical language representation. The diagram of a tensor is simply a geometric shape with a leg sticking out of it for every index in the tensor. For example, + +TODO: add diagram here as the second part of the equation. +.. math:: + T_{ijkl} = + +We can apply this same idea to represent a scalar, a vector and a matrix: + +TODO: add diagrams here. +.. math:: + S = + +.. math:: + V_i = + +.. math:: + G_{ij} = + +Does the last diagram seem familiar? It is because this is the representation of a single-qubit gate! We will see later in this tutorial the relation between quantum circuits and tensor networks. + +Some authors choose to give the shape of the tensor an additional meaning. For example, you might encounter square tensors representing symmetric matrices, to represent its invariance under transposition. + +TODO: add diagram here showing: 1. Square matrix 2. Bending of wires as transpose 3. Final matrix + +In addition, an isometry :math:`V \in \mathbb{C}^{n \times p}` - i.e., a rectangular matrix such that :math:`V V^\dagger = I_p` but :math:`V^\dagger V \neq I_n` - with :math:`n > p` can be depicted using a triangle to emphasize the isometry property: + +TODO: add diagram here showing the isometry definition (identity from the smaller space to the bigger space). See https://www.math3ma.com/blog/matrices-as-tensor-network-diagrams + +For quantum states, it is often useful to adopt the convention that writing the legs towards one direction (e.g. the right) corresponds to a ket, i.e., a vector living in the Hilbert space, while drawing the legs to the opposite direction (e.g. left) means they are a bra vector, i.e., living in the dual space. + +TODO: 1) drawing of a vector with legs towards right diretion (ket) 2) legs towards left (bra) + +TODO: add reference here for category theory. P. Selinger. A survey of graphical languages for monoidal categories. Lecture Notes +in Physics, page 289–355, 2010. +.. tip:: + The diagrammatic representation of tensors is rooted in category theory, which equips the diagrams with all the relevant information so they can used in proofs and formal reasoning! + +- Definition of a tensor as an n-dimensional array. Show notation with n indices, stating that it belongs to C^{d1,...,dn}. Define rank, index, dimension (mention how these terms are sometimes used (wrongly?) interchangeably in literature). - Graphical notation. Mention that there exist certain representations in the literature that allow to represent properties of the tensors (e.g. symmetry, orthogonality). In our case, we can adhere to a general circle. - Specific examples used in a day-to-day: scalars, vectors, matrices. Mention that for quantum states, we can adopt the convention that the legs in one direction mean that the state belongs to one Hilbert space, and the legs to the other side to the dual space. - CODE: include code using numpy creating a >2 dimensional array. +Creating a tensor in code is straightforward, and chances are you have already created one yourself. Using ``Numpy``, all we have to do is to create a ``np.array`` of the desired rank. For instance, we can start crearting a rank-1 tensor (a vector). +""" + +import numpy as np + +tensor_rank1 = np.array([1,2,3]) +print('rank: ', tensor_rank1.ndim) +print('dimensions: ', tensor_rank1.shape) + +############################################################################## +# Then, we can use this to constructor a rank-2 tensor (a matrix). + +tensor_rank2 = np.array([tensor_rank1, tensor_rank1]) +print('rank: ', tensor_rank2.ndim) +print('dimensions: ', tensor_rank2.shape) + +############################################################################## +# As you might have guessed, we can repeat this procedure to create a rank-3 tensor. + +tensor_rank3 = np.array([tensor_rank2]) +print('rank: ', tensor_rank3.ndim) +print('dimensions: ', tensor_rank3.shape) +print('Rank-3 tensor: \n', tensor_rank3) +############################################################################## +# Similarly, we can create a tensor of arbitrary rank following a similar procedure. This recursive approach is instructive to understand a rank-r tensor made up of rank-(r-1) tensors. + +""" From matrix multiplication to tensor contractions ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ From fa780f1d8211100ed47da4aaa72cd4aed4619ef8 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Wed, 21 Aug 2024 17:31:20 -0400 Subject: [PATCH 03/51] apply black and update metadata --- demonstrations/tutorial_tn_basics.metadata.json | 14 ++++---------- demonstrations/tutorial_tn_basics.py | 17 ++++++++--------- 2 files changed, 12 insertions(+), 19 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.metadata.json b/demonstrations/tutorial_tn_basics.metadata.json index b2c5db8670..17359360e6 100644 --- a/demonstrations/tutorial_tn_basics.metadata.json +++ b/demonstrations/tutorial_tn_basics.metadata.json @@ -8,7 +8,7 @@ "id": "ahmed_darwish" } ], - "dateOfPublication": "2022-03-29T00:00:00+00:00", + "dateOfPublication": "2024-08-06T00:00:00+00:00", "dateOfLastModification": "2024-08-06T00:00:00+00:00", "categories": [ "Tensor Networks" @@ -22,7 +22,7 @@ ], "seoDescription": "An introductory tutorial to tensor networks.", "doi": "", - "canonicalURL": "/qml/demos/tutorial_tn_circuits", + "canonicalURL": "/qml/demos/tutorial_tn_basics", "references": [ { "id": "huggins", @@ -43,13 +43,7 @@ "url": "https://www.sciencedirect.com/science/article/pii/S0003491614001596" } ], - "basedOnPapers": ["10.1088/2058-9565/aaea94"], + "basedOnPapers": [], "referencedByPapers": [], - "relatedContent": [ - { - "type": "demonstration", - "id": "tutorial_variational_classifier", - "weight": 1.0 - } - ] + "relatedContent": [] } \ No newline at end of file diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 6973175723..2001d78edc 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -68,24 +68,24 @@ import numpy as np -tensor_rank1 = np.array([1,2,3]) -print('rank: ', tensor_rank1.ndim) -print('dimensions: ', tensor_rank1.shape) +tensor_rank1 = np.array([1, 2, 3]) +print("rank: ", tensor_rank1.ndim) +print("dimensions: ", tensor_rank1.shape) ############################################################################## # Then, we can use this to constructor a rank-2 tensor (a matrix). tensor_rank2 = np.array([tensor_rank1, tensor_rank1]) -print('rank: ', tensor_rank2.ndim) -print('dimensions: ', tensor_rank2.shape) +print("rank: ", tensor_rank2.ndim) +print("dimensions: ", tensor_rank2.shape) ############################################################################## # As you might have guessed, we can repeat this procedure to create a rank-3 tensor. tensor_rank3 = np.array([tensor_rank2]) -print('rank: ', tensor_rank3.ndim) -print('dimensions: ', tensor_rank3.shape) -print('Rank-3 tensor: \n', tensor_rank3) +print("rank: ", tensor_rank3.ndim) +print("dimensions: ", tensor_rank3.shape) +print("Rank-3 tensor: \n", tensor_rank3) ############################################################################## # Similarly, we can create a tensor of arbitrary rank following a similar procedure. This recursive approach is instructive to understand a rank-r tensor made up of rank-(r-1) tensors. @@ -116,4 +116,3 @@ - show examples on https://arxiv.org/pdf/1912.10049 page 8 and 9 showing a quantum circuit for a bell state, defining each component as a tensor and show their contraction. - What else? """ - From b91e44dfe659a3aad679639827c836a0c1702294 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Wed, 21 Aug 2024 17:51:04 -0400 Subject: [PATCH 04/51] add first paragraphs on contraction section --- demonstrations/tutorial_tn_basics.py | 19 +++++++++++++++---- 1 file changed, 15 insertions(+), 4 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 2001d78edc..4d6a85f7f1 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -19,7 +19,7 @@ .. tip:: Some authors refer to the indices of the tensors as their dimensions. In this tutorial, these two concepts will have have different meanings, although related. -For example, a scalar :math:`S` is a rank-0 tensor, a vector :math:`V_i` is a rank-1 tensor and a matrix :math:`G_{ij}` is a rank-2 tensor. +For example, a scalar :math:`s` is a rank-0 tensor, a vector :math:`v_i` is a rank-1 tensor and a matrix :math:`G_{ij}` is a rank-2 tensor. A beautiful and powerful tool accompanying tensors (networks) is their graphical language representation. The diagram of a tensor is simply a geometric shape with a leg sticking out of it for every index in the tensor. For example, @@ -31,10 +31,10 @@ TODO: add diagrams here. .. math:: - S = + s = .. math:: - V_i = + v_i = .. math:: G_{ij} = @@ -87,11 +87,22 @@ print("dimensions: ", tensor_rank3.shape) print("Rank-3 tensor: \n", tensor_rank3) ############################################################################## -# Similarly, we can create a tensor of arbitrary rank following a similar procedure. This recursive approach is instructive to understand a rank-r tensor made up of rank-(r-1) tensors. +# Similarly, we can create a tensor of arbitrary rank following a similar procedure. This recursive approach is instructive to understand a rank-r tensor as made up of rank-(r-1) tensors. """ From matrix multiplication to tensor contractions ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +Matrix-matrix and matrix-vector multiplications are familiar operations within the context of quantum computing. We can now study these opereations under the lens of the tensor notation introduced above. First, a matrix and a vector can be multiplied as + +TODO: add diagram at the end of the equation. +.. math:: + (w)_i = G \cdot v = \sum_j G_ij v_j = + +We see that summing over the shared index :math:`j` is equivalent to **contracting** the corresponding legs from the matrix and vector diagrams. As expected, the result of this multiplication is another rank-1 tensor with dangling leg :math:`i`. Similarly, we can look at the matrix-matrix multiplication: + +TODO: add diagram at the end of the equation. +.. math:: + (G^3)_{jk} = G^1 \cdot G^2 = \sum_j G^{1}_{ij} G^{2}_{jk} = - Show the matrix multiplication in terms of summation over indices, then using the diagrammatic representation. This results in another rank 2 tensor (matrix) - Analagously, we can represent matrix-vector multiplication resulting in a rank 1 tensor (vector). Just as we expected! From 7028d4ae7f977bfdf75d58958c1a2ef3e2fbabf6 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Wed, 21 Aug 2024 19:28:55 -0400 Subject: [PATCH 05/51] add correct category to metadata --- demonstrations/tutorial_tn_basics.metadata.json | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_tn_basics.metadata.json b/demonstrations/tutorial_tn_basics.metadata.json index 17359360e6..cf773425c2 100644 --- a/demonstrations/tutorial_tn_basics.metadata.json +++ b/demonstrations/tutorial_tn_basics.metadata.json @@ -11,7 +11,7 @@ "dateOfPublication": "2024-08-06T00:00:00+00:00", "dateOfLastModification": "2024-08-06T00:00:00+00:00", "categories": [ - "Tensor Networks" + "Getting Started" ], "tags": [], "previewImages": [ From ff4ee64d0abc857c49c78f47ffe04c1d4bddc851 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Fri, 23 Aug 2024 15:48:16 -0400 Subject: [PATCH 06/51] add section on creating a tensor networks --- demonstrations/tutorial_tn_basics.py | 102 +++++++++++++++++++-------- 1 file changed, 72 insertions(+), 30 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 4d6a85f7f1..90c1d3718a 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -9,27 +9,29 @@ ^^^^^^^^^^^^^^^^^^^^^^^^^^^ A common and intuitive way of thinking about tensors is as generalizations of vectors and matrices. That is, we can think of them as multi dimensional arrays - i.e., multi dimensional maps that are linear with respect to every parameter. -A tensor of dimensions :math:`d_1 \times d_2 \times \ldots \times \d_r` can be expressed as +A tensor of dimensions :math:`d_1 \times d_2 \times \ldots \times d_r` can be expressed as .. math:: - T_{i_1, i_2, \ldots, i_r} \in \mathbb{C}^{d_1 x d_2 x \ldots x \d_r}. + T_{i_1, i_2, \ldots, i_r} \in \mathbb{C}^{d_1 \times d_2 \times \ldots \times d_r}. Where each :math:`i_n` is an *index* of dimension :math:`d_n` - it can take values ranging from :math:`1` to :math:`d_n`-, and the number of indices :math:`r` is known as the *rank* of the tensor. We say :math:`T` is a rank-r tensor. .. tip:: Some authors refer to the indices of the tensors as their dimensions. In this tutorial, these two concepts will have have different meanings, although related. -For example, a scalar :math:`s` is a rank-0 tensor, a vector :math:`v_i` is a rank-1 tensor and a matrix :math:`G_{ij}` is a rank-2 tensor. +For example, a scalar :math:`s` is a rank-0 tensor, a vector :math:`v_i` is a rank-1 tensor and a matrix :math:`G_{i,j}` is a rank-2 tensor. A beautiful and powerful tool accompanying tensors (networks) is their graphical language representation. The diagram of a tensor is simply a geometric shape with a leg sticking out of it for every index in the tensor. For example, TODO: add diagram here as the second part of the equation. + .. math:: - T_{ijkl} = + T_{i,j,k,l} = We can apply this same idea to represent a scalar, a vector and a matrix: TODO: add diagrams here. + .. math:: s = @@ -37,7 +39,7 @@ v_i = .. math:: - G_{ij} = + G_{i,j} = Does the last diagram seem familiar? It is because this is the representation of a single-qubit gate! We will see later in this tutorial the relation between quantum circuits and tensor networks. @@ -55,14 +57,10 @@ TODO: add reference here for category theory. P. Selinger. A survey of graphical languages for monoidal categories. Lecture Notes in Physics, page 289–355, 2010. + .. tip:: The diagrammatic representation of tensors is rooted in category theory, which equips the diagrams with all the relevant information so they can used in proofs and formal reasoning! -- Definition of a tensor as an n-dimensional array. Show notation with n indices, stating that it belongs to C^{d1,...,dn}. Define rank, index, dimension (mention how these terms are sometimes used (wrongly?) interchangeably in literature). -- Graphical notation. Mention that there exist certain representations in the literature that allow to represent properties of the tensors (e.g. symmetry, orthogonality). In our case, we can adhere to a general circle. -- Specific examples used in a day-to-day: scalars, vectors, matrices. Mention that for quantum states, we can adopt the convention that the legs in one direction mean that the state belongs to one Hilbert space, and the legs to the other side to the dual space. -- CODE: include code using numpy creating a >2 dimensional array. - Creating a tensor in code is straightforward, and chances are you have already created one yourself. Using ``Numpy``, all we have to do is to create a ``np.array`` of the desired rank. For instance, we can start crearting a rank-1 tensor (a vector). """ @@ -87,28 +85,54 @@ print("dimensions: ", tensor_rank3.shape) print("Rank-3 tensor: \n", tensor_rank3) ############################################################################## -# Similarly, we can create a tensor of arbitrary rank following a similar procedure. This recursive approach is instructive to understand a rank-r tensor as made up of rank-(r-1) tensors. +# Similarly, we can create a tensor of arbitrary rank following a similar procedure. This recursive approach is instructive to understand a rank-r tensor as made up of rank-(r-1) tensors, which translates to the code as adding an additional level in the nested bracket structure ``[tensor_rank_r-1]``. +############################################################################## +# From matrix multiplication to tensor contractions +# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +# Matrix-matrix and matrix-vector multiplications are familiar operations within the context of quantum computing. We can now study these operations under the lens of the tensor notation introduced above. First, a matrix and a vector can be multiplied as +# +# TODO: add diagram at the end of the equation. +# +# .. math:: +# (w)_i = G \cdot v = \sum_j G_ij v_j = +# +# We see that summing over the shared index :math:`j` is equivalent to **contracting** the corresponding legs from the matrix and vector diagrams. As expected, the result of this multiplication is another rank-1 tensor with dangling leg :math:`i`. Similarly, we can look at the matrix-matrix multiplication: +# +# TODO: add diagram at the end of the equation. +# +# .. math:: +# (G^3)_{j,k} = G^1 \cdot G^2 = \sum_j G^{1}_{i,j} G^{2}_{j,k} = +# +# In this case, the resulting tensor has two dangling indices, :math:`i` and :math:`k`, which defines a matrix, as expected! +# +# We can now generalize this concept to tensors, and consequently, to more than a pair of legs being contracted. For example, let us look at three tensors :math:`A_{i,j,k}`, :math:`B_{j,l,m}` and :math:`C_{k,m,n}`. To contract them, all we need to do is to sum over repeated indices (:math:`j`, :math:`k`, :math:`m`), just as we would do in Einstein convention. Thus, the (i,l,n)-th element of the resulting tensor :math:`D` is +# +# .. math:: +# (D)_{i,l,n} = \sum_{j,k,m} `A_{i,j,k} B_{j,l,m} C_{k,m,n} . +# +# Note how the resulting rank-3 tensor is made up of the open legs from all the initial tensors (i,l,n). The diagrammatic representation of this equation is obtained by sticking all the legs with the same indices together. +# +# TODO: add figure here. +# +# With the above contraction we have formed a network of tensors, i.e. a **Tensor Network**! +# +# .. tip:: +# A common question arising when drawing a tensor is "what is the correct order to draw the indices". For instance, in the figure above we have adopted the convention that a tensor :math:`A_{i,j,k}` corresponds to a diagram with the the first leg (i) pointing left, the second leg (j) pointing upwards and the third leg (k) pointing right, and similarly for the other two tensors. However, this need not be the case. We could have defined the first leg to be the one pointing upwards, for example. Based on the use case, and the user, some conventions might seem more natural than others. The only important thing to keep in mind is to be consistent. In other words, once we choose a convetion for the order, we should apply it to all the tensors to avoid contracting the wrong indices ❌. +# +# Remember we pointed out the similarity in the notation for the tensor network contractions? Then it doesn't come as a suprise that we can perform a contraction using the function ``np.einsum``. We start creating the 3 tensors in the network by reshaping a 1D array (coming from ``np.arange``) into rank-3 tensors of the right dimensions. + +# Create the individual rank-3 tensors +A = np.arange(6).reshape(1,2,3) # ijk +B = np.arange(6).reshape(2,3,1) # jlm +C = np.arange(6).reshape(3,1,2) # kmn + +D = np.einsum('ijk, jlm, kmn -> iln', A, B, C) +print(D.shape) + +# The ``np.einsum`` takes as inputs the tensors to be contracted and a string showing the indices of the each tensor and (optionally) the indices of the output tensor. """ -From matrix multiplication to tensor contractions -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ -Matrix-matrix and matrix-vector multiplications are familiar operations within the context of quantum computing. We can now study these opereations under the lens of the tensor notation introduced above. First, a matrix and a vector can be multiplied as - -TODO: add diagram at the end of the equation. -.. math:: - (w)_i = G \cdot v = \sum_j G_ij v_j = - -We see that summing over the shared index :math:`j` is equivalent to **contracting** the corresponding legs from the matrix and vector diagrams. As expected, the result of this multiplication is another rank-1 tensor with dangling leg :math:`i`. Similarly, we can look at the matrix-matrix multiplication: - -TODO: add diagram at the end of the equation. -.. math:: - (G^3)_{jk} = G^1 \cdot G^2 = \sum_j G^{1}_{ij} G^{2}_{jk} = - -- Show the matrix multiplication in terms of summation over indices, then using the diagrammatic representation. This results in another rank 2 tensor (matrix) -- Analagously, we can represent matrix-vector multiplication resulting in a rank 1 tensor (vector). Just as we expected! -- We can generalize this concept to tensors. This is done by summing over repeated indices (just as in einstein convention - external link for it) resulting in another tensor made up of the open legs of all the tensors together. - In diagrammatic notation, this is simply sticking together legs with same indices! (show nice diagram with >3 tensors). We have just formed a network of tensors, i.e. a Tensor Network! -- CODE: Talking about einstein convetion, we can perform this contraction of tensors using `np.einsum`. The cost of contracting a network (bubbling): ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -127,3 +151,21 @@ - show examples on https://arxiv.org/pdf/1912.10049 page 8 and 9 showing a quantum circuit for a bell state, defining each component as a tensor and show their contraction. - What else? """ + +""" +DRAFT: + +1) +- Definition of a tensor as an n-dimensional array. Show notation with n indices, stating that it belongs to C^{d1,...,dn}. Define rank, index, dimension (mention how these terms are sometimes used (wrongly?) interchangeably in literature). +- Graphical notation. Mention that there exist certain representations in the literature that allow to represent properties of the tensors (e.g. symmetry, orthogonality). In our case, we can adhere to a general circle. +- Specific examples used in a day-to-day: scalars, vectors, matrices. Mention that for quantum states, we can adopt the convention that the legs in one direction mean that the state belongs to one Hilbert space, and the legs to the other side to the dual space. +- CODE: include code using numpy creating a >2 dimensional array. + +2) + +- Show the matrix multiplication in terms of summation over indices, then using the diagrammatic representation. This results in another rank 2 tensor (matrix) +- Analagously, we can represent matrix-vector multiplication resulting in a rank 1 tensor (vector). Just as we expected! +- We can generalize this concept to tensors. This is done by summing over repeated indices (just as in einstein convention - external link for it) resulting in another tensor made up of the open legs of all the tensors together. + In diagrammatic notation, this is simply sticking together legs with same indices! (show nice diagram with >3 tensors). We have just formed a network of tensors, i.e. a Tensor Network! +- CODE: Talking about einstein convetion, we can perform this contraction of tensors using `np.einsum`. +""" From b8b91a2a75d8296731b86a5c4ac93675e3ad082f Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Fri, 23 Aug 2024 15:48:51 -0400 Subject: [PATCH 07/51] add section on creating a tensor networks --- demonstrations/tutorial_tn_basics.py | 40 ++++++++++++++-------------- 1 file changed, 20 insertions(+), 20 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 90c1d3718a..5b96b229f9 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -92,43 +92,43 @@ # ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ # Matrix-matrix and matrix-vector multiplications are familiar operations within the context of quantum computing. We can now study these operations under the lens of the tensor notation introduced above. First, a matrix and a vector can be multiplied as -# +# # TODO: add diagram at the end of the equation. -# +# # .. math:: # (w)_i = G \cdot v = \sum_j G_ij v_j = -# +# # We see that summing over the shared index :math:`j` is equivalent to **contracting** the corresponding legs from the matrix and vector diagrams. As expected, the result of this multiplication is another rank-1 tensor with dangling leg :math:`i`. Similarly, we can look at the matrix-matrix multiplication: -# +# # TODO: add diagram at the end of the equation. -# +# # .. math:: -# (G^3)_{j,k} = G^1 \cdot G^2 = \sum_j G^{1}_{i,j} G^{2}_{j,k} = -# +# (G^3)_{j,k} = G^1 \cdot G^2 = \sum_j G^{1}_{i,j} G^{2}_{j,k} = +# # In this case, the resulting tensor has two dangling indices, :math:`i` and :math:`k`, which defines a matrix, as expected! -# +# # We can now generalize this concept to tensors, and consequently, to more than a pair of legs being contracted. For example, let us look at three tensors :math:`A_{i,j,k}`, :math:`B_{j,l,m}` and :math:`C_{k,m,n}`. To contract them, all we need to do is to sum over repeated indices (:math:`j`, :math:`k`, :math:`m`), just as we would do in Einstein convention. Thus, the (i,l,n)-th element of the resulting tensor :math:`D` is -# +# # .. math:: -# (D)_{i,l,n} = \sum_{j,k,m} `A_{i,j,k} B_{j,l,m} C_{k,m,n} . -# +# (D)_{i,l,n} = \sum_{j,k,m} `A_{i,j,k} B_{j,l,m} C_{k,m,n} . +# # Note how the resulting rank-3 tensor is made up of the open legs from all the initial tensors (i,l,n). The diagrammatic representation of this equation is obtained by sticking all the legs with the same indices together. -# +# # TODO: add figure here. -# +# # With the above contraction we have formed a network of tensors, i.e. a **Tensor Network**! -# +# # .. tip:: # A common question arising when drawing a tensor is "what is the correct order to draw the indices". For instance, in the figure above we have adopted the convention that a tensor :math:`A_{i,j,k}` corresponds to a diagram with the the first leg (i) pointing left, the second leg (j) pointing upwards and the third leg (k) pointing right, and similarly for the other two tensors. However, this need not be the case. We could have defined the first leg to be the one pointing upwards, for example. Based on the use case, and the user, some conventions might seem more natural than others. The only important thing to keep in mind is to be consistent. In other words, once we choose a convetion for the order, we should apply it to all the tensors to avoid contracting the wrong indices ❌. -# +# # Remember we pointed out the similarity in the notation for the tensor network contractions? Then it doesn't come as a suprise that we can perform a contraction using the function ``np.einsum``. We start creating the 3 tensors in the network by reshaping a 1D array (coming from ``np.arange``) into rank-3 tensors of the right dimensions. -# Create the individual rank-3 tensors -A = np.arange(6).reshape(1,2,3) # ijk -B = np.arange(6).reshape(2,3,1) # jlm -C = np.arange(6).reshape(3,1,2) # kmn +# Create the individual rank-3 tensors +A = np.arange(6).reshape(1, 2, 3) # ijk +B = np.arange(6).reshape(2, 3, 1) # jlm +C = np.arange(6).reshape(3, 1, 2) # kmn -D = np.einsum('ijk, jlm, kmn -> iln', A, B, C) +D = np.einsum("ijk, jlm, kmn -> iln", A, B, C) print(D.shape) # The ``np.einsum`` takes as inputs the tensors to be contracted and a string showing the indices of the each tensor and (optionally) the indices of the output tensor. From fa36a5dfcd406be2722d8c46c784ee83685c6ca0 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Fri, 23 Aug 2024 16:28:13 -0400 Subject: [PATCH 08/51] add CNOT explanation to contractions --- demonstrations/tutorial_tn_basics.py | 74 +++++++++++++++++++--------- 1 file changed, 52 insertions(+), 22 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 5b96b229f9..2b189e45ee 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -90,7 +90,7 @@ ############################################################################## # From matrix multiplication to tensor contractions # ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - +# # Matrix-matrix and matrix-vector multiplications are familiar operations within the context of quantum computing. We can now study these operations under the lens of the tensor notation introduced above. First, a matrix and a vector can be multiplied as # # TODO: add diagram at the end of the equation. @@ -131,27 +131,57 @@ D = np.einsum("ijk, jlm, kmn -> iln", A, B, C) print(D.shape) -# The ``np.einsum`` takes as inputs the tensors to be contracted and a string showing the indices of the each tensor and (optionally) the indices of the output tensor. -""" - -The cost of contracting a network (bubbling): -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ -- Mention that the resulting tensor network doesn't change but the way we arrive to the final tensor affects how expensive it is to get there. -- Show how can we can calculate the complexity of a contraction by means of a simple example using 2 matrices (rank 2 tensors): dimension_contracted x (dimensions_open). - Intuition behind: we perform one operation (contraction) and repeat many times to "populate" the resulting tensor (dimension_open1 x dimension_open2). Show the equation with indices. -- Show an example with at least three tensors where they all have different dimensions. Walk through it showing that choosing to contract two indices (the ones with lower dimensions) results in a worst computational complexity than contracting other ones (the ones with higher dimensions). -- For this reason there exist heuristics for optimizing contraction path complexity. NP problem -> no perfect solution but great heuristics (https://arxiv.org/pdf/2002.01935). - (optional) mention the idea behind some of them - Link to quimb examples. -- CODE: show this using np.einsum, timeit, and very large dimensions expecting to see a difference. - -From tensor networks to quantum circuits: -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ -- Quantum circuits are a restricted subclass of tensor networks -- show examples on https://arxiv.org/pdf/1912.10049 page 8 and 9 showing a quantum circuit for a bell state, defining each component as a tensor and show their contraction. -- What else? -""" - +############################################################################## +# The ``np.einsum`` takes as inputs the tensors to be contracted and a string showing the indices of the each tensor and (optionally) the indices of the output tensor. To end this section, we want to discuss a common example of a tensor network contraction arising in quantum computing, namely the **CNOT** gate. The CNOT gate can be expressed in the computational basis as +# +# .. math:: +# CNOT = \ket{0}bra{0} \otimes I + \ket{1}bra{1} \otimes X. +# +# That is, if the control qubit is in the :math:`\ket{1}` state, we apply the :math:`X` gate on the target qubit, otherwise, we leave them untouched. Alternatively, we can rewrite this equation as a contraction. To do so, we define two tensors: +# +# .. math:: +# T^1 = \begin{pmatrix} +# \ket{0}\bra{0} \\ +# \ket{1}\bra{1} +# \end{pmatrix} +# +# and +# +# .. math:: +# T^2 = \begin{pmatrix} +# I \\ +# X +# \end{pmatrix} +# +# This means, :math:`(T^1)_{i,j,k}` and :math:`(T^2)_{l,j,m}` are two rank-3 tensors, where the index :math:`j` "*picks*" the elements in the column vector while the first and last indices correspond to the indices of the internal tensors (matrices). For instance, the :math:`0`-th element of :math:`T^1` and :math:`T^2` are :math:`\ket{0}\bra{0}` and :math:`I`, respectively. And similarly for their :math:`1`-st element. This means we can redefine the CNOT expression from above as +# +# .. math:: +# CNOT = \sum_j (T^1)_{i,j,k} \otimes (T^2)_{l,j,m}. +# +# It turns out :math:`T^1` and :math:`T^2` are special tensors known as the COPY and XOR tensors, respectively, and therefore have special diagrammatic representations. +# +# TODO: add a diagram for each of these +# +# Then, their contraction results in the well known CNOT quantum circuit representation. +# +# TODO: add the CNOT diagram here. +############################################################################## +# The cost of contracting a network: +# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +# - Mention that the resulting tensor network doesn't change but the way we arrive to the final tensor affects how expensive it is to get there. +# - Show how can we can calculate the complexity of a contraction by means of a simple example using 2 matrices (rank 2 tensors): dimension_contracted x (dimensions_open). +# Intuition behind: we perform one operation (contraction) and repeat many times to "populate" the resulting tensor (dimension_open1 x dimension_open2). Show the equation with indices. +# - Show an example with at least three tensors where they all have different dimensions. Walk through it showing that choosing to contract two indices (the ones with lower dimensions) results in a worst computational complexity than contracting other ones (the ones with higher dimensions). +# - For this reason there exist heuristics for optimizing contraction path complexity. NP problem -> no perfect solution but great heuristics (https://arxiv.org/pdf/2002.01935). +# (optional) mention the idea behind some of them +# Link to quimb examples. +# - CODE: show this using np.einsum, timeit, and very large dimensions expecting to see a difference. +############################################################################## +# From tensor networks to quantum circuits: +# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +# - Quantum circuits are a restricted subclass of tensor networks +# - show examples on https://arxiv.org/pdf/1912.10049 page 8 and 9 showing a quantum circuit for a bell state, defining each component as a tensor and show their contraction. +# - What else? 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zF}Rf;92^{uZzk2xocV-ufdG+w;8+>>OSa(f)z;PmH;!;tU`+Z5nb}5xFmn~TzPT6W zJPyapBF#p|0W2ClI?2h&pep|o90D(U48)C$GW*{0rIcthDY`7vzB;nO!ar_RwY8Vg z`+}4%e@2L2h1JG^Z_?CmZSQ~fb^pzMR9O-cGlL^d^*FJ3za^A z4a*Dd6=3 Date: Tue, 27 Aug 2024 10:57:38 -0400 Subject: [PATCH 10/51] attempt adding inline graphics --- demonstrations/tutorial_tn_basics.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 61f6ed12f1..5c1cd13565 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -26,7 +26,7 @@ TODO: add diagram here as the second part of the equation. .. math:: - T_{i,j,k,l} = + T_{i,j,k,l} = \includegraphics{../_static/demonstration_assets/tn_basics/tensor_one.png} .. figure:: ../_static/demonstration_assets/tn_basics/tensor_one.png :align: center From 46d43c3bc97619af4bb9a42769be1ed1f429acc7 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Tue, 27 Aug 2024 11:21:13 -0400 Subject: [PATCH 11/51] erase sphinx image insertion --- demonstrations/tutorial_tn_basics.py | 5 ----- 1 file changed, 5 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 5c1cd13565..aaa4af77ca 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -28,11 +28,6 @@ .. math:: T_{i,j,k,l} = \includegraphics{../_static/demonstration_assets/tn_basics/tensor_one.png} -.. figure:: ../_static/demonstration_assets/tn_basics/tensor_one.png - :align: center - :width: 60% - - We can apply this same idea to represent a scalar, a vector and a matrix: TODO: add diagrams here. From 1abeb74206aef1eb5733e1d02dfa59ad2129ceeb Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Tue, 27 Aug 2024 14:03:44 -0400 Subject: [PATCH 12/51] attempt to change the size of image using includegraphics --- demonstrations/tutorial_tn_basics.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index aaa4af77ca..8806338211 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -26,7 +26,7 @@ TODO: add diagram here as the second part of the equation. .. math:: - T_{i,j,k,l} = \includegraphics{../_static/demonstration_assets/tn_basics/tensor_one.png} + T_{i,j,k,l} = \includegraphics[height=0.2\textheight]{../_static/demonstration_assets/tn_basics/tensor_one.png} We can apply this same idea to represent a scalar, a vector and a matrix: From 7e710535cf03dfaea562f888c05d95e68b8ff76f Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Fri, 30 Aug 2024 11:43:44 -0400 Subject: [PATCH 13/51] fix indentation issue --- demonstrations/tutorial_tn_basics.py | 32 ++++++++++++++++++---------- 1 file changed, 21 insertions(+), 11 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 8806338211..692a0b34aa 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -26,7 +26,11 @@ TODO: add diagram here as the second part of the equation. .. math:: - T_{i,j,k,l} = \includegraphics[height=0.2\textheight]{../_static/demonstration_assets/tn_basics/tensor_one.png} + T_{i,j,k,l} = + +.. figure:: ../_static/demonstration_assets/tn_basics/tensor_one.png + :align: center + :width: 60% We can apply this same idea to represent a scalar, a vector and a matrix: @@ -71,7 +75,7 @@ print("dimensions: ", tensor_rank1.shape) ############################################################################## -# Then, we can use this to constructor a rank-2 tensor (a matrix). +# Then, we can use this to construct a rank-2 tensor (a matrix). tensor_rank2 = np.array([tensor_rank1, tensor_rank1]) print("rank: ", tensor_rank2.ndim) @@ -84,9 +88,11 @@ print("rank: ", tensor_rank3.ndim) print("dimensions: ", tensor_rank3.shape) print("Rank-3 tensor: \n", tensor_rank3) + ############################################################################## -# Similarly, we can create a tensor of arbitrary rank following a similar procedure. This recursive approach is instructive to understand a rank-r tensor as made up of rank-(r-1) tensors, which translates to the code as adding an additional level in the nested bracket structure ``[tensor_rank_r-1]``. +# We can create a tensor of arbitrary rank following a similar procedure. This recursive approach is instructive to understand a rank-r tensor as made up of rank-(r-1) tensors, which translates to the code as adding an additional level in the nested bracket structure ``[tensor_rank_r-1]``. # + ############################################################################## # From matrix multiplication to tensor contractions # ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -107,37 +113,40 @@ # # In this case, the resulting tensor has two dangling indices, :math:`i` and :math:`k`, which defines a matrix, as expected! # -# We can now generalize this concept to tensors, and consequently, to more than a pair of legs being contracted. For example, let us look at three tensors :math:`A_{i,j,k}`, :math:`B_{j,l,m}` and :math:`C_{k,m,n}`. To contract them, all we need to do is to sum over repeated indices (:math:`j`, :math:`k`, :math:`m`), just as we would do in Einstein convention. Thus, the (i,l,n)-th element of the resulting tensor :math:`D` is +# We can now generalize this concept to tensors, and consequently, to more than a pair of legs being contracted. For example, let us look at three tensors :math:`A_{i,j,k}`, :math:`B_{j,l,m}` and :math:`C_{k,m,n}`. To contract them, all we need to do is to sum over repeated indices (:math:`j`, :math:`k`, :math:`m`), just as we would do in `Einstein convention `_. Thus, the (i,l,n)-th element of the resulting tensor :math:`D` is # # .. math:: # (D)_{i,l,n} = \sum_{j,k,m} A_{i,j,k} B_{j,l,m} C_{k,m,n} . # -# Note how the resulting rank-3 tensor is made up of the open legs from all the initial tensors :math:`(i,l,n)`. The diagrammatic representation of this equation is obtained by sticking all the legs with the same indices together. +# Note how the resulting rank-3 tensor is made up of the remaining open legs from the initial tensors :math:`(i,l,n)`. The diagrammatic representation of this equation is obtained by sticking all the legs with the same indices together. # # TODO: add figure here. # -# With the above contraction we have formed a network of tensors, i.e. a **Tensor Network**! +# With the above contraction we have formed a network of tensors, i.e., a **Tensor Network**! # # .. tip:: # A common question arising when drawing a tensor is "what is the correct order to draw the indices". For instance, in the figure above we have adopted the convention that a tensor :math:`A_{i,j,k}` corresponds to a diagram with the the first leg (:math:`i`) pointing left, the second leg (:math:`j`) pointing upwards and the third leg (:math:`k`) pointing right, and similarly for the other two tensors. However, this need not be the case. We could have defined the first leg to be the one pointing upwards, for example. Based on the use case, and the user, some conventions might seem more natural than others. The only important thing to keep in mind is to be consistent. In other words, once we choose a convetion for the order, we should apply it to all the tensors to avoid contracting the wrong indices ❌. # -# Remember we pointed out the similarity in the notation between the tensor network contractions and Einstein notation? Then it doesn't come as a suprise that we can perform a contraction using the function ``np.einsum``. To do so, we can start creating the 3 tensors to be contracted by reshaping a 1D array (coming from ``np.arange``) into rank-3 tensors of the correct dimensions. +# Remember we pointed out the similarity in the notation between the tensor network contractions and Einstein notation? Then, it doesn't come as a suprise that we can perform a contraction using the function ``np.einsum``. To do so, we can start creating the 3 tensors to be contracted by reshaping a 1D array (created using``np.arange``) into rank-3 tensors of the correct dimensions. # Create the individual rank-3 tensors A = np.arange(6).reshape(1, 2, 3) # ijk B = np.arange(6).reshape(2, 3, 1) # jlm C = np.arange(6).reshape(3, 1, 2) # kmn +############################################################################## +# The ``np.einsum`` function takes as inputs the tensors to be contracted and a string showing the indices of the each tensor and (optionally) the indices of the output tensor. + D = np.einsum("ijk, jlm, kmn -> iln", A, B, C) print(D.shape) ############################################################################## -# The ``np.einsum`` takes as inputs the tensors to be contracted and a string showing the indices of the each tensor and (optionally) the indices of the output tensor. To end this section, we want to discuss a common example of a tensor network contraction arising in quantum computing, namely the **CNOT** gate. The CNOT gate can be expressed in the computational basis as +# To end this section, we want to discuss a common example of a tensor network contraction arising in quantum computing, namely the **CNOT** gate. The CNOT gate can be expressed in the computational basis as # # .. math:: # CNOT = \ket{0}\bra{0} \otimes I + \ket{1}\bra{1} \otimes X. # -# That is, if the control qubit is in the :math:`\ket{1}` state, we apply the :math:`X` gate on the target qubit, otherwise, we leave them untouched. Alternatively, we can rewrite this equation as a contraction. To do so, we define two tensors: +# That is, if the control qubit is in the :math:`\ket{1}` state, we apply the :math:`X` gate on the target qubit, otherwise, we leave them untouched. However, we can also rewrite this equation as a contraction. To do so, we define two tensors: # # .. math:: # T^1 = \begin{pmatrix} @@ -153,7 +162,7 @@ # X # \end{pmatrix} # -# This means, :math:`(T^1)_{i,j,k}` and :math:`(T^2)_{l,j,m}` are two rank-3 tensors, where the index :math:`j` "*picks*" the elements in the column vector while the first and last indices correspond to the indices of the internal tensors (matrices). For instance, the :math:`0`-th element of :math:`T^1` and :math:`T^2` are :math:`\ket{0}\bra{0}` and :math:`I`, respectively. And similarly for their :math:`1`-st element. This means we can redefine the CNOT expression from above as +# This means :math:`(T^1)_{i,j,k}` and :math:`(T^2)_{l,j,m}` are two rank-3 tensors, where the index :math:`j` "*picks*" the elements in the column vector while the first and last indices correspond to the indices of the internal tensors (matrices). For instance, the :math:`j = 0`-th element of :math:`T^1` and :math:`T^2` are :math:`\ket{0}\bra{0}` and :math:`I`, respectively. And similarly for their :math:`j = 1`-st element. This means we can redefine the CNOT expression from above as # # .. math:: # CNOT = \sum_j (T^1)_{i,j,k} \otimes (T^2)_{l,j,m}. @@ -165,9 +174,10 @@ # Then, their contraction results in the well known CNOT quantum circuit representation. # # TODO: add the CNOT diagram here. + ############################################################################## # The cost of contracting a network: -# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ # - Mention that the resulting tensor network doesn't change but the way we arrive to the final tensor affects how expensive it is to get there. # - Show how can we can calculate the complexity of a contraction by means of a simple example using 2 matrices (rank 2 tensors): dimension_contracted x (dimensions_open). # Intuition behind: we perform one operation (contraction) and repeat many times to "populate" the resulting tensor (dimension_open1 x dimension_open2). Show the equation with indices. From e8224162906787d56633b1f38e2f954f35c348a2 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Fri, 30 Aug 2024 15:12:10 -0400 Subject: [PATCH 14/51] add explanations of complexity of contractions including code example --- demonstrations/tutorial_tn_basics.py | 112 +++++++++++++++++++++------ 1 file changed, 90 insertions(+), 22 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 692a0b34aa..ec71ed71b8 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -109,7 +109,7 @@ # TODO: add diagram at the end of the equation. # # .. math:: -# (G^3)_{j,k} = G^1 \cdot G^2 = \sum_j G^{1}_{i,j} G^{2}_{j,k} = +# (G^3)_{i,k} = G^1 \cdot G^2 = \sum_j G^{1}_{i,j} G^{2}_{j,k} = # # In this case, the resulting tensor has two dangling indices, :math:`i` and :math:`k`, which defines a matrix, as expected! # @@ -178,34 +178,102 @@ ############################################################################## # The cost of contracting a network: # ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ -# - Mention that the resulting tensor network doesn't change but the way we arrive to the final tensor affects how expensive it is to get there. -# - Show how can we can calculate the complexity of a contraction by means of a simple example using 2 matrices (rank 2 tensors): dimension_contracted x (dimensions_open). -# Intuition behind: we perform one operation (contraction) and repeat many times to "populate" the resulting tensor (dimension_open1 x dimension_open2). Show the equation with indices. -# - Show an example with at least three tensors where they all have different dimensions. Walk through it showing that choosing to contract two indices (the ones with lower dimensions) results in a worst computational complexity than contracting other ones (the ones with higher dimensions). +# +# A common task when dealing with tensors is the contraction of large networks resulting in a single tensor (including scalars). To arrive to this final tensor, we can start with a single tensor and contract it with adjacent tensors one-at-a-time. The order in which this is carried out is known as the *contraction path* or *bubbling*. +# While the final tensor is independent of the order of the contraction, the number of operations performed can vary greatly with the order in which we contract the intermediate tensors. Moreover, in a general setup, finding the optimal order of indices to be contracted is not at all a trivial task - actually it is a NP-complete problem [#Arad]_. +# +# For this reason, it is useful to look at how to calculate the computational cost or the *complexity* of a tensor network contraction. First, we look at a simple matrix-matrix contraction. Given rank-2 tensors :math:`G^1_{i,j}` and :math:`G^2_{j,k}`, we have seen the :math:`(i,k)`-th element of the resulting contraction along the :math:`j`-th index is +# +# .. math:: +# (G^3)_{i,k} = G^1 \cdot G^2 = \sum_{j=1}^{d_j} G^{1}_{i,j} G^{2}_{j,k} +# +# where the indices :math:`i, j, k` have dimensions :math:`d_i, d_j, d_k`, respectively. Thus, obtaining the :math:`(i,k)`-th element requires :math:`\mathcal{O}(d_j)` operations. To construct the full tensor :math:`G^3`, we must repeat this procedure :math:`d_i \times d_k` times (once for every possible value of :math:`i` and :math:`k`). Therefore, the total complexity of the contration is +# +# .. math:: +# \mathcal{O}(d_i \times d_j \times d_k) +# +# To illustrate the importance of choosing an efficient contraction path, let us look at a similar contraction between 3 rank-3 tensors as shown in the previous section. +# +# TODO: add diagram here with 3 tensors. Tensor A has dimensions (i,j,k)->(d_i, d_j, d_j), tensor B has dimension (j,l,m)->(d_j, 1, d_m) and tensor C has dimensions (k,m,n)->(d_j, d_m, d_i). Positioned in a traingle-like structure similar to the previous example. +# +# In this case, the tensors are such that :math:`A_{i,j,k} \in \mathcal{C}^{d_i \times d_j \times d_j}`, :math:`B_{j,l,m} \in \mathcal{C}^{d_j \times 1 \times d_m}`, and :math:`C_{k,m,n} \in \mathcal{C}^{d_j \times d_m \times d_i}`. First, we look at the complexity of contracting :math:`(AB)` and then :math:`C`. Following the procedure explained above, the first contraction results in a complexity of +# +# .. math:: +# \sum_{j} A_{i,j,k} B_{j,l,m} \implies \mathcal{O}(d_i \times d_m \times d_j^2 ) +# +# Then, contracting the resulting tensor :mathcal:`(AB)_{i, k, l, m}` with :math:`C_{k,m,n}` requires +# .. math:: +# \sum_{k, m} (AB)_{i, k, l, m} C_{k,m,n} \implies \mathcal{O}(d_j \times d_m \times d_i^2) +# +# operations. Assuming :math:`d_j \leq \d_m \leq d_i`, assymptotially, the whole contraction will have a cost of :math:`\mathcal{O}(d_j \times d_m \times d_i^2)`. Alternatively, we could first contract :math:`B` and :math:`C`, incurring in the cost +# +# .. math:: +# \sum_{m} B_{j,l,m} C_{k,m,n} \implies \mathcal{O}(d_i \times d_m \times d_j^2 ) . +# +# Then, contracting the result with :math:`A` results in +# +# .. math:: +# \sum_{j, k} A_{i,j,k} (BC)_{j,l,k,n} \implies \mathcal{O}(d_i^2 \times d_j^2) . +# +# This means, the second contraction path results in an asymptotic cost of :math:`\mathcal{O}(d_i^2 \times d_j^2)` - lower than the first contraction path. +# To see this in practice, let us perform the above contraction using ``np.einsum``. + +import timeit + +di = 1000 +dm = 100 +dj = 10 + +A = np.arange(di*dj*dj).reshape(di,dj,dj) # ijk +B = np.arange(dj*1*dm).reshape(dj,1,dm) # jlm +C = np.arange(dj*dm*di).reshape(dj,dm,di) # kmn + +iterations = 20 + +contraction = "np.einsum('ijk, jlm -> iklm', A, B)" +execution_time = timeit.timeit(contraction, globals=globals(), number=iterations) + +average_time_ms = execution_time * 1000 / iterations +print(f"Computation cost for AB contraction: {average_time_ms:.8f} ms") + +AB = np.einsum('ijk, jlm -> iklm', A, B) +contraction = "np.einsum('iklm, kmn -> iln', AB, C)" +execution_time = timeit.timeit(contraction, globals=globals(), number=iterations) + +average_time_ms = execution_time * 1000 / iterations +print(f"Computation cost for (AB)C contraction: {average_time_ms:.8f} ms") + +contraction = "np.einsum('jlm, kmn -> jlkn', B, C)" +execution_time = timeit.timeit(contraction, globals=globals(), number=iterations) + +average_time_ms = execution_time * 1000 / iterations +print(f"Computation cost for BC contraction: {average_time_ms:.8f} ms") + +BC = np.einsum('jlm, kmn -> jlkn', B, C) + +contraction = "np.einsum('ijk, jlkn -> iln', A, BC)" +execution_time = timeit.timeit(contraction, globals=globals(), number=iterations) + +average_time_ms = execution_time * 1000 / iterations +print(f"Computation cost for A(BC) contraction: {average_time_ms:.8f} ms") + +############################################################################## # - For this reason there exist heuristics for optimizing contraction path complexity. NP problem -> no perfect solution but great heuristics (https://arxiv.org/pdf/2002.01935). # (optional) mention the idea behind some of them # Link to quimb examples. -# - CODE: show this using np.einsum, timeit, and very large dimensions expecting to see a difference. + ############################################################################## # From tensor networks to quantum circuits: # ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +# # - Quantum circuits are a restricted subclass of tensor networks # - show examples on https://arxiv.org/pdf/1912.10049 page 8 and 9 showing a quantum circuit for a bell state, defining each component as a tensor and show their contraction. # - What else? -""" -DRAFT: - -1) -- Definition of a tensor as an n-dimensional array. Show notation with n indices, stating that it belongs to C^{d1,...,dn}. Define rank, index, dimension (mention how these terms are sometimes used (wrongly?) interchangeably in literature). -- Graphical notation. Mention that there exist certain representations in the literature that allow to represent properties of the tensors (e.g. symmetry, orthogonality). In our case, we can adhere to a general circle. -- Specific examples used in a day-to-day: scalars, vectors, matrices. Mention that for quantum states, we can adopt the convention that the legs in one direction mean that the state belongs to one Hilbert space, and the legs to the other side to the dual space. -- CODE: include code using numpy creating a >2 dimensional array. -2) - -- Show the matrix multiplication in terms of summation over indices, then using the diagrammatic representation. This results in another rank 2 tensor (matrix) -- Analagously, we can represent matrix-vector multiplication resulting in a rank 1 tensor (vector). Just as we expected! -- We can generalize this concept to tensors. This is done by summing over repeated indices (just as in einstein convention - external link for it) resulting in another tensor made up of the open legs of all the tensors together. - In diagrammatic notation, this is simply sticking together legs with same indices! (show nice diagram with >3 tensors). We have just formed a network of tensors, i.e. a Tensor Network! -- CODE: Talking about einstein convetion, we can perform this contraction of tensors using `np.einsum`. -""" +############################################################################## +# References +# ---------- +# .. [#Arad] +# I. Arad and Z. Landau. +# "Quantum computation and the evaluation of tensor networks", +# ``__, 2010. \ No newline at end of file From 08d2593b8a8f8f6d7f4a55db2253bc9a1c19a0ec Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Fri, 30 Aug 2024 16:13:10 -0400 Subject: [PATCH 15/51] fix typos --- demonstrations/tutorial_tn_basics.py | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index ec71ed71b8..36bb4f7d75 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -201,7 +201,8 @@ # .. math:: # \sum_{j} A_{i,j,k} B_{j,l,m} \implies \mathcal{O}(d_i \times d_m \times d_j^2 ) # -# Then, contracting the resulting tensor :mathcal:`(AB)_{i, k, l, m}` with :math:`C_{k,m,n}` requires +# Then, contracting the resulting tensor :math:`(AB)_{i, k, l, m}` with :math:`C_{k,m,n}` requires +# # .. math:: # \sum_{k, m} (AB)_{i, k, l, m} C_{k,m,n} \implies \mathcal{O}(d_j \times d_m \times d_i^2) # From 559dab303c2fb0d465c49527249e044b253c9319 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Fri, 30 Aug 2024 16:23:43 -0400 Subject: [PATCH 16/51] add explanation around contration code and fix bra ket issue --- demonstrations/tutorial_tn_basics.py | 28 +++++++++++++++++++++------- 1 file changed, 21 insertions(+), 7 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 36bb4f7d75..e7494e7ffb 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -144,14 +144,14 @@ # To end this section, we want to discuss a common example of a tensor network contraction arising in quantum computing, namely the **CNOT** gate. The CNOT gate can be expressed in the computational basis as # # .. math:: -# CNOT = \ket{0}\bra{0} \otimes I + \ket{1}\bra{1} \otimes X. +# CNOT = |0\rangle \langle 0 | \otimes I + |1\rangle \langle 1 | \otimes X. # -# That is, if the control qubit is in the :math:`\ket{1}` state, we apply the :math:`X` gate on the target qubit, otherwise, we leave them untouched. However, we can also rewrite this equation as a contraction. To do so, we define two tensors: +# That is, if the control qubit is in the :math:`|1\rangle` state, we apply the :math:`X` gate on the target qubit, otherwise, we leave them untouched. However, we can also rewrite this equation as a contraction. To do so, we define two tensors: # # .. math:: # T^1 = \begin{pmatrix} -# \ket{0}\bra{0} \\ -# \ket{1}\bra{1} +# |0\rangle \langle 0\\ +# |1\rangle \langle 1 | # \end{pmatrix} # # and @@ -162,7 +162,7 @@ # X # \end{pmatrix} # -# This means :math:`(T^1)_{i,j,k}` and :math:`(T^2)_{l,j,m}` are two rank-3 tensors, where the index :math:`j` "*picks*" the elements in the column vector while the first and last indices correspond to the indices of the internal tensors (matrices). For instance, the :math:`j = 0`-th element of :math:`T^1` and :math:`T^2` are :math:`\ket{0}\bra{0}` and :math:`I`, respectively. And similarly for their :math:`j = 1`-st element. This means we can redefine the CNOT expression from above as +# This means :math:`(T^1)_{i,j,k}` and :math:`(T^2)_{l,j,m}` are two rank-3 tensors, where the index :math:`j` "*picks*" the elements in the column vector while the first and last indices correspond to the indices of the internal tensors (matrices). For instance, the :math:`j = 0`-th element of :math:`T^1` and :math:`T^2` are :math:`|0\rangle \langle 0 |` and :math:`I`, respectively. And similarly for their :math:`j = 1`-st element. This means we can redefine the CNOT expression from above as # # .. math:: # CNOT = \sum_j (T^1)_{i,j,k} \otimes (T^2)_{l,j,m}. @@ -206,7 +206,7 @@ # .. math:: # \sum_{k, m} (AB)_{i, k, l, m} C_{k,m,n} \implies \mathcal{O}(d_j \times d_m \times d_i^2) # -# operations. Assuming :math:`d_j \leq \d_m \leq d_i`, assymptotially, the whole contraction will have a cost of :math:`\mathcal{O}(d_j \times d_m \times d_i^2)`. Alternatively, we could first contract :math:`B` and :math:`C`, incurring in the cost +# operations. Assuming :math:`d_j \leq d_m \leq d_i`, assymptotially, the whole contraction will have a cost of :math:`\mathcal{O}(d_j \times d_m \times d_i^2)`. Alternatively, we could first contract :math:`B` and :math:`C`, incurring in the cost # # .. math:: # \sum_{m} B_{j,l,m} C_{k,m,n} \implies \mathcal{O}(d_i \times d_m \times d_j^2 ) . @@ -217,7 +217,7 @@ # \sum_{j, k} A_{i,j,k} (BC)_{j,l,k,n} \implies \mathcal{O}(d_i^2 \times d_j^2) . # # This means, the second contraction path results in an asymptotic cost of :math:`\mathcal{O}(d_i^2 \times d_j^2)` - lower than the first contraction path. -# To see this in practice, let us perform the above contraction using ``np.einsum``. +# To see this in practice, let us perform the above contraction using ``np.einsum``. First, we crate 3 tensors with the correct dimensions. import timeit @@ -229,6 +229,9 @@ B = np.arange(dj*1*dm).reshape(dj,1,dm) # jlm C = np.arange(dj*dm*di).reshape(dj,dm,di) # kmn +############################################################################## +# Then, we can perform the individual contractions between pairs of tensors and time it using ``timeit``. We repeat the contraction ``20`` times and average the computation time to account for smaller fluctuations. First, we start contracting :math:`A` and :math:`B`. + iterations = 20 contraction = "np.einsum('ijk, jlm -> iklm', A, B)" @@ -237,6 +240,9 @@ average_time_ms = execution_time * 1000 / iterations print(f"Computation cost for AB contraction: {average_time_ms:.8f} ms") +############################################################################## +# Then, we contract the result with :math:`C`. + AB = np.einsum('ijk, jlm -> iklm', A, B) contraction = "np.einsum('iklm, kmn -> iln', AB, C)" execution_time = timeit.timeit(contraction, globals=globals(), number=iterations) @@ -244,12 +250,18 @@ average_time_ms = execution_time * 1000 / iterations print(f"Computation cost for (AB)C contraction: {average_time_ms:.8f} ms") +############################################################################## +# As expected, the last contraction is much more costly than the first one. We now repeat the procedure, contracting :math:`B` and :math:`C` first. + contraction = "np.einsum('jlm, kmn -> jlkn', B, C)" execution_time = timeit.timeit(contraction, globals=globals(), number=iterations) average_time_ms = execution_time * 1000 / iterations print(f"Computation cost for BC contraction: {average_time_ms:.8f} ms") +############################################################################## +# We see this contraction is in the same order of magnitude as the contraction between :math:`A` and :math:`B`, as expected by the complexity analysis since they both yield :math:`\mathcal{O}(d_i \times d_m \times d_j^2 )`. Then, we perform the contraction between the resulting tensor and :math:`A`. + BC = np.einsum('jlm, kmn -> jlkn', B, C) contraction = "np.einsum('ijk, jlkn -> iln', A, BC)" @@ -259,6 +271,8 @@ print(f"Computation cost for A(BC) contraction: {average_time_ms:.8f} ms") ############################################################################## +# From this we see tha the second contraction path yields a much lower complexity compared to the first one, just as we expected! +# # - For this reason there exist heuristics for optimizing contraction path complexity. NP problem -> no perfect solution but great heuristics (https://arxiv.org/pdf/2002.01935). # (optional) mention the idea behind some of them # Link to quimb examples. From 8d0f3fbb1d97c81b7d6732d393016e9f3ab3b085 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Fri, 30 Aug 2024 16:40:52 -0400 Subject: [PATCH 17/51] small comment --- demonstrations/tutorial_tn_basics.py | 12 ++++++++---- 1 file changed, 8 insertions(+), 4 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index e7494e7ffb..b2dcac5bc3 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -180,7 +180,11 @@ # ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ # # A common task when dealing with tensors is the contraction of large networks resulting in a single tensor (including scalars). To arrive to this final tensor, we can start with a single tensor and contract it with adjacent tensors one-at-a-time. The order in which this is carried out is known as the *contraction path* or *bubbling*. -# While the final tensor is independent of the order of the contraction, the number of operations performed can vary greatly with the order in which we contract the intermediate tensors. Moreover, in a general setup, finding the optimal order of indices to be contracted is not at all a trivial task - actually it is a NP-complete problem [#Arad]_. +# While the final tensor is independent of the order of the contraction, the number of operations performed can vary greatly with the order in which we contract the intermediate tensors. Moreover, in a general setup, finding the optimal order of indices to be contracted is not at all a trivial task. +# +# .. note:: +# +# Actually finding the optimal contraction path of a tensor network is a NP-complete problem [#Arad]_. # # For this reason, it is useful to look at how to calculate the computational cost or the *complexity* of a tensor network contraction. First, we look at a simple matrix-matrix contraction. Given rank-2 tensors :math:`G^1_{i,j}` and :math:`G^2_{j,k}`, we have seen the :math:`(i,k)`-th element of the resulting contraction along the :math:`j`-th index is # @@ -274,8 +278,8 @@ # From this we see tha the second contraction path yields a much lower complexity compared to the first one, just as we expected! # # - For this reason there exist heuristics for optimizing contraction path complexity. NP problem -> no perfect solution but great heuristics (https://arxiv.org/pdf/2002.01935). -# (optional) mention the idea behind some of them -# Link to quimb examples. +# - Link to quimb examples. +# -The minimum required info would be that there are heuristics that work reasonably well, where one can find these heuristics and ideally how to set them in default.tensor. Examples: Faster identification of optimal contraction sequences for tensor networks, ############################################################################## # From tensor networks to quantum circuits: @@ -283,7 +287,7 @@ # # - Quantum circuits are a restricted subclass of tensor networks # - show examples on https://arxiv.org/pdf/1912.10049 page 8 and 9 showing a quantum circuit for a bell state, defining each component as a tensor and show their contraction. -# - What else? +# - Include suggestions from Korbinian: https://github.com/PennyLaneAI/qml/pull/1193#discussion_r1719502008. ############################################################################## # References From 0796e84d2040ea232bfdf2e7686338a6ec19466c Mon Sep 17 00:00:00 2001 From: Emiliano Godinez Date: Mon, 30 Sep 2024 18:53:55 +0200 Subject: [PATCH 18/51] bring changes from fork into main repo --- .../tn_basics/01-tensor.png | Bin 0 -> 36908 bytes .../tn_basics/02-tensors.png | Bin 0 -> 69933 bytes 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zrfrs-g$(Woo)fVB-ylXyHhepbwWS+u4&&VSpj5d3v4(y|913m<`fk;Ks<~jPH#IaEzrOyogq2~< zF}Rf;92^{uZzk2xocV-ufdG+w;8+>>OSa(f)z;PmH;!;tU`+Z5nb}5xFmn~TzPT6W zJPyapBF#p|0W2ClI?2h&pep|o90D(U48)C$GW*{0rIcthDY`7vzB;nO!ar_RwY8Vg z`+}4%e@2L2h1JG^Z_?CmZSQ~fb^pzMR9O-cGlL^d^*FJ3za^A z4a*Dd6=3 p` can be depicted using a triangle to emphasize the isometry property: - -TODO: add diagram here showing the isometry definition (identity from the smaller space to the bigger space). See https://www.math3ma.com/blog/matrices-as-tensor-network-diagrams - -For quantum states, it is often useful to adopt the convention that writing the legs towards one direction (e.g. the right) corresponds to a ket, i.e., a vector living in the Hilbert space, while drawing the legs to the opposite direction (e.g. left) means they are a bra vector, i.e., living in the dual space. - -TODO: 1) drawing of a vector with legs towards right diretion (ket) 2) legs towards left (bra) +When working within the quantum computing notation, we adopt the convention that writing the leg of a quantum state (i.e., a vector) towards the right direction corresponds to a ket, i.e., a vector living in the Hilbert space, while drawing the legs to the left means they are a bra vector, i.e., living in the dual space. -TODO: add reference here for category theory. P. Selinger. A survey of graphical languages for monoidal categories. Lecture Notes -in Physics, page 289–355, 2010. +.. figure:: ../_static/demonstration_assets/tn_basics/03-braket.png + :align: center + :width: 50% .. tip:: - The diagrammatic representation of tensors is rooted in category theory, which equips the diagrams with all the relevant information so they can used in proofs and formal reasoning! + The diagrammatic representation of tensors is rooted in category theory, which equips the diagrams with all the relevant information so they can used in proofs and formal reasoning! [#Selinger2010]_ Creating a tensor in code is straightforward, and chances are you have already created one yourself. Using ``Numpy``, all we have to do is to create a ``np.array`` of the desired rank. For instance, we can start crearting a rank-1 tensor (a vector). """ @@ -99,17 +81,27 @@ # # Matrix-matrix and matrix-vector multiplications are familiar operations within the context of quantum computing. We can now study these operations under the lens of the tensor notation introduced above. First, a matrix and a vector can be multiplied as # -# TODO: add diagram at the end of the equation. # # .. math:: -# (w)_i = G \cdot v = \sum_j G_ij v_j = +# (w)_j = G \cdot v = \sum_i G_ij v_i # -# We see that summing over the shared index :math:`j` is equivalent to **contracting** the corresponding legs from the matrix and vector diagrams. As expected, the result of this multiplication is another rank-1 tensor with dangling leg :math:`i`. Similarly, we can look at the matrix-matrix multiplication: +# .. figure:: ../_static/demonstration_assets/tn_basics/04-matrix-vector.png +# :align: center +# :width: 50% +# +# .. note:: +# +# Recall we are adopting the convention of writing a ket vector with its leg pointing right, as done in quantum circuits. In turn, this means the "input" index of a matrix - its column index - points towards the left, while the "output" index - its row index - points towards the right. +# +# We see that summing over the shared index :math:`i` is equivalent to **contracting** the corresponding legs from the matrix and vector diagrams. As expected, the result of this multiplication is another rank-1 tensor with dangling leg :math:`j`. Similarly, we can look at the matrix-matrix multiplication: # -# TODO: add diagram at the end of the equation. # # .. math:: -# (G^3)_{i,k} = G^1 \cdot G^2 = \sum_j G^{1}_{i,j} G^{2}_{j,k} = +# (G^3)_{i,k} = G^2 \cdot G^1 = \sum_j G^{2}_{j,k} G^{1}_{i,j} +# +# .. figure:: ../_static/demonstration_assets/tn_basics/05-matrix-matrix.png +# :align: center +# :width: 50% # # In this case, the resulting tensor has two dangling indices, :math:`i` and :math:`k`, which defines a matrix, as expected! # @@ -120,7 +112,9 @@ # # Note how the resulting rank-3 tensor is made up of the remaining open legs from the initial tensors :math:`(i,l,n)`. The diagrammatic representation of this equation is obtained by sticking all the legs with the same indices together. # -# TODO: add figure here. +# .. figure:: ../_static/demonstration_assets/tn_basics/06-tensor-tensor.png +# :align: center +# :width: 50% # # With the above contraction we have formed a network of tensors, i.e., a **Tensor Network**! # @@ -169,11 +163,15 @@ # # It turns out :math:`T^1` and :math:`T^2` are special tensors known as the COPY and XOR tensors, respectively, and therefore have special diagrammatic representations. # -# TODO: add a diagram for each of these +# .. figure:: ../_static/demonstration_assets/tn_basics/07-t1-t2.png +# :align: center +# :width: 50% # # Then, their contraction results in the well known CNOT quantum circuit representation. # -# TODO: add the CNOT diagram here. +# .. figure:: ../_static/demonstration_assets/tn_basics/08-cnot.png +# :align: center +# :width: 50% ############################################################################## # The cost of contracting a network: @@ -189,7 +187,7 @@ # For this reason, it is useful to look at how to calculate the computational cost or the *complexity* of a tensor network contraction. First, we look at a simple matrix-matrix contraction. Given rank-2 tensors :math:`G^1_{i,j}` and :math:`G^2_{j,k}`, we have seen the :math:`(i,k)`-th element of the resulting contraction along the :math:`j`-th index is # # .. math:: -# (G^3)_{i,k} = G^1 \cdot G^2 = \sum_{j=1}^{d_j} G^{1}_{i,j} G^{2}_{j,k} +# (G^3)_{i,k} = G^2 \cdot G^1 = \sum_{j=1}^{d_j} G^{2}_{j,k} G^{1}_{i,j} # # where the indices :math:`i, j, k` have dimensions :math:`d_i, d_j, d_k`, respectively. Thus, obtaining the :math:`(i,k)`-th element requires :math:`\mathcal{O}(d_j)` operations. To construct the full tensor :math:`G^3`, we must repeat this procedure :math:`d_i \times d_k` times (once for every possible value of :math:`i` and :math:`k`). Therefore, the total complexity of the contration is # @@ -198,7 +196,9 @@ # # To illustrate the importance of choosing an efficient contraction path, let us look at a similar contraction between 3 rank-3 tensors as shown in the previous section. # -# TODO: add diagram here with 3 tensors. Tensor A has dimensions (i,j,k)->(d_i, d_j, d_j), tensor B has dimension (j,l,m)->(d_j, 1, d_m) and tensor C has dimensions (k,m,n)->(d_j, d_m, d_i). Positioned in a traingle-like structure similar to the previous example. +# .. figure:: ../_static/demonstration_assets/tn_basics/09-contraction.png +# :align: center +# :width: 50% # # In this case, the tensors are such that :math:`A_{i,j,k} \in \mathcal{C}^{d_i \times d_j \times d_j}`, :math:`B_{j,l,m} \in \mathcal{C}^{d_j \times 1 \times d_m}`, and :math:`C_{k,m,n} \in \mathcal{C}^{d_j \times d_m \times d_i}`. First, we look at the complexity of contracting :math:`(AB)` and then :math:`C`. Following the procedure explained above, the first contraction results in a complexity of # @@ -221,7 +221,7 @@ # \sum_{j, k} A_{i,j,k} (BC)_{j,l,k,n} \implies \mathcal{O}(d_i^2 \times d_j^2) . # # This means, the second contraction path results in an asymptotic cost of :math:`\mathcal{O}(d_i^2 \times d_j^2)` - lower than the first contraction path. -# To see this in practice, let us perform the above contraction using ``np.einsum``. First, we crate 3 tensors with the correct dimensions. +# To see this in practice, let us perform the above contraction using ``np.einsum``. First, we create 3 tensors with the correct dimensions. import timeit @@ -275,11 +275,54 @@ print(f"Computation cost for A(BC) contraction: {average_time_ms:.8f} ms") ############################################################################## -# From this we see tha the second contraction path yields a much lower complexity compared to the first one, just as we expected! +# From this we see that the second contraction path yields a much lower complexity compared to the first one, just as we expected! + +############################################################################## +# Contraction paths: +# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +# +# In the previous section, through a toy example, we explored how the choice of the contraction path affects the computational cost of the tensor network contraction. As shown in [#Lam1997]_ , finding an optimal contraction path is equivalent to solving the "multiplication problem" and thus, it is NP-hard. In this section, we provide a general description of wide-spread techniques used to tackle this ubiquitous task. +# +# First, we set up the framework of the problem. While multiway contractions - contractions between more than 2 tensors at a time - are possible, we will consider only pairwise contractions since the former can always be decomposed in terms of the latter. In addition, contracting a tensor network need not result in a single tensor. However, here we consider only the single tensor case as it underlies the more general scenario [#Gray2021]_. +# +# The underlying idea behind finding a contraction path is based on the construction of the computational graph, i.e., a rooted binary tree - also known as the **contraction tree** - that specifies the sequence of pairwise contractions to be executed. In this tree structure, a leaf node corresponds to a tensor from the original network and the pairwise contractions give rise to the intermediate tensors corresponding to the rest of the nodes in the tree. +# +# TODO: here include a drawing similar to one in Fig2.a of https://journals.aps.org/prx/pdf/10.1103/PhysRevX.14.011009 # -# - For this reason there exist heuristics for optimizing contraction path complexity. NP problem -> no perfect solution but great heuristics (https://arxiv.org/pdf/2002.01935). -# - Link to quimb examples. -# -The minimum required info would be that there are heuristics that work reasonably well, where one can find these heuristics and ideally how to set them in default.tensor. Examples: Faster identification of optimal contraction sequences for tensor networks, +# Transforming a tensor network with an arbitrary structure into this binary tree can be achieved by a *tree embedding* of the tensor network graph [#Bienstock1990]_. Thus, optimization of the contraction path is equivalent to a search over tree embeddings of the network. +# +# .. note:: +# +# Besides finding a contraction path that minimizes the computational cost, we can also attempt to find a path that optimizes the memory cost. That is, a contraction path in which all intermediate tensors are below a certain size. +# +# Now, how do we traverse the space of trees to optimize over? The most straightforward idea is to perform an exhaustic search through all of them. As explained in [#Pfeifer2014]_, this can be done (with some additional improvements) using the following well known algorithms: +# +# - Depth-first search +# - Breadth-first search +# - Dynamic programming +# +# While the exhaustive approach scales like :math:`\mathcal{O}(N!)`, with :math:`N` the number of tensors in the network, it can handle a handful of tensors within seconds, providing a good benchmark. In addition, compared to the following algorithms, the exhaustive search guarantees to find the global minimum optimizing the desired metric - space and/or time. +# +# .. note:: +# +# A recursive implementation of the depth-first search is used by default in the well known package ``opt_einsum`` `(see docs) `_. +# +# Further approaches introduced in [#Gray2021]_ are based on alternative common graph theoretic tasks, rather than searching over the contraction tree space, such as the `balanced bipartitioning `_ and `community detection `_ algorithms. And even though, these are only heuristics that do not guarantee an optimal contraction path, they can often achieve an arbitrarliy close to optimal performance. +# +# An extra level of optimization, known as *hyper-optimization* is introduced by the use of different algorithms to find the optimal contraction based on the specific tensor network, as some algorithms are better suited for certain network structures. For an in-depth exploration of these heuristics, please refer to [#Gray2021]_. +# +# .. note:: +# +# The size of (intermediate) tensors can grow exponential with the number of indices and dimensions, specially for large-scale tensor networks. Thus, we might run into memory problems when performing the contractions. A useful additional technique to split these tensors into more manageable pieces is known as *slicing*. The idea is to change space for computation time, by temporarily fixing the values of some indices in the tensors, performing independently the contraction for each fixed value and summing the results [#Gray2021]_. +# +# As we will explore in the next section, we can use tensor networks to simulate quantum circuits. In particular, the calculation of an expectation value corresponds to the contraction of the tensor network into a single tensor (scalar) as discussed in this section. In ``Pennylane``, this simulation can be performed using the ``DefaultTensor`` device and the method used to find the contraction path can be chosen via the ``contraction_optimizer`` keyword argument. + +import pennylane as qml + +dev = qml.device("default.tensor", method="tn", contraction_optimizer="auto-hq") + +############################################################################## +# The different types of values accepted for ``contraction_optimizer`` are determined by the ``optimize`` parameter in ``Quimb`` (see `docs `_). See `this tutorial `_ to learn more about the ``DefaultTensor`` device. ############################################################################## # From tensor networks to quantum circuits: @@ -292,7 +335,33 @@ ############################################################################## # References # ---------- +# +# .. [#Selinger2010] +# P. Selinger. +# "A Survey of Graphical Languages for Monoidal Categories", +# ``__, in *New Structures for Physics*, Springer Berlin Heidelberg, pp. 289–355, 2010. +# # .. [#Arad] # I. Arad and Z. Landau. # "Quantum computation and the evaluation of tensor networks", -# ``__, 2010. \ No newline at end of file +# ``__, 2010. +# +# .. [#Lam1997] +# C.-C. Lam, P. Sadayappan, and R. Wenger. +# "On Optimizing a Class of Multi-Dimensional Loops with Reduction for Parallel Execution", +# ``__, Parallel Processing Letters, Vol. 7, No. 2, pp. 157-168, 1997. +# +# .. [#Gray2021] +# J. Gray, S. Kourtis, and S. Choi. +# "Hyper-optimized tensor network contraction", +# ``__, Quantum, Vol. 5, pp. 410, 2021. +# +# .. [#Bienstock1990] +# D. 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z*}3y|;%5Y;p?ItB1S8#_qho1lNlrPp9|R3VG?_0uAY%X|4ppw~U9s8%(NPW%P~7mk z=;wD@3AU`NrE}9r=+x?TtB*Ef#ulr?2L8~)y~fX6gZO7@VhildS|MXCpccStBj@Kw zhKHMh)DegY`UDyXNbZC61$gFkb)QnFVHWE!?&(b};NI+X_ymtUc(4O0Rf>^ zK-8$%f1#S%(7<|w^K1yO1YR0&R1%3qyU(=SfR*GUV(#GJ09gYV3_dkAweK-J7r!AX zRuV3UX;u%q1IJv{y{b>JB$CBrlV|7Vl70xm`4}+GpMUQ*FoHM2Ky$x*>EmEQ`EGoa z$I5O$PxH;YO!xK?gQV`9hz_`_^Sn((SWvKzN;O|iga;7cR2Q=0gI_*cDC2&8LmQK7x@2(cVBOW|1}M^0sbeH%f^7qw_Xg&_)=LfuebAGc)d`c0G{P~K{@}w(qEGJ c_&+JrH6*SMfle0hnB_~s+}PTv4C9Ue7aQd-+yDRo literal 0 HcmV?d00001 diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 7c2541b53e..799c592994 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -22,9 +22,6 @@ For example, a scalar :math:`s` is a rank-0 tensor, a vector :math:`v_i` is a rank-1 tensor and a matrix :math:`G_{i,j}` is a rank-2 tensor. A beautiful and powerful tool accompanying tensors (networks) is their graphical language representation. The diagram of a tensor is simply a geometric shape with a leg sticking out of it for every index in the tensor. For example, - -.. math:: - T_{i,j,k,l} = .. figure:: ../_static/demonstration_assets/tn_basics/01-tensor.png :align: center @@ -52,21 +49,21 @@ import numpy as np -tensor_rank1 = np.array([1, 2, 3]) +tensor_rank1 = np.array([1, 2, 3, 4]) print("rank: ", tensor_rank1.ndim) print("dimensions: ", tensor_rank1.shape) ############################################################################## # Then, we can use this to construct a rank-2 tensor (a matrix). -tensor_rank2 = np.array([tensor_rank1, tensor_rank1]) +tensor_rank2 = np.array([tensor_rank1, tensor_rank1, tensor_rank1]) print("rank: ", tensor_rank2.ndim) print("dimensions: ", tensor_rank2.shape) ############################################################################## # As you might have guessed, we can repeat this procedure to create a rank-3 tensor. -tensor_rank3 = np.array([tensor_rank2]) +tensor_rank3 = np.array([tensor_rank2, tensor_rank2]) print("rank: ", tensor_rank3.ndim) print("dimensions: ", tensor_rank3.shape) print("Rank-3 tensor: \n", tensor_rank3) @@ -285,9 +282,11 @@ # # First, we set up the framework of the problem. While multiway contractions - contractions between more than 2 tensors at a time - are possible, we will consider only pairwise contractions since the former can always be decomposed in terms of the latter. In addition, contracting a tensor network need not result in a single tensor. However, here we consider only the single tensor case as it underlies the more general scenario [#Gray2021]_. # -# The underlying idea behind finding a contraction path is based on the construction of the computational graph, i.e., a rooted binary tree - also known as the **contraction tree** - that specifies the sequence of pairwise contractions to be executed. In this tree structure, a leaf node corresponds to a tensor from the original network and the pairwise contractions give rise to the intermediate tensors corresponding to the rest of the nodes in the tree. +# The underlying idea behind finding a contraction path is based on the construction of the computational graph, i.e., a rooted binary tree - also known as the **contraction tree** - that specifies the sequence of pairwise contractions to be executed. In this tree structure, a leaf node corresponds to a tensor from the original network (blue tensors) and the pairwise contractions (red lines) give rise to the intermediate tensors (red tensors) corresponding to the rest of the nodes in the contraction tree. # -# TODO: here include a drawing similar to one in Fig2.a of https://journals.aps.org/prx/pdf/10.1103/PhysRevX.14.011009 +# .. figure:: ../_static/demonstration_assets/tn_basics/10-contraction-tree.png +# :align: center +# :width: 50% # # Transforming a tensor network with an arbitrary structure into this binary tree can be achieved by a *tree embedding* of the tensor network graph [#Bienstock1990]_. Thus, optimization of the contraction path is equivalent to a search over tree embeddings of the network. # From eeb36bb5f7609e89b766c8c655246fa8dd5981e2 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Wed, 9 Oct 2024 11:46:13 +0200 Subject: [PATCH 20/51] add reference to SVD from mps tutorial --- demonstrations/tutorial_tn_basics.py | 6 ++++++ 1 file changed, 6 insertions(+) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 799c592994..1fb309a5c9 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -169,6 +169,12 @@ # .. figure:: ../_static/demonstration_assets/tn_basics/08-cnot.png # :align: center # :width: 50% +# +# This demonstrates the relation between the CNOT acting as a rank-4 tensor with dimensions :math:`2 \times 2 \times 2 \times 2` and its decomposition in terms of two rank-3 local tensors (:math:`T^1` and :math:`T^2`) of dimensions :math:`2 \times 2 \times 2`. +# +# .. note:: +# +# More generally, we can find decompositions of multi-qubit gates into local tensors by means of the ubiquitous singular valued decomposition (SVD). This method is explained in detail in our :doc:`MPS tutorial `. ############################################################################## # The cost of contracting a network: From e992d16def7ff74080cf91b69c612559439a17a0 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Mon, 14 Oct 2024 23:44:03 +0200 Subject: [PATCH 21/51] add expectation value section --- demonstrations/tutorial_tn_basics.py | 81 ++++++++++++++++++++++------ 1 file changed, 66 insertions(+), 15 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 1fb309a5c9..ab42b8dc96 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -6,7 +6,7 @@ - Define the scope of this tutorial: basics of tensor networks, their relation to quantum circuits, and the importance of choosing the contraction paths. From matrices to tensors -^^^^^^^^^^^^^^^^^^^^^^^^^^^ +------------------------------- A common and intuitive way of thinking about tensors is as generalizations of vectors and matrices. That is, we can think of them as multi dimensional arrays - i.e., multi dimensional maps that are linear with respect to every parameter. A tensor of dimensions :math:`d_1 \times d_2 \times \ldots \times d_r` can be expressed as @@ -74,7 +74,7 @@ ############################################################################## # From matrix multiplication to tensor contractions -# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +# ------------------------------------------------- # # Matrix-matrix and matrix-vector multiplications are familiar operations within the context of quantum computing. We can now study these operations under the lens of the tensor notation introduced above. First, a matrix and a vector can be multiplied as # @@ -178,7 +178,7 @@ ############################################################################## # The cost of contracting a network: -# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +# ---------------------------------- # # A common task when dealing with tensors is the contraction of large networks resulting in a single tensor (including scalars). To arrive to this final tensor, we can start with a single tensor and contract it with adjacent tensors one-at-a-time. The order in which this is carried out is known as the *contraction path* or *bubbling*. # While the final tensor is independent of the order of the contraction, the number of operations performed can vary greatly with the order in which we contract the intermediate tensors. Moreover, in a general setup, finding the optimal order of indices to be contracted is not at all a trivial task. @@ -282,7 +282,7 @@ ############################################################################## # Contraction paths: -# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +# ------------------ # # In the previous section, through a toy example, we explored how the choice of the contraction path affects the computational cost of the tensor network contraction. As shown in [#Lam1997]_ , finding an optimal contraction path is equivalent to solving the "multiplication problem" and thus, it is NP-hard. In this section, we provide a general description of wide-spread techniques used to tackle this ubiquitous task. # @@ -315,11 +315,7 @@ # Further approaches introduced in [#Gray2021]_ are based on alternative common graph theoretic tasks, rather than searching over the contraction tree space, such as the `balanced bipartitioning `_ and `community detection `_ algorithms. And even though, these are only heuristics that do not guarantee an optimal contraction path, they can often achieve an arbitrarliy close to optimal performance. # # An extra level of optimization, known as *hyper-optimization* is introduced by the use of different algorithms to find the optimal contraction based on the specific tensor network, as some algorithms are better suited for certain network structures. For an in-depth exploration of these heuristics, please refer to [#Gray2021]_. -# -# .. note:: -# -# The size of (intermediate) tensors can grow exponential with the number of indices and dimensions, specially for large-scale tensor networks. Thus, we might run into memory problems when performing the contractions. A useful additional technique to split these tensors into more manageable pieces is known as *slicing*. The idea is to change space for computation time, by temporarily fixing the values of some indices in the tensors, performing independently the contraction for each fixed value and summing the results [#Gray2021]_. -# +# # As we will explore in the next section, we can use tensor networks to simulate quantum circuits. In particular, the calculation of an expectation value corresponds to the contraction of the tensor network into a single tensor (scalar) as discussed in this section. In ``Pennylane``, this simulation can be performed using the ``DefaultTensor`` device and the method used to find the contraction path can be chosen via the ``contraction_optimizer`` keyword argument. import pennylane as qml @@ -328,15 +324,65 @@ ############################################################################## # The different types of values accepted for ``contraction_optimizer`` are determined by the ``optimize`` parameter in ``Quimb`` (see `docs `_). See `this tutorial `_ to learn more about the ``DefaultTensor`` device. +# +# Slicing: +# ^^^^^^^^ +# +# The size of (intermediate) tensors can grow exponential with the number of indices and dimensions, specially for large-scale tensor networks. Thus, we might run into memory problems when performing the contractions. A useful additional technique to split these tensors into more manageable pieces is known as **slicing**. +# +# The idea is to change space for computation time, by temporarily fixing the values of some indices in the tensors, performing independently the contraction for each fixed value and summing the results [#Gray2021]_. + ############################################################################## # From tensor networks to quantum circuits: -# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +# ----------------------------------------- +# +# Until now, we have looked at general tensor networks, while ✨sparkling✨ the discussions with examples related to quantum circuits. Here, we leverage the components we have learned to explore this relation more in depth. +# +# First, it is important to emphasize that quantum circuits don't just "look" or "behave" like tensor networks, but rather they *are* tensor networks! Quantum Circuits are a special class of tensor networks where each horizontal wire corresponds to the Hilbert space of a single qubit and the tensors acting on these subsystems are restricted to be unitary operators. +# +# A general quantum circuit acting on :math:`N` qubits can be expressed in terms of the initial quantum state :math:`\psi_0` and the unitary propagator :math:`U` such that the evolved state is +# +# .. math:: +# \psi = U \psi_0, +# +# which can also be depicted diagramatically as +# +# TODO: include a diagram here for the full U and the decomposition +# +# In the right hand side of the equality we have assumed a specific form for the U tensor in terms of local 2-qubit gates, which is often the case when dealing with real quantum hardware. In addition, it is common for the initial state to be a product state such as :math:`|0\rangle^{\otimes N}`, hence the form of the tensor. +# +# Now we can ask the important question: what quantities can we compute from this tensor network? +# +# Expectation values: +# ^^^^^^^^^^^^^^^^^^^ +# +# As anticipated in the previous section, a natural quantity to compute using the tensor network arising from a quantum circuit is the expectation value of an observable :math:`O` evaluated at the quantum state :math:`\psi` +# +# .. math:: +# \langle O \rangle = \langle \psi | O | \psi \rangle = \langle \psi_0 U^\dagger | O | U \psi_0 \rangle . +# +# If the observable is a linear combination of hermitian operators (e.g., a Hamiltonian) +# +# .. math:: +# O = \sum_i c_i O_i +# +# we can calculate the total expectation value "naïvely" by computing the inner product for each component :math:`O_i` and summing up the weighted results. However, it is possible to represent this operation more efficiently by means of a tensor network structure called Matrix Product Operator (MPO) [#Pirvu2010]_. Constructing these networks efficiently for hamiltonians of arbitrary structure is an interesting task, which goes beyond the scope of this tutorial. +# +# When the observable of interest is *local*, i.e., it acts on a few neighbouring qubits, we can calculate the expectation value by considering only the section of the quantum circuit within the *reverse light cone* (causal cone) of the observable :math:`O_l`. +# +# TODO: add a figure here showing the light cone. +# +# Then, the sections outside of the light cone can be ignored as these correspond to contractions resulting in the identity: :math:`G G^\dagger = I` (see grayed area in the drawing above). This helps us decrease the size of the tensor to be contracted, and consequently, the computational expense, by focusing on the part of the evolved state with support inside the light cone of the observable +# +# .. math:: +# \langle O_l \rangle = \langle \psi_l | O | \psi_l \rangle. +# +# Sampling: +# ^^^^^^^^^^^^^^^^^^^ +# +# Explain how we can use the "perfect sampling algorithm" with quantum circuits. # -# - Quantum circuits are a restricted subclass of tensor networks -# - show examples on https://arxiv.org/pdf/1912.10049 page 8 and 9 showing a quantum circuit for a bell state, defining each component as a tensor and show their contraction. -# - Include suggestions from Korbinian: https://github.com/PennyLaneAI/qml/pull/1193#discussion_r1719502008. - ############################################################################## # References # ---------- @@ -369,4 +415,9 @@ # .. [#Pfeifer2014] # R. N. C. Pfeifer, J. Haegeman, and F. Verstraete. # "Faster identification of optimal contraction sequences for tensor networks", -# ``__, Physical Review \ No newline at end of file +# ``__, Physical Review +# +# .. [#Pirvu2010] +# B. Pirvu, V. Murg, J. I. Cirac, and F. Verstraete. +# "Matrix product operator representations," +# ``__, New Journal of Physics, vol. 12, no. 2, p. 025012, 2010. From 1772d56f343830e9c06db8b5b30fbbf602eff0cc Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Tue, 15 Oct 2024 18:26:56 +0200 Subject: [PATCH 22/51] add explanation on sampling --- demonstrations/tutorial_tn_basics.py | 68 ++++++++++++++++++++++++---- 1 file changed, 58 insertions(+), 10 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index ab42b8dc96..a4244ab64c 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -157,8 +157,9 @@ # # .. math:: # CNOT = \sum_j (T^1)_{i,j,k} \otimes (T^2)_{l,j,m}. -# -# It turns out :math:`T^1` and :math:`T^2` are special tensors known as the COPY and XOR tensors, respectively, and therefore have special diagrammatic representations. +# +# .. note:: +# It turns out :math:`T^1` and :math:`T^2` are special tensors known as the COPY and XOR tensors, respectively, and therefore have special diagrammatic representations. The former is defined as a tensor equal to unity when all the indices have the same value (0 or 1) and vanishes otherwise, while the latter is equal to unity when the values of the three indices contain an even number of 1's and vanishes otherwise. We anticipate that the COPY tensor can be used to obtain the diagonal of a matrix. This will be useful in the last section of this tutorial. # # .. figure:: ../_static/demonstration_assets/tn_basics/07-t1-t2.png # :align: center @@ -316,14 +317,14 @@ # # An extra level of optimization, known as *hyper-optimization* is introduced by the use of different algorithms to find the optimal contraction based on the specific tensor network, as some algorithms are better suited for certain network structures. For an in-depth exploration of these heuristics, please refer to [#Gray2021]_. # -# As we will explore in the next section, we can use tensor networks to simulate quantum circuits. In particular, the calculation of an expectation value corresponds to the contraction of the tensor network into a single tensor (scalar) as discussed in this section. In ``Pennylane``, this simulation can be performed using the ``DefaultTensor`` device and the method used to find the contraction path can be chosen via the ``contraction_optimizer`` keyword argument. +# As we will explore in the next section, we can use tensor networks to simulate quantum circuits. In particular, the calculation of an expectation value corresponds to the contraction of the tensor network into a single tensor (scalar) as discussed in this section. In ``Pennylane``, this simulation can be performed using the :class:`~pennylane.devices.default_tensor.DefaultTensor` device and the method used to find the contraction path can be chosen via the ``contraction_optimizer`` keyword argument. import pennylane as qml dev = qml.device("default.tensor", method="tn", contraction_optimizer="auto-hq") ############################################################################## -# The different types of values accepted for ``contraction_optimizer`` are determined by the ``optimize`` parameter in ``Quimb`` (see `docs `_). See `this tutorial `_ to learn more about the ``DefaultTensor`` device. +# The different types of values accepted for ``contraction_optimizer`` are determined by the ``optimize`` parameter in ``Quimb`` (see `docs `_) as this is the backend behind the :class:`~pennylane.devices.default_tensor.DefaultTensor` device. See `this tutorial `_ to learn more the use of this device in ``Pennylane``. # # Slicing: # ^^^^^^^^ @@ -332,14 +333,13 @@ # # The idea is to change space for computation time, by temporarily fixing the values of some indices in the tensors, performing independently the contraction for each fixed value and summing the results [#Gray2021]_. - ############################################################################## # From tensor networks to quantum circuits: # ----------------------------------------- # # Until now, we have looked at general tensor networks, while ✨sparkling✨ the discussions with examples related to quantum circuits. Here, we leverage the components we have learned to explore this relation more in depth. # -# First, it is important to emphasize that quantum circuits don't just "look" or "behave" like tensor networks, but rather they *are* tensor networks! Quantum Circuits are a special class of tensor networks where each horizontal wire corresponds to the Hilbert space of a single qubit and the tensors acting on these subsystems are restricted to be unitary operators. +# First, it is important to emphasize that quantum circuits don't just "look" or "behave" like tensor networks, but rather they *are* tensor networks! Quantum Circuits are a special class of tensor networks where each horizontal wire corresponds to the Hilbert space of a single qubit and the tensors acting on these subsystems are restricted to be unitary operators, denoting the time evolution of the state (from left to right). # # A general quantum circuit acting on :math:`N` qubits can be expressed in terms of the initial quantum state :math:`\psi_0` and the unitary propagator :math:`U` such that the evolved state is # @@ -350,7 +350,7 @@ # # TODO: include a diagram here for the full U and the decomposition # -# In the right hand side of the equality we have assumed a specific form for the U tensor in terms of local 2-qubit gates, which is often the case when dealing with real quantum hardware. In addition, it is common for the initial state to be a product state such as :math:`|0\rangle^{\otimes N}`, hence the form of the tensor. +# In the right hand side of the equality we have assumed a specific form for the U tensor in terms of local 2-qubit gates, which is often the case when dealing with real quantum hardware. In addition, it is common for the initial state to be a product state such as :math:`|0\rangle^{\otimes N}`, hence the form of the tensor in the diagram. # # Now we can ask the important question: what quantities can we compute from this tensor network? # @@ -373,15 +373,57 @@ # # TODO: add a figure here showing the light cone. # -# Then, the sections outside of the light cone can be ignored as these correspond to contractions resulting in the identity: :math:`G G^\dagger = I` (see grayed area in the drawing above). This helps us decrease the size of the tensor to be contracted, and consequently, the computational expense, by focusing on the part of the evolved state with support inside the light cone of the observable +# Then, the sections outside of the light cone can be ignored as these correspond to contractions resulting in the identity: :math:`G G^\dagger = I` (see grayed area in the drawing above). This helps us decrease the size of the tensor to be contracted, and consequently, the computational expense, by focusing on the part of the circuit with support inside the light cone of the observable - i.e., the gates that affect the calculation of the expectation value. # # .. math:: # \langle O_l \rangle = \langle \psi_l | O | \psi_l \rangle. # +# where :math:`\psi_l` is the section of the evolved state within the light cone of the observable. +# # Sampling: -# ^^^^^^^^^^^^^^^^^^^ +# ^^^^^^^^^ +# +# We can also use the tensor network arising from a quantum circuit to sample bitstrings from the evolved probability distribution :math:`| \psi \rangle`- emulating what you would obtain from a real quantum computer. Since this is an ubiquitous task in quantum information, several algorithms have been proposed to generate samples from probability distributions represented as tensor networks. In particular, here we discuss the "Perfect Sampling Algorithm" applied to unitary tensor networks [#Ferris2012]_, as this generates uncorrelated samples, unlike markov-based approaches. +# +# .. note:: +# The method used in `Quimb `_ (the backend behind :class:`~pennylane.devices.default_tensor.DefaultTensor`) to generate samples from the quantum circuit is also based on this algorithm. +# +# A cornerstone behind this algorithm is the well known `chain rule ` which allows us to write the join probability of an event using only conditional probabilities. Using this, we can express the probability of sampling the bitstring :math:`(x_1, x_2, x_3, \ldots, x_N)` from :math:`| \psi \rangle` as +# +# .. math:: +# p(x_1, x_2, x_3, \ldots, x_N) = p(x_1) p(x_2|x_1) p(x_3| x_1 x_2) \ldots p(x_N | x_1 x_2 x_3 \ldots x_{N-1}) +# +# Thus, to obtain a sample from the joint distribution on the left side of the equation, we can compute the terms on the right side by means of marginal distributions. First, we start by computing the marginal probability :math:`p(x_1)`. To do so, we compute the reduced density matrix :math:`\rho_{1}` by tracing out all the other qubits. +# +# .. math:: +# \rho_{1} = tr_{2,3,\ldots,N}(| \psi \rangle \langle \psi |) # -# Explain how we can use the "perfect sampling algorithm" with quantum circuits. +# Then, the diagonal of this :math:`2 \times 2` density matrix gives us the probability of sampling 0 or 1, i.e., :math:`p(x_1 = 0)` and :math:`p(x_2 = 1)`. This diagonal corresponds to the following probability vector +# +# .. math:: +# | p_{x_1} \rangle = \sum_{i=0}^{1} p(x_1=i) | i \rangle = diag\left( tr_{2,3,\ldots,N}(| \psi \rangle \langle \psi |) \right). +# +# The tensor network corresponding to the computation of this vector is +# +# TODO: add diagram here showing the whole contraction resulting in a single vector. +# +# In this diagram, we have extracted the diagonal of the reduced density matrix by contracting it with the COPY that we introduced earlier on this tutorial! +# +# Once we obtain the probability vector, we can generate a random sample weighted by these probabilities, i.e., we generate a random number :math:`r \in [0,1]` and choose :math:`x_1 = 0` if :math:`r < p(x_1=0)` and :math:`x_1 = 1` otherwise. We save this sample as :math:`\hat{x_1}`. +# +# Next, we can calculate the following term :math:`p(x_2|\hat{x_1})` conditioned on the sample we have just obtained. To do so, we **project** the first qubit to be :math:`hat{x_1}`. We can do this by contracting the computational basis state :math:`| \hat{x_1} \rangle` with :math:`|\psi \rangle`, resulting in a smaller state :math:`|\psi_{x_1} \rangle`. Then, we can proceed exactly as we did in the previous step, calculating the reduced density matrix over the remaining qubits, and computing the probability vector +# +# .. math:: +# | p_{x_2 | \hat{x_1}} \rangle diag\left( tr_{3,4, \ldots,N}(| \psi_{x_1} \rangle \langle \psi_{x_1} |) \right) +# +# from which we sample the next value :math:`\hat{x_2}` and use it to compute the next term p(x_3| \hat{x_1} \hat{x_2}) using the same procedure. The following diagram shows the full tensor network for this step including the projection onto the copmutational basis state :math:`| \hat{x_1} \rangle`. +# +# TODO: include this diagram here +# +# Here, similarly as done previously, we should only include in the contraction the parts of the circuit within the light cone of botht the projection and the diagonal computation. This procedure can be repeated recursively until we obtain the final sample :math:`(\hat{x}_1, \hat{x}_2, \hat{x}_3, \ldots, \hat{x}_N)`. To obtain more samples, we repeat the procedure from the beginning. +# +# .. note:: +# TODO: add a note on cacheing the results. # ############################################################################## # References @@ -421,3 +463,9 @@ # B. Pirvu, V. Murg, J. I. Cirac, and F. Verstraete. # "Matrix product operator representations," # ``__, New Journal of Physics, vol. 12, no. 2, p. 025012, 2010. +# +# .. [#Ferris2012] +# A. J. Ferris and G. Vidal. +# "Perfect sampling with unitary tensor networks," +# ``__, Physical Review B, vol. 85, no. 16, 2012. + From 44804c7908eb9d7c18e2b271717c55f64678ee63 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Tue, 15 Oct 2024 22:50:56 +0200 Subject: [PATCH 23/51] attempt to fix formatting issue --- demonstrations/tutorial_tn_basics.py | 29 +++++++++++++++------------- 1 file changed, 16 insertions(+), 13 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index a4244ab64c..30020b017c 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -158,8 +158,7 @@ # .. math:: # CNOT = \sum_j (T^1)_{i,j,k} \otimes (T^2)_{l,j,m}. # -# .. note:: -# It turns out :math:`T^1` and :math:`T^2` are special tensors known as the COPY and XOR tensors, respectively, and therefore have special diagrammatic representations. The former is defined as a tensor equal to unity when all the indices have the same value (0 or 1) and vanishes otherwise, while the latter is equal to unity when the values of the three indices contain an even number of 1's and vanishes otherwise. We anticipate that the COPY tensor can be used to obtain the diagonal of a matrix. This will be useful in the last section of this tutorial. +# It turns out :math:`T^1` and :math:`T^2` are special tensors known as the COPY and XOR tensors, respectively, and therefore have special diagrammatic representations. # # .. figure:: ../_static/demonstration_assets/tn_basics/07-t1-t2.png # :align: center @@ -171,6 +170,9 @@ # :align: center # :width: 50% # +# .. note:: +# By looking at the elements of the COPY tensor, we can interpret it as being equal to unity when all the indices have the same value (0 or 1) and vanishing otherwise. On the other hand, the XOR tensor can be understood as being equal to unity when the values of the three indices contain an even number of 1's and vanishing otherwise. We anticipate that the COPY tensor can be used to obtain the diagonal of a matrix. This will be useful in the last section of this tutorial. +# # This demonstrates the relation between the CNOT acting as a rank-4 tensor with dimensions :math:`2 \times 2 \times 2 \times 2` and its decomposition in terms of two rank-3 local tensors (:math:`T^1` and :math:`T^2`) of dimensions :math:`2 \times 2 \times 2`. # # .. note:: @@ -178,8 +180,8 @@ # More generally, we can find decompositions of multi-qubit gates into local tensors by means of the ubiquitous singular valued decomposition (SVD). This method is explained in detail in our :doc:`MPS tutorial `. ############################################################################## -# The cost of contracting a network: -# ---------------------------------- +# The cost of contracting a network +# --------------------------------- # # A common task when dealing with tensors is the contraction of large networks resulting in a single tensor (including scalars). To arrive to this final tensor, we can start with a single tensor and contract it with adjacent tensors one-at-a-time. The order in which this is carried out is known as the *contraction path* or *bubbling*. # While the final tensor is independent of the order of the contraction, the number of operations performed can vary greatly with the order in which we contract the intermediate tensors. Moreover, in a general setup, finding the optimal order of indices to be contracted is not at all a trivial task. @@ -282,8 +284,8 @@ # From this we see that the second contraction path yields a much lower complexity compared to the first one, just as we expected! ############################################################################## -# Contraction paths: -# ------------------ +# Contraction paths +# ----------------- # # In the previous section, through a toy example, we explored how the choice of the contraction path affects the computational cost of the tensor network contraction. As shown in [#Lam1997]_ , finding an optimal contraction path is equivalent to solving the "multiplication problem" and thus, it is NP-hard. In this section, we provide a general description of wide-spread techniques used to tackle this ubiquitous task. # @@ -326,15 +328,16 @@ ############################################################################## # The different types of values accepted for ``contraction_optimizer`` are determined by the ``optimize`` parameter in ``Quimb`` (see `docs `_) as this is the backend behind the :class:`~pennylane.devices.default_tensor.DefaultTensor` device. See `this tutorial `_ to learn more the use of this device in ``Pennylane``. # -# Slicing: -# ^^^^^^^^ +# Slicing +# ~~~~~~~~ # # The size of (intermediate) tensors can grow exponential with the number of indices and dimensions, specially for large-scale tensor networks. Thus, we might run into memory problems when performing the contractions. A useful additional technique to split these tensors into more manageable pieces is known as **slicing**. # # The idea is to change space for computation time, by temporarily fixing the values of some indices in the tensors, performing independently the contraction for each fixed value and summing the results [#Gray2021]_. ############################################################################## -# From tensor networks to quantum circuits: +# +# From tensor networks to quantum circuits # ----------------------------------------- # # Until now, we have looked at general tensor networks, while ✨sparkling✨ the discussions with examples related to quantum circuits. Here, we leverage the components we have learned to explore this relation more in depth. @@ -354,8 +357,8 @@ # # Now we can ask the important question: what quantities can we compute from this tensor network? # -# Expectation values: -# ^^^^^^^^^^^^^^^^^^^ +# Expectation values +# ~~~~~~~~~~~~~~~~~~ # # As anticipated in the previous section, a natural quantity to compute using the tensor network arising from a quantum circuit is the expectation value of an observable :math:`O` evaluated at the quantum state :math:`\psi` # @@ -380,8 +383,8 @@ # # where :math:`\psi_l` is the section of the evolved state within the light cone of the observable. # -# Sampling: -# ^^^^^^^^^ +# Sampling +# ~~~~~~~~~~~~~~~~~~~~~~~~ # # We can also use the tensor network arising from a quantum circuit to sample bitstrings from the evolved probability distribution :math:`| \psi \rangle`- emulating what you would obtain from a real quantum computer. Since this is an ubiquitous task in quantum information, several algorithms have been proposed to generate samples from probability distributions represented as tensor networks. In particular, here we discuss the "Perfect Sampling Algorithm" applied to unitary tensor networks [#Ferris2012]_, as this generates uncorrelated samples, unlike markov-based approaches. # From f1ce7eb0bd55717264a723ab3990964e83f905ef Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Tue, 15 Oct 2024 22:55:54 +0200 Subject: [PATCH 24/51] fix latex typos --- demonstrations/tutorial_tn_basics.py | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 30020b017c..533e593f3c 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -342,12 +342,12 @@ # # Until now, we have looked at general tensor networks, while ✨sparkling✨ the discussions with examples related to quantum circuits. Here, we leverage the components we have learned to explore this relation more in depth. # -# First, it is important to emphasize that quantum circuits don't just "look" or "behave" like tensor networks, but rather they *are* tensor networks! Quantum Circuits are a special class of tensor networks where each horizontal wire corresponds to the Hilbert space of a single qubit and the tensors acting on these subsystems are restricted to be unitary operators, denoting the time evolution of the state (from left to right). +# First, it is important to emphasize that quantum circuits don't just "look" or "behave" like tensor networks, but rather they **are** tensor networks! Quantum Circuits are a special class of tensor networks where each horizontal wire corresponds to the Hilbert space of a single qubit and the tensors acting on these subsystems are restricted to be unitary operators, denoting the time evolution of the state (from left to right). # -# A general quantum circuit acting on :math:`N` qubits can be expressed in terms of the initial quantum state :math:`\psi_0` and the unitary propagator :math:`U` such that the evolved state is +# A general quantum circuit acting on :math:`N` qubits can be expressed in terms of the initial quantum state :math:`| \psi_0 \rangle` and the unitary propagator :math:`U` such that the evolved state is # # .. math:: -# \psi = U \psi_0, +# | \psi \rangle = U |\psi_0 \rangle, # # which can also be depicted diagramatically as # @@ -399,12 +399,12 @@ # Thus, to obtain a sample from the joint distribution on the left side of the equation, we can compute the terms on the right side by means of marginal distributions. First, we start by computing the marginal probability :math:`p(x_1)`. To do so, we compute the reduced density matrix :math:`\rho_{1}` by tracing out all the other qubits. # # .. math:: -# \rho_{1} = tr_{2,3,\ldots,N}(| \psi \rangle \langle \psi |) +# \rho_{1} = \mathrm{Tr}_{2,3,\ldots,N}(| \psi \rangle \langle \psi |) # # Then, the diagonal of this :math:`2 \times 2` density matrix gives us the probability of sampling 0 or 1, i.e., :math:`p(x_1 = 0)` and :math:`p(x_2 = 1)`. This diagonal corresponds to the following probability vector # # .. math:: -# | p_{x_1} \rangle = \sum_{i=0}^{1} p(x_1=i) | i \rangle = diag\left( tr_{2,3,\ldots,N}(| \psi \rangle \langle \psi |) \right). +# | p_{x_1} \rangle = \sum_{i=0}^{1} p(x_1=i) | i \rangle = diag\left( \mathrm{Tr}_{2,3,\ldots,N}(| \psi \rangle \langle \psi |) \right). # # The tensor network corresponding to the computation of this vector is # @@ -417,7 +417,7 @@ # Next, we can calculate the following term :math:`p(x_2|\hat{x_1})` conditioned on the sample we have just obtained. To do so, we **project** the first qubit to be :math:`hat{x_1}`. We can do this by contracting the computational basis state :math:`| \hat{x_1} \rangle` with :math:`|\psi \rangle`, resulting in a smaller state :math:`|\psi_{x_1} \rangle`. Then, we can proceed exactly as we did in the previous step, calculating the reduced density matrix over the remaining qubits, and computing the probability vector # # .. math:: -# | p_{x_2 | \hat{x_1}} \rangle diag\left( tr_{3,4, \ldots,N}(| \psi_{x_1} \rangle \langle \psi_{x_1} |) \right) +# | p_{x_2 | \hat{x_1}} \rangle = \mathrm{diag} \left( \mathrm{Tr}_{3,4, \ldots,N}(| \psi_{x_1} \rangle \langle \psi_{x_1} |) \right) # # from which we sample the next value :math:`\hat{x_2}` and use it to compute the next term p(x_3| \hat{x_1} \hat{x_2}) using the same procedure. The following diagram shows the full tensor network for this step including the projection onto the copmutational basis state :math:`| \hat{x_1} \rangle`. # From 2ff8136fd1b79d6169d9e7114e068e365056852a Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Tue, 15 Oct 2024 23:45:10 +0200 Subject: [PATCH 25/51] fixing typos hoping to fix title formatting issue --- demonstrations/tutorial_tn_basics.py | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 533e593f3c..36b1fa522f 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -329,7 +329,7 @@ # The different types of values accepted for ``contraction_optimizer`` are determined by the ``optimize`` parameter in ``Quimb`` (see `docs `_) as this is the backend behind the :class:`~pennylane.devices.default_tensor.DefaultTensor` device. See `this tutorial `_ to learn more the use of this device in ``Pennylane``. # # Slicing -# ~~~~~~~~ +# ~~~~~~~ # # The size of (intermediate) tensors can grow exponential with the number of indices and dimensions, specially for large-scale tensor networks. Thus, we might run into memory problems when performing the contractions. A useful additional technique to split these tensors into more manageable pieces is known as **slicing**. # @@ -338,7 +338,7 @@ ############################################################################## # # From tensor networks to quantum circuits -# ----------------------------------------- +# ---------------------------------------- # # Until now, we have looked at general tensor networks, while ✨sparkling✨ the discussions with examples related to quantum circuits. Here, we leverage the components we have learned to explore this relation more in depth. # @@ -391,7 +391,7 @@ # .. note:: # The method used in `Quimb `_ (the backend behind :class:`~pennylane.devices.default_tensor.DefaultTensor`) to generate samples from the quantum circuit is also based on this algorithm. # -# A cornerstone behind this algorithm is the well known `chain rule ` which allows us to write the join probability of an event using only conditional probabilities. Using this, we can express the probability of sampling the bitstring :math:`(x_1, x_2, x_3, \ldots, x_N)` from :math:`| \psi \rangle` as +# A cornerstone behind this algorithm is the well known `chain rule `_ which allows us to write the join probability of an event using only conditional probabilities. Using this, we can express the probability of sampling the bitstring :math:`(x_1, x_2, x_3, \ldots, x_N)` from :math:`| \psi \rangle` as # # .. math:: # p(x_1, x_2, x_3, \ldots, x_N) = p(x_1) p(x_2|x_1) p(x_3| x_1 x_2) \ldots p(x_N | x_1 x_2 x_3 \ldots x_{N-1}) @@ -404,7 +404,7 @@ # Then, the diagonal of this :math:`2 \times 2` density matrix gives us the probability of sampling 0 or 1, i.e., :math:`p(x_1 = 0)` and :math:`p(x_2 = 1)`. This diagonal corresponds to the following probability vector # # .. math:: -# | p_{x_1} \rangle = \sum_{i=0}^{1} p(x_1=i) | i \rangle = diag\left( \mathrm{Tr}_{2,3,\ldots,N}(| \psi \rangle \langle \psi |) \right). +# | p_{x_1} \rangle = \sum_{i=0}^{1} p(x_1=i) | i \rangle = \mathrm{diag}\left( \mathrm{Tr}_{2,3,\ldots,N}(| \psi \rangle \langle \psi |) \right). # # The tensor network corresponding to the computation of this vector is # @@ -412,14 +412,14 @@ # # In this diagram, we have extracted the diagonal of the reduced density matrix by contracting it with the COPY that we introduced earlier on this tutorial! # -# Once we obtain the probability vector, we can generate a random sample weighted by these probabilities, i.e., we generate a random number :math:`r \in [0,1]` and choose :math:`x_1 = 0` if :math:`r < p(x_1=0)` and :math:`x_1 = 1` otherwise. We save this sample as :math:`\hat{x_1}`. +# Once we obtain the probability vector, we can generate a random sample weighted by these probabilities, i.e., we generate a random number :math:`r \in [0,1]` and choose :math:`x_1 = 0` if :math:`r < p(x_1=0)` and :math:`x_1 = 1` otherwise. We save this sample as :math:`\hat{x}_1`. # -# Next, we can calculate the following term :math:`p(x_2|\hat{x_1})` conditioned on the sample we have just obtained. To do so, we **project** the first qubit to be :math:`hat{x_1}`. We can do this by contracting the computational basis state :math:`| \hat{x_1} \rangle` with :math:`|\psi \rangle`, resulting in a smaller state :math:`|\psi_{x_1} \rangle`. Then, we can proceed exactly as we did in the previous step, calculating the reduced density matrix over the remaining qubits, and computing the probability vector +# Next, we can calculate the following term :math:`p(x_2|\hat{x}_1)` conditioned on the sample we have just obtained. To do so, we **project** the first qubit to be :math:`\hat{x}_1`. We can do this by contracting the computational basis state :math:`| \hat{x}_1 \rangle` with :math:`|\psi \rangle`, resulting in a smaller state :math:`|\psi_{x_1} \rangle`. Then, we can proceed exactly as we did in the previous step, calculating the reduced density matrix over the remaining qubits, and computing the probability vector # # .. math:: -# | p_{x_2 | \hat{x_1}} \rangle = \mathrm{diag} \left( \mathrm{Tr}_{3,4, \ldots,N}(| \psi_{x_1} \rangle \langle \psi_{x_1} |) \right) +# | p_{x_2 | \hat{x}_1} \rangle = \mathrm{diag} \left( \mathrm{Tr}_{3,4, \ldots,N}(| \psi_{x}_1 \rangle \langle \psi_{x}_1 |) \right) # -# from which we sample the next value :math:`\hat{x_2}` and use it to compute the next term p(x_3| \hat{x_1} \hat{x_2}) using the same procedure. The following diagram shows the full tensor network for this step including the projection onto the copmutational basis state :math:`| \hat{x_1} \rangle`. +# from which we sample the next value :math:`\hat{x}_2` and use it to compute the next term :math:`p(x_3| \hat{x}_1 \hat{x}_2)` using the same procedure. The following diagram shows the full tensor network for this step including the projection onto the copmutational basis state :math:`| \hat{x}_1 \rangle`. # # TODO: include this diagram here # From e5718311b7fe5bc4a71ee275300779a4ed6eed09 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Wed, 16 Oct 2024 00:50:18 +0200 Subject: [PATCH 26/51] add illustrations to quantum computing section --- .../tn_basics/11-circuit.png | Bin 0 -> 194806 bytes .../tn_basics/12-expectation-local.png | Bin 0 -> 225187 bytes .../tn_basics/13-sample.png | Bin 0 -> 187684 bytes .../tn_basics/14-sample-cntd.png | Bin 0 -> 213941 bytes demonstrations/tutorial_tn_basics.py | 23 +++++++++++------- 5 files changed, 14 insertions(+), 9 deletions(-) create mode 100644 _static/demonstration_assets/tn_basics/11-circuit.png create mode 100644 _static/demonstration_assets/tn_basics/12-expectation-local.png create mode 100644 _static/demonstration_assets/tn_basics/13-sample.png create mode 100644 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zAhdUOb=iOoAM~wm$H;h>#f*rdGVuWW34-vD{T{)G^M3vMYySdZ;Q(OREUTB(*IW^H zu$>PNGR=rEHwU5hPIR5Vef;+fy}Q}*2d4F=Sh?>{j6G^-g!=stFdoJLTK(7WZkO@@ t)&KcfaOEjnARes?@c($J|F3_3c#gqfU}PnjnG^lG;-WGSGw*AA{vWF71~vcy literal 0 HcmV?d00001 diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 36b1fa522f..8f175ce188 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -328,9 +328,6 @@ ############################################################################## # The different types of values accepted for ``contraction_optimizer`` are determined by the ``optimize`` parameter in ``Quimb`` (see `docs `_) as this is the backend behind the :class:`~pennylane.devices.default_tensor.DefaultTensor` device. See `this tutorial `_ to learn more the use of this device in ``Pennylane``. # -# Slicing -# ~~~~~~~ -# # The size of (intermediate) tensors can grow exponential with the number of indices and dimensions, specially for large-scale tensor networks. Thus, we might run into memory problems when performing the contractions. A useful additional technique to split these tensors into more manageable pieces is known as **slicing**. # # The idea is to change space for computation time, by temporarily fixing the values of some indices in the tensors, performing independently the contraction for each fixed value and summing the results [#Gray2021]_. @@ -351,7 +348,9 @@ # # which can also be depicted diagramatically as # -# TODO: include a diagram here for the full U and the decomposition +# .. figure:: ../_static/demonstration_assets/tn_basics/11-circuit.png +# :align: center +# :width: 35% # # In the right hand side of the equality we have assumed a specific form for the U tensor in terms of local 2-qubit gates, which is often the case when dealing with real quantum hardware. In addition, it is common for the initial state to be a product state such as :math:`|0\rangle^{\otimes N}`, hence the form of the tensor in the diagram. # @@ -374,7 +373,9 @@ # # When the observable of interest is *local*, i.e., it acts on a few neighbouring qubits, we can calculate the expectation value by considering only the section of the quantum circuit within the *reverse light cone* (causal cone) of the observable :math:`O_l`. # -# TODO: add a figure here showing the light cone. +# .. figure:: ../_static/demonstration_assets/tn_basics/12-expectation-local.png +# :align: center +# :width: 35% # # Then, the sections outside of the light cone can be ignored as these correspond to contractions resulting in the identity: :math:`G G^\dagger = I` (see grayed area in the drawing above). This helps us decrease the size of the tensor to be contracted, and consequently, the computational expense, by focusing on the part of the circuit with support inside the light cone of the observable - i.e., the gates that affect the calculation of the expectation value. # @@ -384,7 +385,7 @@ # where :math:`\psi_l` is the section of the evolved state within the light cone of the observable. # # Sampling -# ~~~~~~~~~~~~~~~~~~~~~~~~ +# ~~~~~~~~ # # We can also use the tensor network arising from a quantum circuit to sample bitstrings from the evolved probability distribution :math:`| \psi \rangle`- emulating what you would obtain from a real quantum computer. Since this is an ubiquitous task in quantum information, several algorithms have been proposed to generate samples from probability distributions represented as tensor networks. In particular, here we discuss the "Perfect Sampling Algorithm" applied to unitary tensor networks [#Ferris2012]_, as this generates uncorrelated samples, unlike markov-based approaches. # @@ -408,7 +409,9 @@ # # The tensor network corresponding to the computation of this vector is # -# TODO: add diagram here showing the whole contraction resulting in a single vector. +# .. figure:: ../_static/demonstration_assets/tn_basics/13-sample.png +# :align: center +# :width: 35% # # In this diagram, we have extracted the diagonal of the reduced density matrix by contracting it with the COPY that we introduced earlier on this tutorial! # @@ -419,9 +422,11 @@ # .. math:: # | p_{x_2 | \hat{x}_1} \rangle = \mathrm{diag} \left( \mathrm{Tr}_{3,4, \ldots,N}(| \psi_{x}_1 \rangle \langle \psi_{x}_1 |) \right) # -# from which we sample the next value :math:`\hat{x}_2` and use it to compute the next term :math:`p(x_3| \hat{x}_1 \hat{x}_2)` using the same procedure. The following diagram shows the full tensor network for this step including the projection onto the copmutational basis state :math:`| \hat{x}_1 \rangle`. +# from which we sample the next value :math:`\hat{x}_2` and use it to compute the next term :math:`p(x_3| \hat{x}_1 \hat{x}_2)` using the same procedure. The following diagram shows the full tensor network for this step including the projection onto the computational basis state :math:`| \hat{x}_1 \rangle`. # -# TODO: include this diagram here +# .. figure:: ../_static/demonstration_assets/tn_basics/14-sample-contd.png +# :align: center +# :width: 35% # # Here, similarly as done previously, we should only include in the contraction the parts of the circuit within the light cone of botht the projection and the diagonal computation. This procedure can be repeated recursively until we obtain the final sample :math:`(\hat{x}_1, \hat{x}_2, \hat{x}_3, \ldots, \hat{x}_N)`. To obtain more samples, we repeat the procedure from the beginning. # From b80beb64ca98559b97a80baa7ad8f804b4e0fc91 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Wed, 16 Oct 2024 11:48:36 +0200 Subject: [PATCH 27/51] triggering the deployment again --- demonstrations/tutorial_tn_basics.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 8f175ce188..256af0f7fb 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -426,7 +426,7 @@ # # .. figure:: ../_static/demonstration_assets/tn_basics/14-sample-contd.png # :align: center -# :width: 35% +# :width: 30% # # Here, similarly as done previously, we should only include in the contraction the parts of the circuit within the light cone of botht the projection and the diagonal computation. This procedure can be repeated recursively until we obtain the final sample :math:`(\hat{x}_1, \hat{x}_2, \hat{x}_3, \ldots, \hat{x}_N)`. To obtain more samples, we repeat the procedure from the beginning. # From 82ee6cabde3958c6c62a1d78909935d294f94b15 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Wed, 16 Oct 2024 13:31:52 +0200 Subject: [PATCH 28/51] change title name to see if it compiles finally --- demonstrations/tutorial_tn_basics.py | 7 +++---- 1 file changed, 3 insertions(+), 4 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 256af0f7fb..37492ac087 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -333,9 +333,8 @@ # The idea is to change space for computation time, by temporarily fixing the values of some indices in the tensors, performing independently the contraction for each fixed value and summing the results [#Gray2021]_. ############################################################################## -# -# From tensor networks to quantum circuits -# ---------------------------------------- +# Quantum circuits are tensor networks +# ------------------------------------ # # Until now, we have looked at general tensor networks, while ✨sparkling✨ the discussions with examples related to quantum circuits. Here, we leverage the components we have learned to explore this relation more in depth. # @@ -424,7 +423,7 @@ # # from which we sample the next value :math:`\hat{x}_2` and use it to compute the next term :math:`p(x_3| \hat{x}_1 \hat{x}_2)` using the same procedure. The following diagram shows the full tensor network for this step including the projection onto the computational basis state :math:`| \hat{x}_1 \rangle`. # -# .. figure:: ../_static/demonstration_assets/tn_basics/14-sample-contd.png +# .. figure:: ../_static/demonstration_assets/tn_basics/14-sample-cntd.png # :align: center # :width: 30% # From ccd3eaca47e7a42dbe7b310e598b73098a6c0c17 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Thu, 17 Oct 2024 23:47:00 +0200 Subject: [PATCH 29/51] erasing a section to try and fix the indentation issue --- demonstrations/tutorial_tn_basics.py | 49 ---------------------------- 1 file changed, 49 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 37492ac087..845c1c0191 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -283,55 +283,6 @@ ############################################################################## # From this we see that the second contraction path yields a much lower complexity compared to the first one, just as we expected! -############################################################################## -# Contraction paths -# ----------------- -# -# In the previous section, through a toy example, we explored how the choice of the contraction path affects the computational cost of the tensor network contraction. As shown in [#Lam1997]_ , finding an optimal contraction path is equivalent to solving the "multiplication problem" and thus, it is NP-hard. In this section, we provide a general description of wide-spread techniques used to tackle this ubiquitous task. -# -# First, we set up the framework of the problem. While multiway contractions - contractions between more than 2 tensors at a time - are possible, we will consider only pairwise contractions since the former can always be decomposed in terms of the latter. In addition, contracting a tensor network need not result in a single tensor. However, here we consider only the single tensor case as it underlies the more general scenario [#Gray2021]_. -# -# The underlying idea behind finding a contraction path is based on the construction of the computational graph, i.e., a rooted binary tree - also known as the **contraction tree** - that specifies the sequence of pairwise contractions to be executed. In this tree structure, a leaf node corresponds to a tensor from the original network (blue tensors) and the pairwise contractions (red lines) give rise to the intermediate tensors (red tensors) corresponding to the rest of the nodes in the contraction tree. -# -# .. figure:: ../_static/demonstration_assets/tn_basics/10-contraction-tree.png -# :align: center -# :width: 50% -# -# Transforming a tensor network with an arbitrary structure into this binary tree can be achieved by a *tree embedding* of the tensor network graph [#Bienstock1990]_. Thus, optimization of the contraction path is equivalent to a search over tree embeddings of the network. -# -# .. note:: -# -# Besides finding a contraction path that minimizes the computational cost, we can also attempt to find a path that optimizes the memory cost. That is, a contraction path in which all intermediate tensors are below a certain size. -# -# Now, how do we traverse the space of trees to optimize over? The most straightforward idea is to perform an exhaustic search through all of them. As explained in [#Pfeifer2014]_, this can be done (with some additional improvements) using the following well known algorithms: -# -# - Depth-first search -# - Breadth-first search -# - Dynamic programming -# -# While the exhaustive approach scales like :math:`\mathcal{O}(N!)`, with :math:`N` the number of tensors in the network, it can handle a handful of tensors within seconds, providing a good benchmark. In addition, compared to the following algorithms, the exhaustive search guarantees to find the global minimum optimizing the desired metric - space and/or time. -# -# .. note:: -# -# A recursive implementation of the depth-first search is used by default in the well known package ``opt_einsum`` `(see docs) `_. -# -# Further approaches introduced in [#Gray2021]_ are based on alternative common graph theoretic tasks, rather than searching over the contraction tree space, such as the `balanced bipartitioning `_ and `community detection `_ algorithms. And even though, these are only heuristics that do not guarantee an optimal contraction path, they can often achieve an arbitrarliy close to optimal performance. -# -# An extra level of optimization, known as *hyper-optimization* is introduced by the use of different algorithms to find the optimal contraction based on the specific tensor network, as some algorithms are better suited for certain network structures. For an in-depth exploration of these heuristics, please refer to [#Gray2021]_. -# -# As we will explore in the next section, we can use tensor networks to simulate quantum circuits. In particular, the calculation of an expectation value corresponds to the contraction of the tensor network into a single tensor (scalar) as discussed in this section. In ``Pennylane``, this simulation can be performed using the :class:`~pennylane.devices.default_tensor.DefaultTensor` device and the method used to find the contraction path can be chosen via the ``contraction_optimizer`` keyword argument. - -import pennylane as qml - -dev = qml.device("default.tensor", method="tn", contraction_optimizer="auto-hq") - -############################################################################## -# The different types of values accepted for ``contraction_optimizer`` are determined by the ``optimize`` parameter in ``Quimb`` (see `docs `_) as this is the backend behind the :class:`~pennylane.devices.default_tensor.DefaultTensor` device. See `this tutorial `_ to learn more the use of this device in ``Pennylane``. -# -# The size of (intermediate) tensors can grow exponential with the number of indices and dimensions, specially for large-scale tensor networks. Thus, we might run into memory problems when performing the contractions. A useful additional technique to split these tensors into more manageable pieces is known as **slicing**. -# -# The idea is to change space for computation time, by temporarily fixing the values of some indices in the tensors, performing independently the contraction for each fixed value and summing the results [#Gray2021]_. - ############################################################################## # Quantum circuits are tensor networks # ------------------------------------ From c7bca323ed1e8ca4ace29093a2597b3ab0531417 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Fri, 18 Oct 2024 00:35:12 +0200 Subject: [PATCH 30/51] triggering the deployment again --- demonstrations/tutorial_tn_basics.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 845c1c0191..924814ac7a 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -284,8 +284,8 @@ # From this we see that the second contraction path yields a much lower complexity compared to the first one, just as we expected! ############################################################################## -# Quantum circuits are tensor networks -# ------------------------------------ +# Quantum circuits are tensor networks! +# ------------------------------------- # # Until now, we have looked at general tensor networks, while ✨sparkling✨ the discussions with examples related to quantum circuits. Here, we leverage the components we have learned to explore this relation more in depth. # From 8b692ffa40dfd0bd40991ed7e69b54052c571c4d Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Fri, 18 Oct 2024 23:51:57 +0200 Subject: [PATCH 31/51] bringing back changes and changing title to see if the section header is fixed --- demonstrations/tutorial_tn_basics.py | 53 ++++++++++++++++++++++++++-- 1 file changed, 50 insertions(+), 3 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 924814ac7a..f0b4cd54a5 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -284,10 +284,57 @@ # From this we see that the second contraction path yields a much lower complexity compared to the first one, just as we expected! ############################################################################## -# Quantum circuits are tensor networks! -# ------------------------------------- +# Contraction paths +# ----------------- # -# Until now, we have looked at general tensor networks, while ✨sparkling✨ the discussions with examples related to quantum circuits. Here, we leverage the components we have learned to explore this relation more in depth. +# In the previous section, through a toy example, we explored how the choice of the contraction path affects the computational cost of the tensor network contraction. As shown in [#Lam1997]_ , finding an optimal contraction path is equivalent to solving the "multiplication problem" and thus, it is NP-hard. In this section, we provide a general description of wide-spread techniques used to tackle this ubiquitous task. +# +# First, we set up the framework of the problem. While multiway contractions - contractions between more than 2 tensors at a time - are possible, we will consider only pairwise contractions since the former can always be decomposed in terms of the latter. In addition, contracting a tensor network need not result in a single tensor. However, here we consider only the single tensor case as it underlies the more general scenario [#Gray2021]_. +# +# The underlying idea behind finding a contraction path is based on the construction of the computational graph, i.e., a rooted binary tree - also known as the **contraction tree** - that specifies the sequence of pairwise contractions to be executed. In this tree structure, a leaf node corresponds to a tensor from the original network (blue tensors) and the pairwise contractions (red lines) give rise to the intermediate tensors (red tensors) corresponding to the rest of the nodes in the contraction tree. +# +# .. figure:: ../_static/demonstration_assets/tn_basics/10-contraction-tree.png +# :align: center +# :width: 50% +# +# Transforming a tensor network with an arbitrary structure into this binary tree can be achieved by a *tree embedding* of the tensor network graph [#Bienstock1990]_. Thus, optimization of the contraction path is equivalent to a search over tree embeddings of the network. +# +# .. note:: +# +# Besides finding a contraction path that minimizes the computational cost, we can also attempt to find a path that optimizes the memory cost. That is, a contraction path in which all intermediate tensors are below a certain size. +# +# Now, how do we traverse the space of trees to optimize over? The most straightforward idea is to perform an exhaustic search through all of them. As explained in [#Pfeifer2014]_, this can be done (with some additional improvements) using the following well known algorithms: +# +# - Depth-first search +# - Breadth-first search +# - Dynamic programming +# +# While the exhaustive approach scales like :math:`\mathcal{O}(N!)`, with :math:`N` the number of tensors in the network, it can handle a handful of tensors within seconds, providing a good benchmark. In addition, compared to the following algorithms, the exhaustive search guarantees to find the global minimum optimizing the desired metric - space and/or time. +# +# .. note:: +# +# A recursive implementation of the depth-first search is used by default in the well known package ``opt_einsum`` `(see docs) `_. +# +# Further approaches introduced in [#Gray2021]_ are based on alternative common graph theoretic tasks, rather than searching over the contraction tree space, such as the `balanced bipartitioning `_ and `community detection `_ algorithms. And even though, these are only heuristics that do not guarantee an optimal contraction path, they can often achieve an arbitrarliy close to optimal performance. +# +# An extra level of optimization, known as *hyper-optimization* is introduced by the use of different algorithms to find the optimal contraction based on the specific tensor network, as some algorithms are better suited for certain network structures. For an in-depth exploration of these heuristics, please refer to [#Gray2021]_. +# +# As we will explore in the next section, we can use tensor networks to simulate quantum circuits. In particular, the calculation of an expectation value corresponds to the contraction of the tensor network into a single tensor (scalar) as discussed in this section. In ``Pennylane``, this simulation can be performed using the :class:`~pennylane.devices.default_tensor.DefaultTensor` device and the method used to find the contraction path can be chosen via the ``contraction_optimizer`` keyword argument. + +import pennylane as qml + +dev = qml.device("default.tensor", method="tn", contraction_optimizer="auto-hq") + +############################################################################## +# The different types of values accepted for ``contraction_optimizer`` are determined by the ``optimize`` parameter in ``Quimb`` (see `docs `_) as this is the backend behind the :class:`~pennylane.devices.default_tensor.DefaultTensor` device. See `this tutorial `_ to learn more the use of this device in ``Pennylane``. +# +# The size of (intermediate) tensors can grow exponential with the number of indices and dimensions, specially for large-scale tensor networks. Thus, we might run into memory problems when performing the contractions. A useful additional technique to split these tensors into more manageable pieces is known as **slicing**. +# +# The idea is to change space for computation time, by temporarily fixing the values of some indices in the tensors, performing independently the contraction for each fixed value and summing the results [#Gray2021]_. + +############################################################################## +# The cost of contracting a network +# --------------------------------- # # First, it is important to emphasize that quantum circuits don't just "look" or "behave" like tensor networks, but rather they **are** tensor networks! Quantum Circuits are a special class of tensor networks where each horizontal wire corresponds to the Hilbert space of a single qubit and the tensors acting on these subsystems are restricted to be unitary operators, denoting the time evolution of the state (from left to right). # From 72784d7e9c611fa007f07740ebfad007df8ce87d Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Sat, 19 Oct 2024 00:33:11 +0200 Subject: [PATCH 32/51] adding missing white space to fix title rendering error --- demonstrations/tutorial_tn_basics.py | 9 +++++---- 1 file changed, 5 insertions(+), 4 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index f0b4cd54a5..b44d4977e9 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -333,9 +333,11 @@ # The idea is to change space for computation time, by temporarily fixing the values of some indices in the tensors, performing independently the contraction for each fixed value and summing the results [#Gray2021]_. ############################################################################## -# The cost of contracting a network +# Quantum circuits are tensor networks # --------------------------------- # +# Until now, we have looked at general tensor networks, while ✨sparkling✨ the discussions with examples related to quantum circuits. Here, we leverage the components we have learned to explore this relation more in depth. +# # First, it is important to emphasize that quantum circuits don't just "look" or "behave" like tensor networks, but rather they **are** tensor networks! Quantum Circuits are a special class of tensor networks where each horizontal wire corresponds to the Hilbert space of a single qubit and the tensors acting on these subsystems are restricted to be unitary operators, denoting the time evolution of the state (from left to right). # # A general quantum circuit acting on :math:`N` qubits can be expressed in terms of the initial quantum state :math:`| \psi_0 \rangle` and the unitary propagator :math:`U` such that the evolved state is @@ -429,7 +431,7 @@ # # .. note:: # TODO: add a note on cacheing the results. -# + ############################################################################## # References # ---------- @@ -472,5 +474,4 @@ # .. [#Ferris2012] # A. J. Ferris and G. Vidal. # "Perfect sampling with unitary tensor networks," -# ``__, Physical Review B, vol. 85, no. 16, 2012. - +# ``__, Physical Review B, vol. 85, no. 16, 2012. \ No newline at end of file From 65b714ed17cf49ca4dcb7817042f398cba308380 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Sat, 19 Oct 2024 15:12:09 +0200 Subject: [PATCH 33/51] add caching results comment modify size of diagrams --- demonstrations/tutorial_tn_basics.py | 58 ++++++++++++++++------------ 1 file changed, 33 insertions(+), 25 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index b44d4977e9..7d2cf7548a 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -25,13 +25,13 @@ .. figure:: ../_static/demonstration_assets/tn_basics/01-tensor.png :align: center - :width: 50% + :width: 40% We can apply this same idea to represent a scalar, a vector and a matrix: .. figure:: ../_static/demonstration_assets/tn_basics/02-tensors.png :align: center - :width: 50% + :width: 40% Does the last diagram seem familiar? It is because this is the representation of a single-qubit gate! We will see later in this tutorial the relation between quantum circuits and tensor networks. @@ -162,13 +162,15 @@ # # .. figure:: ../_static/demonstration_assets/tn_basics/07-t1-t2.png # :align: center -# :width: 50% +# :width: 55% # # Then, their contraction results in the well known CNOT quantum circuit representation. # +# TODO: add a tensor with for legs with the label CNOT after everything. +# # .. figure:: ../_static/demonstration_assets/tn_basics/08-cnot.png # :align: center -# :width: 50% +# :width: 45% # # .. note:: # By looking at the elements of the COPY tensor, we can interpret it as being equal to unity when all the indices have the same value (0 or 1) and vanishing otherwise. On the other hand, the XOR tensor can be understood as being equal to unity when the values of the three indices contain an even number of 1's and vanishing otherwise. We anticipate that the COPY tensor can be used to obtain the diagonal of a matrix. This will be useful in the last section of this tutorial. @@ -349,7 +351,7 @@ # # .. figure:: ../_static/demonstration_assets/tn_basics/11-circuit.png # :align: center -# :width: 35% +# :width: 45% # # In the right hand side of the equality we have assumed a specific form for the U tensor in terms of local 2-qubit gates, which is often the case when dealing with real quantum hardware. In addition, it is common for the initial state to be a product state such as :math:`|0\rangle^{\otimes N}`, hence the form of the tensor in the diagram. # @@ -358,40 +360,46 @@ # Expectation values # ~~~~~~~~~~~~~~~~~~ # -# As anticipated in the previous section, a natural quantity to compute using the tensor network arising from a quantum circuit is the expectation value of an observable :math:`O` evaluated at the quantum state :math:`\psi` +# As anticipated in the previous section, a natural quantity to compute using the tensor network arising from a quantum circuit is the expectation value of an observable :math:`O` evaluated at the quantum state :math:`|\psi \rangle` # # .. math:: -# \langle O \rangle = \langle \psi | O | \psi \rangle = \langle \psi_0 U^\dagger | O | U \psi_0 \rangle . +# \langle O \rangle = \langle \psi | O | \psi \rangle = \langle \psi_0| U^\dagger O U |\psi_0 \rangle . # # If the observable is a linear combination of hermitian operators (e.g., a Hamiltonian) # # .. math:: # O = \sum_i c_i O_i # -# we can calculate the total expectation value "naïvely" by computing the inner product for each component :math:`O_i` and summing up the weighted results. However, it is possible to represent this operation more efficiently by means of a tensor network structure called Matrix Product Operator (MPO) [#Pirvu2010]_. Constructing these networks efficiently for hamiltonians of arbitrary structure is an interesting task, which goes beyond the scope of this tutorial. +# we can calculate the total expectation value "naïvely" by computing the inner product for each component :math:`O_i` and summing up the weighted results. +# +# .. math:: +# O = \sum_i c_i \langle O_i \rangle = \sum_i c_i \langle \psi | O_i | \psi \rangle. +# +# However, it is possible to perform this operation more efficiently using tensor networks by means of a structure called Matrix Product Operator (MPO) [#Pirvu2010]_. The idea is to construct an efficient representation of the observable :math:`O` which can be contracted with the tensor network from :math:`|\psi \rangle`. Constructing these networks efficiently for hamiltonians of arbitrary structure is an interesting task, which goes beyond the scope of this tutorial. # # When the observable of interest is *local*, i.e., it acts on a few neighbouring qubits, we can calculate the expectation value by considering only the section of the quantum circuit within the *reverse light cone* (causal cone) of the observable :math:`O_l`. +# TODO: here and in the other figures showing light cones, I can add = diagram without gates out of the light cone to show really how it decreases. # # .. figure:: ../_static/demonstration_assets/tn_basics/12-expectation-local.png # :align: center -# :width: 35% +# :width: 45% # -# Then, the sections outside of the light cone can be ignored as these correspond to contractions resulting in the identity: :math:`G G^\dagger = I` (see grayed area in the drawing above). This helps us decrease the size of the tensor to be contracted, and consequently, the computational expense, by focusing on the part of the circuit with support inside the light cone of the observable - i.e., the gates that affect the calculation of the expectation value. +# Then, the sections outside of the light cone (grayed-out gates in the figure above) can be ignored since these are contractions resulting in the identity: :math:`G G^\dagger = I`. This helps us decrease the size of the tensor to be contracted, and consequently, the computational expense, by focusing on the section of the circuit with support inside the light cone of the observable - i.e., the gates that affect the calculation of the expectation value. # # .. math:: -# \langle O_l \rangle = \langle \psi_l | O | \psi_l \rangle. +# \langle O_l \rangle = \langle \psi_l | O_l | \psi_l \rangle. # -# where :math:`\psi_l` is the section of the evolved state within the light cone of the observable. +# where :math:`| \psi_l \rangle` is the section of the evolved state within the light cone of :math:`O_l`. # # Sampling # ~~~~~~~~ # -# We can also use the tensor network arising from a quantum circuit to sample bitstrings from the evolved probability distribution :math:`| \psi \rangle`- emulating what you would obtain from a real quantum computer. Since this is an ubiquitous task in quantum information, several algorithms have been proposed to generate samples from probability distributions represented as tensor networks. In particular, here we discuss the "Perfect Sampling Algorithm" applied to unitary tensor networks [#Ferris2012]_, as this generates uncorrelated samples, unlike markov-based approaches. +# In addition to calculating expectation values, we can also use the tensor network arising from a quantum circuit to sample bitstrings from the evolved probability distribution :math:`| \psi \rangle` - emulating what you would obtain from a real quantum computer. Since this is an ubiquitous task in quantum information, several algorithms have been proposed to generate samples from probability distributions represented as tensor networks. In particular, here we discuss the "Perfect Sampling Algorithm" applied to unitary tensor networks [#Ferris2012]_, as this generates uncorrelated samples, unlike markov-based approaches. # # .. note:: # The method used in `Quimb `_ (the backend behind :class:`~pennylane.devices.default_tensor.DefaultTensor`) to generate samples from the quantum circuit is also based on this algorithm. # -# A cornerstone behind this algorithm is the well known `chain rule `_ which allows us to write the join probability of an event using only conditional probabilities. Using this, we can express the probability of sampling the bitstring :math:`(x_1, x_2, x_3, \ldots, x_N)` from :math:`| \psi \rangle` as +# A cornerstone behind this algorithm is the well known `chain rule `_ which allows us to write the joint probability of an event using only conditional probabilities. Using this, we can express the probability of sampling the bitstring :math:`(x_1, x_2, x_3, \ldots, x_N)` from :math:`| \psi \rangle` as # # .. math:: # p(x_1, x_2, x_3, \ldots, x_N) = p(x_1) p(x_2|x_1) p(x_3| x_1 x_2) \ldots p(x_N | x_1 x_2 x_3 \ldots x_{N-1}) @@ -404,33 +412,33 @@ # Then, the diagonal of this :math:`2 \times 2` density matrix gives us the probability of sampling 0 or 1, i.e., :math:`p(x_1 = 0)` and :math:`p(x_2 = 1)`. This diagonal corresponds to the following probability vector # # .. math:: -# | p_{x_1} \rangle = \sum_{i=0}^{1} p(x_1=i) | i \rangle = \mathrm{diag}\left( \mathrm{Tr}_{2,3,\ldots,N}(| \psi \rangle \langle \psi |) \right). +# | p_{x_1} \rangle = \sum_{i=0}^{1} p(x_1=i) | i \rangle = \mathrm{diag}(\rho_1) = \mathrm{diag}\left( \mathrm{Tr}_{2,3,\ldots,N}(| \psi \rangle \langle \psi |) \right). # # The tensor network corresponding to the computation of this vector is # # .. figure:: ../_static/demonstration_assets/tn_basics/13-sample.png # :align: center -# :width: 35% -# -# In this diagram, we have extracted the diagonal of the reduced density matrix by contracting it with the COPY that we introduced earlier on this tutorial! +# :width: 45% +# .. note:: +# In this diagram, we have extracted the diagonal of the reduced density matrix by contracting it with the COPY tensor introduced earlier on this tutorial! # -# Once we obtain the probability vector, we can generate a random sample weighted by these probabilities, i.e., we generate a random number :math:`r \in [0,1]` and choose :math:`x_1 = 0` if :math:`r < p(x_1=0)` and :math:`x_1 = 1` otherwise. We save this sample as :math:`\hat{x}_1`. +# Once we obtain the probability vector, we can generate a random sample weighted by these probabilities. To do so, we generate a random number :math:`r \in [0,1]` and choose :math:`x_1 = 0` if :math:`r < p(x_1=0)` and :math:`x_1 = 1` otherwise. We save this sample as :math:`\hat{x}_1`. # -# Next, we can calculate the following term :math:`p(x_2|\hat{x}_1)` conditioned on the sample we have just obtained. To do so, we **project** the first qubit to be :math:`\hat{x}_1`. We can do this by contracting the computational basis state :math:`| \hat{x}_1 \rangle` with :math:`|\psi \rangle`, resulting in a smaller state :math:`|\psi_{x_1} \rangle`. Then, we can proceed exactly as we did in the previous step, calculating the reduced density matrix over the remaining qubits, and computing the probability vector +# Next, we can calculate the following term :math:`p(x_2|\hat{x}_1)` conditioned on the sample we have just obtained. To accomplish this, we **project** the first qubit to be :math:`\hat{x}_1`. We can do this by contracting the computational basis state :math:`| \hat{x}_1 \rangle` with :math:`|\psi \rangle`, resulting in a smaller state :math:`|\psi_{x_1} \rangle`. Then, we can proceed exactly as we did in the previous step, calculating the reduced density matrix :math:`\rho_2` by tracing out the remaining qubits :math:`3,4,\ldots,N` and computing the probability vector from its diagonal # # .. math:: -# | p_{x_2 | \hat{x}_1} \rangle = \mathrm{diag} \left( \mathrm{Tr}_{3,4, \ldots,N}(| \psi_{x}_1 \rangle \langle \psi_{x}_1 |) \right) +# | p_{x_2 | \hat{x}_1} \rangle = \mathrm{diag} \left( \mathrm{Tr}_{3,4, \ldots,N}(| \psi_{x_1} \rangle \langle \psi_{x_1} |) \right). # -# from which we sample the next value :math:`\hat{x}_2` and use it to compute the next term :math:`p(x_3| \hat{x}_1 \hat{x}_2)` using the same procedure. The following diagram shows the full tensor network for this step including the projection onto the computational basis state :math:`| \hat{x}_1 \rangle`. +# From this vector, we sample the next value :math:`\hat{x}_2` (just like we sampled :math:`\hat{x}_1}`) and use it to compute the next term :math:`p(x_3| \hat{x}_1 \hat{x}_2)` using the same procedure. The following diagram shows the full tensor network for this step including the projection onto the computational basis state :math:`| \hat{x}_1 \rangle`. # # .. figure:: ../_static/demonstration_assets/tn_basics/14-sample-cntd.png # :align: center -# :width: 30% +# :width: 45% # -# Here, similarly as done previously, we should only include in the contraction the parts of the circuit within the light cone of botht the projection and the diagonal computation. This procedure can be repeated recursively until we obtain the final sample :math:`(\hat{x}_1, \hat{x}_2, \hat{x}_3, \ldots, \hat{x}_N)`. To obtain more samples, we repeat the procedure from the beginning. +# Analogously as done with the expectation values, these contractions only involve the sections of the circuit within the light cone of **both** the projection with :math:`| \hat{x}_1 \rangle` and the contraction with the COPY tensor (diagonal computation). This procedure can be repeated recursively using the chain rule equation until we obtain the full bitstring :math:`(\hat{x}_1, \hat{x}_2, \hat{x}_3, \ldots, \hat{x}_N)`. To obtain more samples, we repeat the procedure from the beginning - this is what makes every sample memoryless or a perfect sample from the probability distribution. # # .. note:: -# TODO: add a note on cacheing the results. +# We can reduce the computational cost of this algorithm by **caching** results from previous contractions. When we draw a new sample that partially matches a previously explored configuration (marginal probability), we can reuse the cached results and avoid contracting this part of the network over again. For example, let's assume we have performed the perfect sampling algorithm once and obtained the sample :math:`0110`. If the next sample we need to generate starts with the substring :math:`01`, we can reuse the marginal probabilies up to :math:`p(x_3|01)` and only calculate the new parts of the sequence. The same caching idea can be applied to other tensor network algorithms involving many contractions. ############################################################################## # References From 4aacaf42c5b89aa3bd61e8272a43b57be56e4c69 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Sat, 19 Oct 2024 15:50:57 +0200 Subject: [PATCH 34/51] add introduction --- demonstrations/tutorial_tn_basics.py | 36 +++++++++++++++++++--------- 1 file changed, 25 insertions(+), 11 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 7d2cf7548a..a9ec27a6a3 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -1,9 +1,13 @@ -r"""Tutorial: ABC of tensor networks +r"""Tutorial: ABC of tensor networks for quantum circuits ============================================================= -Intro: -- introduce where are tensor networks used (physics, math, machine learning), why are they so widely spread, and mention how they are not just a data structure but also a diagrammatic language for formal reasoning and even proofs! -- Define the scope of this tutorial: basics of tensor networks, their relation to quantum circuits, and the importance of choosing the contraction paths. +If you are in the domain of quantum computing or quantum information, chances are that you have heard (maybe a lot) tensor networks being brough to the conversation. In fact, they are not only useful in these areas, but actually there a rather widespread "tool" with applications ranging from physics, math and computer science. + +Part of the excitement surrounding tensor networks is due to their ability to represent complex data using an efficient representation, which allows for - among other things- fast classical simulations. In addition, the diagramatic language accompanying tensor networks makes working with them intuitive and suitable to describe a vast range of mathematical concepts, among them quantum circuits. + +In this tutorial, we aim to provide an introduction to tensor networks with a focus on its applications for quantum computing. We choose to start by discussing the basic notions and definitions of tensors and tensor networks, and work our way up to more advanced topics such as contraction paths and algorithms used to simulate quantum computers using tensor networks. + +Without further a do, let us dive right in! 🤓📚 From matrices to tensors ------------------------------- @@ -14,7 +18,7 @@ .. math:: T_{i_1, i_2, \ldots, i_r} \in \mathbb{C}^{d_1 \times d_2 \times \ldots \times d_r}. -Where each :math:`i_n` is an *index* of dimension :math:`d_n` - it can take values ranging from :math:`1` to :math:`d_n`-, and the number of indices :math:`r` is known as the *rank* of the tensor. We say :math:`T` is a rank-r tensor. +Where each :math:`i_n` is an *index* of dimension :math:`d_n` - it takes integer values such that :math:`i_n \in [1, d_n] - and the number of indices :math:`r` is known as the *rank* of the tensor. We say :math:`T` is a rank-:math:`r` tensor. .. tip:: Some authors refer to the indices of the tensors as their dimensions. In this tutorial, these two concepts will have have different meanings, although related. @@ -39,7 +43,7 @@ .. figure:: ../_static/demonstration_assets/tn_basics/03-braket.png :align: center - :width: 50% + :width: 55% .. tip:: The diagrammatic representation of tensors is rooted in category theory, which equips the diagrams with all the relevant information so they can used in proofs and formal reasoning! [#Selinger2010]_ @@ -84,7 +88,7 @@ # # .. figure:: ../_static/demonstration_assets/tn_basics/04-matrix-vector.png # :align: center -# :width: 50% +# :width: 55% # # .. note:: # @@ -98,7 +102,7 @@ # # .. figure:: ../_static/demonstration_assets/tn_basics/05-matrix-matrix.png # :align: center -# :width: 50% +# :width: 55% # # In this case, the resulting tensor has two dangling indices, :math:`i` and :math:`k`, which defines a matrix, as expected! # @@ -111,7 +115,7 @@ # # .. figure:: ../_static/demonstration_assets/tn_basics/06-tensor-tensor.png # :align: center -# :width: 50% +# :width: 55% # # With the above contraction we have formed a network of tensors, i.e., a **Tensor Network**! # @@ -132,6 +136,9 @@ print(D.shape) ############################################################################## +# The CNOT gate +# ~~~~~~~~~~~~~ +# # To end this section, we want to discuss a common example of a tensor network contraction arising in quantum computing, namely the **CNOT** gate. The CNOT gate can be expressed in the computational basis as # # .. math:: @@ -162,7 +169,7 @@ # # .. figure:: ../_static/demonstration_assets/tn_basics/07-t1-t2.png # :align: center -# :width: 55% +# :width: 60% # # Then, their contraction results in the well known CNOT quantum circuit representation. # @@ -429,7 +436,7 @@ # .. math:: # | p_{x_2 | \hat{x}_1} \rangle = \mathrm{diag} \left( \mathrm{Tr}_{3,4, \ldots,N}(| \psi_{x_1} \rangle \langle \psi_{x_1} |) \right). # -# From this vector, we sample the next value :math:`\hat{x}_2` (just like we sampled :math:`\hat{x}_1}`) and use it to compute the next term :math:`p(x_3| \hat{x}_1 \hat{x}_2)` using the same procedure. The following diagram shows the full tensor network for this step including the projection onto the computational basis state :math:`| \hat{x}_1 \rangle`. +# From this vector, we sample the next value :math:`\hat{x}_2` (just like we sampled :math:`\hat{x}_1`) and use it to compute the next term :math:`p(x_3| \hat{x}_1 \hat{x}_2)` using the same procedure. The following diagram shows the full tensor network for this step including the projection onto the computational basis state :math:`| \hat{x}_1 \rangle`. # # .. figure:: ../_static/demonstration_assets/tn_basics/14-sample-cntd.png # :align: center @@ -440,6 +447,13 @@ # .. note:: # We can reduce the computational cost of this algorithm by **caching** results from previous contractions. When we draw a new sample that partially matches a previously explored configuration (marginal probability), we can reuse the cached results and avoid contracting this part of the network over again. For example, let's assume we have performed the perfect sampling algorithm once and obtained the sample :math:`0110`. If the next sample we need to generate starts with the substring :math:`01`, we can reuse the marginal probabilies up to :math:`p(x_3|01)` and only calculate the new parts of the sequence. The same caching idea can be applied to other tensor network algorithms involving many contractions. +############################################################################## +# +# Conclusion +# ---------- +# +# + ############################################################################## # References # ---------- From 2147d59ef3f17c7f2cec7d2015ed8e969266de80 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Sat, 19 Oct 2024 16:23:51 +0200 Subject: [PATCH 35/51] add conclusion --- demonstrations/tutorial_tn_basics.py | 20 ++++++++++++-------- 1 file changed, 12 insertions(+), 8 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index a9ec27a6a3..dccf3bfa43 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -1,7 +1,7 @@ r"""Tutorial: ABC of tensor networks for quantum circuits ============================================================= -If you are in the domain of quantum computing or quantum information, chances are that you have heard (maybe a lot) tensor networks being brough to the conversation. In fact, they are not only useful in these areas, but actually there a rather widespread "tool" with applications ranging from physics, math and computer science. +If you are in the domain of quantum computing or quantum information, chances are that you have heard (maybe a lot) tensor networks being brough to the conversation. In fact, they are not only useful in these areas, but actually they are rather a widespread "tool" with applications ranging from physics, math and computer science. Part of the excitement surrounding tensor networks is due to their ability to represent complex data using an efficient representation, which allows for - among other things- fast classical simulations. In addition, the diagramatic language accompanying tensor networks makes working with them intuitive and suitable to describe a vast range of mathematical concepts, among them quantum circuits. @@ -18,7 +18,7 @@ .. math:: T_{i_1, i_2, \ldots, i_r} \in \mathbb{C}^{d_1 \times d_2 \times \ldots \times d_r}. -Where each :math:`i_n` is an *index* of dimension :math:`d_n` - it takes integer values such that :math:`i_n \in [1, d_n] - and the number of indices :math:`r` is known as the *rank* of the tensor. We say :math:`T` is a rank-:math:`r` tensor. +Where each :math:`i_n` is an *index* of dimension :math:`d_n` - it takes integer values such that :math:`i_n \in [1, d_n]` - and the number of indices :math:`r` is known as the *rank* of the tensor. We say :math:`T` is a rank-:math:`r` tensor. .. tip:: Some authors refer to the indices of the tensors as their dimensions. In this tutorial, these two concepts will have have different meanings, although related. @@ -46,7 +46,7 @@ :width: 55% .. tip:: - The diagrammatic representation of tensors is rooted in category theory, which equips the diagrams with all the relevant information so they can used in proofs and formal reasoning! [#Selinger2010]_ + The diagrammatic representation of tensors is rooted in category theory, which equips the diagrams with all the relevant information so they can used in proofs and formal reasoning! 💡 [#Selinger2010]_ Creating a tensor in code is straightforward, and chances are you have already created one yourself. Using ``Numpy``, all we have to do is to create a ``np.array`` of the desired rank. For instance, we can start crearting a rank-1 tensor (a vector). """ @@ -73,7 +73,7 @@ print("Rank-3 tensor: \n", tensor_rank3) ############################################################################## -# We can create a tensor of arbitrary rank following a similar procedure. This recursive approach is instructive to understand a rank-r tensor as made up of rank-(r-1) tensors, which translates to the code as adding an additional level in the nested bracket structure ``[tensor_rank_r-1]``. +# We can create a tensor of arbitrary rank following a similar procedure. This recursive approach is instructive to understand a rank-:math:`r` tensor as made up of rank-:math:`(r-1)` tensors, which translates to the code as adding an additional level in the nested bracket structure ``[tensor_rank_r-1]``. # ############################################################################## @@ -106,7 +106,7 @@ # # In this case, the resulting tensor has two dangling indices, :math:`i` and :math:`k`, which defines a matrix, as expected! # -# We can now generalize this concept to tensors, and consequently, to more than a pair of legs being contracted. For example, let us look at three tensors :math:`A_{i,j,k}`, :math:`B_{j,l,m}` and :math:`C_{k,m,n}`. To contract them, all we need to do is to sum over repeated indices (:math:`j`, :math:`k`, :math:`m`), just as we would do in `Einstein convention `_. Thus, the (i,l,n)-th element of the resulting tensor :math:`D` is +# We can now generalize this concept to tensors, and consequently, to more than a pair of legs being contracted. For example, let us look at three tensors :math:`A_{i,j,k}`, :math:`B_{j,l,m}` and :math:`C_{k,m,n}`. To contract them, all we need to do is to sum over repeated indices (:math:`j`, :math:`k`, :math:`m`), just as we would do in `Einstein convention `_. Thus, the :math:`(i,l,n)`-th element of the resulting tensor :math:`D` is # # .. math:: # (D)_{i,l,n} = \sum_{j,k,m} A_{i,j,k} B_{j,l,m} C_{k,m,n} . @@ -122,7 +122,7 @@ # .. tip:: # A common question arising when drawing a tensor is "what is the correct order to draw the indices". For instance, in the figure above we have adopted the convention that a tensor :math:`A_{i,j,k}` corresponds to a diagram with the the first leg (:math:`i`) pointing left, the second leg (:math:`j`) pointing upwards and the third leg (:math:`k`) pointing right, and similarly for the other two tensors. However, this need not be the case. We could have defined the first leg to be the one pointing upwards, for example. Based on the use case, and the user, some conventions might seem more natural than others. The only important thing to keep in mind is to be consistent. In other words, once we choose a convetion for the order, we should apply it to all the tensors to avoid contracting the wrong indices ❌. # -# Remember we pointed out the similarity in the notation between the tensor network contractions and Einstein notation? Then, it doesn't come as a suprise that we can perform a contraction using the function ``np.einsum``. To do so, we can start creating the 3 tensors to be contracted by reshaping a 1D array (created using``np.arange``) into rank-3 tensors of the correct dimensions. +# Remember we pointed out the similarity in the notation between the tensor network contractions and Einstein notation? Then, it doesn't come as a suprise that we can perform a contraction using the function ``np.einsum``. To do so, we can start creating the 3 tensors to be contracted by reshaping a 1D array (created using ``np.arange``) into rank-3 tensors of the correct dimensions. # Create the individual rank-3 tensors A = np.arange(6).reshape(1, 2, 3) # ijk @@ -130,7 +130,7 @@ C = np.arange(6).reshape(3, 1, 2) # kmn ############################################################################## -# The ``np.einsum`` function takes as inputs the tensors to be contracted and a string showing the indices of the each tensor and (optionally) the indices of the output tensor. +# The ``np.einsum`` function takes as inputs the tensors to be contracted and a string showing the indices of each tensor and (optionally) the indices of the output tensor. D = np.einsum("ijk, jlm, kmn -> iln", A, B, C) print(D.shape) @@ -160,7 +160,7 @@ # X # \end{pmatrix} # -# This means :math:`(T^1)_{i,j,k}` and :math:`(T^2)_{l,j,m}` are two rank-3 tensors, where the index :math:`j` "*picks*" the elements in the column vector while the first and last indices correspond to the indices of the internal tensors (matrices). For instance, the :math:`j = 0`-th element of :math:`T^1` and :math:`T^2` are :math:`|0\rangle \langle 0 |` and :math:`I`, respectively. And similarly for their :math:`j = 1`-st element. This means we can redefine the CNOT expression from above as +# This means :math:`(T^1)_{i,j,k}` and :math:`(T^2)_{l,j,m}` are two rank-3 tensors, where the index :math:`j` *"picks"* the elements in the column vector while the first and last indices correspond to the indices of the internal tensors (matrices). For instance, the :math:`j = 0`-th element of :math:`T^1` and :math:`T^2` are :math:`|0\rangle \langle 0 |` and :math:`I`, respectively. And similarly for their :math:`j = 1`-st element. This means we can redefine the CNOT expression from above as # # .. math:: # CNOT = \sum_j (T^1)_{i,j,k} \otimes (T^2)_{l,j,m}. @@ -452,7 +452,11 @@ # Conclusion # ---------- # +# And that is it for this tutorial! # +# Although the world of tensor networks and their relation to quantum computing is vastly wider than what we could ever cover in one tutorial, we hope that after this explanations you now feel equipped with the tools needed dive deeper into this topic by yourself. 🔍 +# +# If you want to learn more about how to use tensor networks as a diagramatic tool, check out `these amazing lectures notes `_ by J.Biamonte. In addition, check out `this website `_ for great explanations on many important algorithms and tensor network structures by Flatiron Institute. ############################################################################## # References From 6027f891404e7e8a2bba7e01f321e8840b937f5e Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Sat, 19 Oct 2024 16:54:41 +0200 Subject: [PATCH 36/51] add about the author title --- demonstrations/tutorial_tn_basics.py | 13 +++++++++---- 1 file changed, 9 insertions(+), 4 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index dccf3bfa43..5e0e6bb329 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -452,11 +452,11 @@ # Conclusion # ---------- # -# And that is it for this tutorial! -# -# Although the world of tensor networks and their relation to quantum computing is vastly wider than what we could ever cover in one tutorial, we hope that after this explanations you now feel equipped with the tools needed dive deeper into this topic by yourself. 🔍 +# Although the world of tensor networks and their relation to quantum computing is vastly wider than what we could ever cover in one tutorial, we hope that after this explanations you now feel equipped with the tools needed to dive deeper into this topic by yourself. 🔍 # # If you want to learn more about how to use tensor networks as a diagramatic tool, check out `these amazing lectures notes `_ by J.Biamonte. In addition, check out `this website `_ for great explanations on many important algorithms and tensor network structures by Flatiron Institute. +# +# And that is it for this tutorial! 🎉 ############################################################################## # References @@ -500,4 +500,9 @@ # .. [#Ferris2012] # A. J. Ferris and G. Vidal. # "Perfect sampling with unitary tensor networks," -# ``__, Physical Review B, vol. 85, no. 16, 2012. \ No newline at end of file +# ``__, Physical Review B, vol. 85, no. 16, 2012. + +############################################################################## +# About the author +# ---------------- +# \ No newline at end of file From 1a97b163bb5a1262d65fd7c58ae1a6b2bd175ba3 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Mon, 21 Oct 2024 19:28:44 +0200 Subject: [PATCH 37/51] update figures --- .../tn_basics/05-matrix-matrix.png | Bin 67067 -> 63183 bytes .../tn_basics/08-cnot.png | Bin 17396 -> 23593 bytes .../tn_basics/11-circuit.png | Bin 194806 -> 207990 bytes .../tn_basics/12-expectation-local.png | Bin 225187 -> 188815 bytes .../tn_basics/13-sample.png | Bin 187684 -> 162290 bytes .../tn_basics/14-sample-cntd.png | Bin 213941 -> 199403 bytes 6 files changed, 0 insertions(+), 0 deletions(-) diff --git a/_static/demonstration_assets/tn_basics/05-matrix-matrix.png 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zAhdUOb=iOoAM~wm$H;h>#f*rdGVuWW34-vD{T{)G^M3vMYySdZ;Q(OREUTB(*IW^H zu$>PNGR=rEHwU5hPIR5Vef;+fy}Q}*2d4F=Sh?>{j6G^-g!=stFdoJLTK(7WZkO@@ t)&KcfaOEjnARes?@c($J|F3_3c#gqfU}PnjnG^lG;-WGSGw*AA{vWF71~vcy From 9b6e0691c5f088b402ee6a08d51fc6cbe06643cb Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Mon, 21 Oct 2024 19:59:51 +0200 Subject: [PATCH 38/51] erase TODO's marks --- demonstrations/tutorial_tn_basics.py | 2 -- 1 file changed, 2 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 5e0e6bb329..3dbc2597d2 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -173,7 +173,6 @@ # # Then, their contraction results in the well known CNOT quantum circuit representation. # -# TODO: add a tensor with for legs with the label CNOT after everything. # # .. figure:: ../_static/demonstration_assets/tn_basics/08-cnot.png # :align: center @@ -385,7 +384,6 @@ # However, it is possible to perform this operation more efficiently using tensor networks by means of a structure called Matrix Product Operator (MPO) [#Pirvu2010]_. The idea is to construct an efficient representation of the observable :math:`O` which can be contracted with the tensor network from :math:`|\psi \rangle`. Constructing these networks efficiently for hamiltonians of arbitrary structure is an interesting task, which goes beyond the scope of this tutorial. # # When the observable of interest is *local*, i.e., it acts on a few neighbouring qubits, we can calculate the expectation value by considering only the section of the quantum circuit within the *reverse light cone* (causal cone) of the observable :math:`O_l`. -# TODO: here and in the other figures showing light cones, I can add = diagram without gates out of the light cone to show really how it decreases. # # .. figure:: ../_static/demonstration_assets/tn_basics/12-expectation-local.png # :align: center From bef0876a80c66f50c1c3bfe7d760e049bf3e647b Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Fri, 25 Oct 2024 14:13:31 +0200 Subject: [PATCH 39/51] correct spelling mistakes --- demonstrations/tutorial_tn_basics.py | 30 ++++++++++++++-------------- 1 file changed, 15 insertions(+), 15 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 3dbc2597d2..f6d1fd0ece 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -1,37 +1,37 @@ r"""Tutorial: ABC of tensor networks for quantum circuits ============================================================= -If you are in the domain of quantum computing or quantum information, chances are that you have heard (maybe a lot) tensor networks being brough to the conversation. In fact, they are not only useful in these areas, but actually they are rather a widespread "tool" with applications ranging from physics, math and computer science. +If you are in the domain of quantum computing or quantum information, chances are that you have heard (maybe a lot) about tensor networks being brought into the conversation. In fact, they are not only useful in these areas, but actually they are rather a widespread "tool" with applications ranging across physics, math, and computer science. -Part of the excitement surrounding tensor networks is due to their ability to represent complex data using an efficient representation, which allows for - among other things- fast classical simulations. In addition, the diagramatic language accompanying tensor networks makes working with them intuitive and suitable to describe a vast range of mathematical concepts, among them quantum circuits. +Part of the excitement surrounding tensor networks is due to their ability to represent complex data efficiently, which allows for — among other things — fast classical simulations. In addition, the diagrammatic language accompanying tensor networks makes working with them intuitive and suitable for describing a vast range of mathematical concepts, including quantum circuits. -In this tutorial, we aim to provide an introduction to tensor networks with a focus on its applications for quantum computing. We choose to start by discussing the basic notions and definitions of tensors and tensor networks, and work our way up to more advanced topics such as contraction paths and algorithms used to simulate quantum computers using tensor networks. +In this tutorial, we aim to provide an introduction to tensor networks with a focus on their applications in quantum computing. We choose to start by discussing the basic notions and definitions of tensors and tensor networks, and work our way up to more advanced topics such as contraction paths and algorithms used to simulate quantum computers using tensor networks. + +Without further ado, let’s dive right in! 🤓📚 -Without further a do, let us dive right in! 🤓📚 From matrices to tensors ------------------------------- -A common and intuitive way of thinking about tensors is as generalizations of vectors and matrices. That is, we can think of them as multi dimensional arrays - i.e., multi dimensional maps that are linear with respect to every parameter. -A tensor of dimensions :math:`d_1 \times d_2 \times \ldots \times d_r` can be expressed as +A common and intuitive way of thinking about tensors is as generalizations of vectors and matrices. That is, we can think of them as multidimensional arrays—i.e., multidimensional maps that are linear with respect to every parameter. A tensor of dimensions :math:`d_1 \times d_2 \times \ldots \times d_r` can be expressed as .. math:: - T_{i_1, i_2, \ldots, i_r} \in \mathbb{C}^{d_1 \times d_2 \times \ldots \times d_r}. + T_{i_1, i_2, \ldots, i_r} \in \mathbb{C}^{d_1 \times d_2 \times \ldots \times d_r}, -Where each :math:`i_n` is an *index* of dimension :math:`d_n` - it takes integer values such that :math:`i_n \in [1, d_n]` - and the number of indices :math:`r` is known as the *rank* of the tensor. We say :math:`T` is a rank-:math:`r` tensor. +where each :math:`i_n` is an *index* of dimension :math:`d_n`—it takes integer values such that :math:`i_n \in [1, d_n]`—and the number of indices :math:`r` is known as the *rank* of the tensor. We say :math:`T` is a rank-:math:`r` tensor. .. tip:: - Some authors refer to the indices of the tensors as their dimensions. In this tutorial, these two concepts will have have different meanings, although related. + Some authors refer to the indices of the tensors as their dimensions. In this tutorial, these two concepts will have different meanings, although related. -For example, a scalar :math:`s` is a rank-0 tensor, a vector :math:`v_i` is a rank-1 tensor and a matrix :math:`G_{i,j}` is a rank-2 tensor. +For example, a scalar :math:`s` is a rank-0 tensor, a vector :math:`v_i` is a rank-1 tensor, and a matrix :math:`G_{i,j}` is a rank-2 tensor. A beautiful and powerful tool accompanying tensors (networks) is their graphical language representation. The diagram of a tensor is simply a geometric shape with a leg sticking out of it for every index in the tensor. For example, - + .. figure:: ../_static/demonstration_assets/tn_basics/01-tensor.png :align: center :width: 40% -We can apply this same idea to represent a scalar, a vector and a matrix: +We can apply this same idea to represent a scalar, a vector, and a matrix: .. figure:: ../_static/demonstration_assets/tn_basics/02-tensors.png :align: center @@ -39,16 +39,16 @@ Does the last diagram seem familiar? It is because this is the representation of a single-qubit gate! We will see later in this tutorial the relation between quantum circuits and tensor networks. -When working within the quantum computing notation, we adopt the convention that writing the leg of a quantum state (i.e., a vector) towards the right direction corresponds to a ket, i.e., a vector living in the Hilbert space, while drawing the legs to the left means they are a bra vector, i.e., living in the dual space. +When working within the quantum computing notation, we adopt the convention that drawing the leg of a quantum state (i.e., a vector) to the right corresponds to a ket, i.e., a vector living in the Hilbert space, while drawing the legs to the left means they are a bra vector, i.e., living in the dual space. .. figure:: ../_static/demonstration_assets/tn_basics/03-braket.png :align: center :width: 55% .. tip:: - The diagrammatic representation of tensors is rooted in category theory, which equips the diagrams with all the relevant information so they can used in proofs and formal reasoning! 💡 [#Selinger2010]_ + The diagrammatic representation of tensors is rooted in category theory, which equips the diagrams with all the relevant information so they can be used in proofs and formal reasoning! 💡 [#Selinger2010]_ -Creating a tensor in code is straightforward, and chances are you have already created one yourself. Using ``Numpy``, all we have to do is to create a ``np.array`` of the desired rank. For instance, we can start crearting a rank-1 tensor (a vector). +Creating a tensor in code is straightforward, and chances are you have already created one yourself. Using ``Numpy``, all we have to do is create a ``np.array`` of the desired rank. For instance, we can start by creating a rank-1 tensor (a vector). """ import numpy as np From 876e71e720aff18b0ff30d2b8f331baf1c907788 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Sat, 26 Oct 2024 19:25:16 +0200 Subject: [PATCH 40/51] adding spelling corrections --- demonstrations/tutorial_tn_basics.py | 107 +++++++++++++-------------- 1 file changed, 53 insertions(+), 54 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index f6d1fd0ece..77dc38d7c6 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -80,49 +80,48 @@ # From matrix multiplication to tensor contractions # ------------------------------------------------- # -# Matrix-matrix and matrix-vector multiplications are familiar operations within the context of quantum computing. We can now study these operations under the lens of the tensor notation introduced above. First, a matrix and a vector can be multiplied as -# +# Matrix-matrix and matrix-vector multiplications are familiar operations within the context of quantum computing. We can now study these operations through the lens of the tensor notation introduced above. First, a matrix and a vector can be multiplied as # # .. math:: -# (w)_j = G \cdot v = \sum_i G_ij v_i +# (w)_j = G \cdot v = \sum_i G_{i, j} v_i # # .. figure:: ../_static/demonstration_assets/tn_basics/04-matrix-vector.png # :align: center # :width: 55% -# +# # .. note:: -# -# Recall we are adopting the convention of writing a ket vector with its leg pointing right, as done in quantum circuits. In turn, this means the "input" index of a matrix - its column index - points towards the left, while the "output" index - its row index - points towards the right. -# -# We see that summing over the shared index :math:`i` is equivalent to **contracting** the corresponding legs from the matrix and vector diagrams. As expected, the result of this multiplication is another rank-1 tensor with dangling leg :math:`j`. Similarly, we can look at the matrix-matrix multiplication: # +# Recall we are adopting the convention of drawing a ket vector with its leg pointing right, as done in quantum circuits. In turn, this means the "input" index of a matrix —its column index— points towards the left, while the 'output' index —its row index— points to the right. +# +# We see that summing over the shared index :math:`i` is equivalent to **contracting** the corresponding legs from the matrix and vector diagrams. As expected, the result of this multiplication is another rank-1 tensor with dangling leg :math:`j`. Similarly, we can look at the matrix-matrix multiplication: # # .. math:: # (G^3)_{i,k} = G^2 \cdot G^1 = \sum_j G^{2}_{j,k} G^{1}_{i,j} -# +# # .. figure:: ../_static/demonstration_assets/tn_basics/05-matrix-matrix.png # :align: center # :width: 55% # -# In this case, the resulting tensor has two dangling indices, :math:`i` and :math:`k`, which defines a matrix, as expected! +# Here, the resulting tensor has two dangling indices, :math:i and :math:k, defining a matrix, as expected! # -# We can now generalize this concept to tensors, and consequently, to more than a pair of legs being contracted. For example, let us look at three tensors :math:`A_{i,j,k}`, :math:`B_{j,l,m}` and :math:`C_{k,m,n}`. To contract them, all we need to do is to sum over repeated indices (:math:`j`, :math:`k`, :math:`m`), just as we would do in `Einstein convention `_. Thus, the :math:`(i,l,n)`-th element of the resulting tensor :math:`D` is +# We can now generalize this concept to tensors, and consequently, to more than two legs being contracted. For example, let us look at three tensors :math:`A_{i,j,k}`, :math:`B_{j,l,m}` and :math:`C_{k,m,n}`. To contract them, all we need to do is to sum over repeated indices (:math:`j`, :math:`k`, :math:`m`), just as we would do in `Einstein convention `_. Thus, the :math:`(i,l,n)`-th element of the resulting tensor :math:`D` is # # .. math:: # (D)_{i,l,n} = \sum_{j,k,m} A_{i,j,k} B_{j,l,m} C_{k,m,n} . # -# Note how the resulting rank-3 tensor is made up of the remaining open legs from the initial tensors :math:`(i,l,n)`. The diagrammatic representation of this equation is obtained by sticking all the legs with the same indices together. +# Notice how the resulting rank-3 tensor consists of the remaining open legs from the initial tensors :math:`(i,l,n)`. The diagrammatic representation of this equation is obtained by connecting all the legs with the same indices. # # .. figure:: ../_static/demonstration_assets/tn_basics/06-tensor-tensor.png # :align: center # :width: 55% # -# With the above contraction we have formed a network of tensors, i.e., a **Tensor Network**! +# With the above contraction, we have formed a network of tensors, i.e., a **Tensor Network**! # # .. tip:: -# A common question arising when drawing a tensor is "what is the correct order to draw the indices". For instance, in the figure above we have adopted the convention that a tensor :math:`A_{i,j,k}` corresponds to a diagram with the the first leg (:math:`i`) pointing left, the second leg (:math:`j`) pointing upwards and the third leg (:math:`k`) pointing right, and similarly for the other two tensors. However, this need not be the case. We could have defined the first leg to be the one pointing upwards, for example. Based on the use case, and the user, some conventions might seem more natural than others. The only important thing to keep in mind is to be consistent. In other words, once we choose a convetion for the order, we should apply it to all the tensors to avoid contracting the wrong indices ❌. +# A common question arising when drawing a tensor is "what is the correct order to draw the indices?". For instance, in the figure above, we have adopted the convention that a tensor :math:`A_{i,j,k}` corresponds to a diagram with the first leg (:math:`i`) pointing left, the second leg (:math:`j`) pointing upwards, and the third leg (:math:`k`) pointing right, and similarly for the other two tensors. However, this need not be the case. We could have defined the first leg to be the one pointing upwards, for example. Based on the use case, and the user, some conventions might seem more natural than others. The only important thing to keep in mind is to be consistent. In other words, once we choose a convention for the order, we should apply it to all the tensors to avoid contracting the wrong indices ❌. # -# Remember we pointed out the similarity in the notation between the tensor network contractions and Einstein notation? Then, it doesn't come as a suprise that we can perform a contraction using the function ``np.einsum``. To do so, we can start creating the 3 tensors to be contracted by reshaping a 1D array (created using ``np.arange``) into rank-3 tensors of the correct dimensions. +# Remember the similarity between tensor network contractions and Einstein notation? Then, it doesn't come as a surprise that we can perform a contraction using the function ``np.einsum``. To do so, we can start by creating the 3 tensors to be contracted by reshaping a 1D array (created using ``np.arange``) into rank-3 tensors of the correct dimensions. + # Create the individual rank-3 tensors A = np.arange(6).reshape(1, 2, 3) # ijk @@ -144,7 +143,7 @@ # .. math:: # CNOT = |0\rangle \langle 0 | \otimes I + |1\rangle \langle 1 | \otimes X. # -# That is, if the control qubit is in the :math:`|1\rangle` state, we apply the :math:`X` gate on the target qubit, otherwise, we leave them untouched. However, we can also rewrite this equation as a contraction. To do so, we define two tensors: +# That is, if the control qubit is in the :math:`|1\rangle` state, we apply the :math:`X` gate on the target qubit; otherwise, we leave it untouched. However, we can also rewrite this equation as a contraction. To do so, we define two tensors: # # .. math:: # T^1 = \begin{pmatrix} @@ -160,7 +159,7 @@ # X # \end{pmatrix} # -# This means :math:`(T^1)_{i,j,k}` and :math:`(T^2)_{l,j,m}` are two rank-3 tensors, where the index :math:`j` *"picks"* the elements in the column vector while the first and last indices correspond to the indices of the internal tensors (matrices). For instance, the :math:`j = 0`-th element of :math:`T^1` and :math:`T^2` are :math:`|0\rangle \langle 0 |` and :math:`I`, respectively. And similarly for their :math:`j = 1`-st element. This means we can redefine the CNOT expression from above as +# This means :math:`(T^1)_{i,j,k}` and :math:`(T^2)_{l,j,m}` are two rank-3 tensors, where the index :math:`j` *"picks"* the elements in the column vector while the first and last indices correspond to the indices of the internal tensors (matrices). For instance, the :math:`j = 0`-th element of :math:`T^1` and :math:`T^2` are :math:`|0\rangle \langle 0 |` and :math:`I`, respectively, and similarly for their :math:`j = 1`-st element. This means we can redefine the CNOT expression from above as # # .. math:: # CNOT = \sum_j (T^1)_{i,j,k} \otimes (T^2)_{l,j,m}. @@ -171,7 +170,7 @@ # :align: center # :width: 60% # -# Then, their contraction results in the well known CNOT quantum circuit representation. +# Then, their contraction results in the well-known CNOT quantum circuit representation. # # # .. figure:: ../_static/demonstration_assets/tn_basics/08-cnot.png @@ -179,36 +178,36 @@ # :width: 45% # # .. note:: -# By looking at the elements of the COPY tensor, we can interpret it as being equal to unity when all the indices have the same value (0 or 1) and vanishing otherwise. On the other hand, the XOR tensor can be understood as being equal to unity when the values of the three indices contain an even number of 1's and vanishing otherwise. We anticipate that the COPY tensor can be used to obtain the diagonal of a matrix. This will be useful in the last section of this tutorial. +# By looking at the elements of the COPY tensor, we can interpret it as being equal to unity when all the indices have the same value (0 or 1) and vanishing otherwise. On the other hand, the XOR tensor can be understood as being equal to unity when the values of the three indices contain an even number of 1's, and vanishing otherwise. We anticipate that the COPY tensor can be used to obtain the diagonal of a matrix. This will be useful in the last section of this tutorial. # # This demonstrates the relation between the CNOT acting as a rank-4 tensor with dimensions :math:`2 \times 2 \times 2 \times 2` and its decomposition in terms of two rank-3 local tensors (:math:`T^1` and :math:`T^2`) of dimensions :math:`2 \times 2 \times 2`. # # .. note:: # -# More generally, we can find decompositions of multi-qubit gates into local tensors by means of the ubiquitous singular valued decomposition (SVD). This method is explained in detail in our :doc:`MPS tutorial `. +# More generally, we can find decompositions of multi-qubit gates into local tensors by means of the ubiquitous singular value decomposition (SVD). This method is explained in detail in our :doc:`MPS tutorial `. ############################################################################## # The cost of contracting a network # --------------------------------- # -# A common task when dealing with tensors is the contraction of large networks resulting in a single tensor (including scalars). To arrive to this final tensor, we can start with a single tensor and contract it with adjacent tensors one-at-a-time. The order in which this is carried out is known as the *contraction path* or *bubbling*. -# While the final tensor is independent of the order of the contraction, the number of operations performed can vary greatly with the order in which we contract the intermediate tensors. Moreover, in a general setup, finding the optimal order of indices to be contracted is not at all a trivial task. +# A common task when dealing with tensors is the contraction of large networks resulting in a single tensor (including scalars). To arrive at this final tensor, we can start with a single tensor and contract it with adjacent tensors one-at-a-time. The order in which this is carried out is known as the *contraction path* or *bubbling*. +# While the final tensor is independent of the order of the contraction, the number of operations performed can vary greatly depending on the order in which we contract the intermediate tensors. Moreover, in a general setup, finding the optimal order of indices to be contracted is not a trivial task. # # .. note:: # -# Actually finding the optimal contraction path of a tensor network is a NP-complete problem [#Arad]_. +# Actually, finding the optimal contraction path of a tensor network is an NP-complete problem [#Arad]_. # -# For this reason, it is useful to look at how to calculate the computational cost or the *complexity* of a tensor network contraction. First, we look at a simple matrix-matrix contraction. Given rank-2 tensors :math:`G^1_{i,j}` and :math:`G^2_{j,k}`, we have seen the :math:`(i,k)`-th element of the resulting contraction along the :math:`j`-th index is +# For this reason, it is useful to understand how to calculate the computational cost or the *complexity* of a tensor network contraction. First, we look at a simple matrix-matrix contraction. Given rank-2 tensors :math:`G^1_{i,j}` and :math:`G^2_{j,k}`, we have seen that the :math:`(i,k)`-th element of the resulting contraction along the :math:`j`-th index is # # .. math:: # (G^3)_{i,k} = G^2 \cdot G^1 = \sum_{j=1}^{d_j} G^{2}_{j,k} G^{1}_{i,j} # -# where the indices :math:`i, j, k` have dimensions :math:`d_i, d_j, d_k`, respectively. Thus, obtaining the :math:`(i,k)`-th element requires :math:`\mathcal{O}(d_j)` operations. To construct the full tensor :math:`G^3`, we must repeat this procedure :math:`d_i \times d_k` times (once for every possible value of :math:`i` and :math:`k`). Therefore, the total complexity of the contration is +# where the indices :math:`i, j, k` have dimensions :math:`d_i, d_j, d_k`, respectively. Thus, obtaining the :math:`(i,k)`-th element requires :math:`\mathcal{O}(d_j)` operations. To construct the full tensor :math:`G^3`, we must repeat this procedure :math:`d_i \times d_k` times (once for every possible value of :math:`i` and :math:`k`). Therefore, the total complexity of the contraction is # # .. math:: # \mathcal{O}(d_i \times d_j \times d_k) # -# To illustrate the importance of choosing an efficient contraction path, let us look at a similar contraction between 3 rank-3 tensors as shown in the previous section. +# To illustrate the importance of choosing an efficient contraction path, let us look at a similar contraction between 3 rank-3 tensors, as shown in the previous section. # # .. figure:: ../_static/demonstration_assets/tn_basics/09-contraction.png # :align: center @@ -224,7 +223,7 @@ # .. math:: # \sum_{k, m} (AB)_{i, k, l, m} C_{k,m,n} \implies \mathcal{O}(d_j \times d_m \times d_i^2) # -# operations. Assuming :math:`d_j \leq d_m \leq d_i`, assymptotially, the whole contraction will have a cost of :math:`\mathcal{O}(d_j \times d_m \times d_i^2)`. Alternatively, we could first contract :math:`B` and :math:`C`, incurring in the cost +# operations. Assuming :math:`d_j \leq d_m \leq d_i`, asymptotically, the whole contraction will have a cost of :math:`\mathcal{O}(d_j \times d_m \times d_i^2)`. Alternatively, we could first contract :math:`B` and :math:`C`, incurring the cost # # .. math:: # \sum_{m} B_{j,l,m} C_{k,m,n} \implies \mathcal{O}(d_i \times d_m \times d_j^2 ) . @@ -234,7 +233,7 @@ # .. math:: # \sum_{j, k} A_{i,j,k} (BC)_{j,l,k,n} \implies \mathcal{O}(d_i^2 \times d_j^2) . # -# This means, the second contraction path results in an asymptotic cost of :math:`\mathcal{O}(d_i^2 \times d_j^2)` - lower than the first contraction path. +# This means the second contraction path results in an asymptotic cost of :math:`\mathcal{O}(d_i^2 \times d_j^2)`—lower than the first contraction path. # To see this in practice, let us perform the above contraction using ``np.einsum``. First, we create 3 tensors with the correct dimensions. import timeit @@ -248,7 +247,7 @@ C = np.arange(dj*dm*di).reshape(dj,dm,di) # kmn ############################################################################## -# Then, we can perform the individual contractions between pairs of tensors and time it using ``timeit``. We repeat the contraction ``20`` times and average the computation time to account for smaller fluctuations. First, we start contracting :math:`A` and :math:`B`. +# Then, we perform the individual contractions between pairs of tensors and time them using ``timeit``. We repeat the contraction ``20`` times and average the computation time to account for smaller fluctuations. First, we start by contracting :math:`A` and :math:`B`. iterations = 20 @@ -278,7 +277,7 @@ print(f"Computation cost for BC contraction: {average_time_ms:.8f} ms") ############################################################################## -# We see this contraction is in the same order of magnitude as the contraction between :math:`A` and :math:`B`, as expected by the complexity analysis since they both yield :math:`\mathcal{O}(d_i \times d_m \times d_j^2 )`. Then, we perform the contraction between the resulting tensor and :math:`A`. +# We see that this contraction is of the same order of magnitude as the contraction between :math:`A` and :math:`B`, as expected from the complexity analysis, since they both yield :math:`\mathcal{O}(d_i \times d_m \times d_j^2 )`. Then, we perform the contraction between the resulting tensor and :math:`A`. BC = np.einsum('jlm, kmn -> jlkn', B, C) @@ -289,15 +288,15 @@ print(f"Computation cost for A(BC) contraction: {average_time_ms:.8f} ms") ############################################################################## -# From this we see that the second contraction path yields a much lower complexity compared to the first one, just as we expected! +# From this, we see that the second contraction path results in a much lower complexity compared to the first one, just as we expected! ############################################################################## # Contraction paths # ----------------- # -# In the previous section, through a toy example, we explored how the choice of the contraction path affects the computational cost of the tensor network contraction. As shown in [#Lam1997]_ , finding an optimal contraction path is equivalent to solving the "multiplication problem" and thus, it is NP-hard. In this section, we provide a general description of wide-spread techniques used to tackle this ubiquitous task. +# In the previous section, we explored how the choice of the contraction path affects the computational cost of the tensor network contraction through a toy example. As shown in [#Lam1997]_, finding an optimal contraction path is equivalent to solving the "multiplication problem," and thus, it is NP-hard. In this section, we provide a general description of the widespread techniques used to tackle this ubiquitous task. # -# First, we set up the framework of the problem. While multiway contractions - contractions between more than 2 tensors at a time - are possible, we will consider only pairwise contractions since the former can always be decomposed in terms of the latter. In addition, contracting a tensor network need not result in a single tensor. However, here we consider only the single tensor case as it underlies the more general scenario [#Gray2021]_. +# First, we set up the framework of the problem. While multiway contractions - contractions between more than 2 tensors at a time - are possible, we will consider only pairwise contractions since the former can always be decomposed in terms of the latter. In addition, contracting a tensor network doesn't need to result in a single tensor. However, here we consider only the single tensor case as it underlies the more general scenario [#Gray2021]_. # # The underlying idea behind finding a contraction path is based on the construction of the computational graph, i.e., a rooted binary tree - also known as the **contraction tree** - that specifies the sequence of pairwise contractions to be executed. In this tree structure, a leaf node corresponds to a tensor from the original network (blue tensors) and the pairwise contractions (red lines) give rise to the intermediate tensors (red tensors) corresponding to the rest of the nodes in the contraction tree. # @@ -309,25 +308,25 @@ # # .. note:: # -# Besides finding a contraction path that minimizes the computational cost, we can also attempt to find a path that optimizes the memory cost. That is, a contraction path in which all intermediate tensors are below a certain size. +# Besides finding a contraction path that minimizes the computational cost, we can also attempt to find a path that optimizes the memory cost. That is a contraction path in which all intermediate tensors are below a certain size. # -# Now, how do we traverse the space of trees to optimize over? The most straightforward idea is to perform an exhaustic search through all of them. As explained in [#Pfeifer2014]_, this can be done (with some additional improvements) using the following well known algorithms: +# Now, how do we traverse the space of trees to optimize over? The most straightforward idea is to perform an exhaustive search through all of them. As explained in [#Pfeifer2014]_, this can be done (with some additional improvements) using the following well-known algorithms: # # - Depth-first search # - Breadth-first search # - Dynamic programming # -# While the exhaustive approach scales like :math:`\mathcal{O}(N!)`, with :math:`N` the number of tensors in the network, it can handle a handful of tensors within seconds, providing a good benchmark. In addition, compared to the following algorithms, the exhaustive search guarantees to find the global minimum optimizing the desired metric - space and/or time. +# While the exhaustive approach scales like :math:`\mathcal{O}(N!)`, with :math:`N` the number of tensors in the network, it can handle a handful of tensors within seconds, providing a good benchmark. In addition, compared to the following algorithms, the exhaustive search guarantees finding the global minimum and optimizing the desired metric - space and/or time. # # .. note:: # -# A recursive implementation of the depth-first search is used by default in the well known package ``opt_einsum`` `(see docs) `_. +# A recursive implementation of the depth-first search is used by default in the well-known package ``opt_einsum`` `(see docs) `_. # -# Further approaches introduced in [#Gray2021]_ are based on alternative common graph theoretic tasks, rather than searching over the contraction tree space, such as the `balanced bipartitioning `_ and `community detection `_ algorithms. And even though, these are only heuristics that do not guarantee an optimal contraction path, they can often achieve an arbitrarliy close to optimal performance. +# Further approaches introduced in [#Gray2021]_ are based on alternative common graph-theoretic tasks, rather than searching over the contraction tree space, such as the `balanced bi-partitioning `_ and `community detection `_ algorithms. And even though these are only heuristics that do not guarantee an optimal contraction path, they can often achieve an arbitrarily close to optimal performance. # # An extra level of optimization, known as *hyper-optimization* is introduced by the use of different algorithms to find the optimal contraction based on the specific tensor network, as some algorithms are better suited for certain network structures. For an in-depth exploration of these heuristics, please refer to [#Gray2021]_. # -# As we will explore in the next section, we can use tensor networks to simulate quantum circuits. In particular, the calculation of an expectation value corresponds to the contraction of the tensor network into a single tensor (scalar) as discussed in this section. In ``Pennylane``, this simulation can be performed using the :class:`~pennylane.devices.default_tensor.DefaultTensor` device and the method used to find the contraction path can be chosen via the ``contraction_optimizer`` keyword argument. +# As we will explore in the next section, we can use tensor networks to simulate quantum circuits. In particular, the calculation of an expectation value corresponds to the contraction of the tensor network into a single tensor (scalar) as discussed in this section. In ``Pennylane``, this simulation can be performed using the :class:`~pennylane.devices.default_tensor.DefaultTensor` device, and the method used to find the contraction path can be chosen via the ``contraction_optimizer`` keyword argument. import pennylane as qml @@ -336,15 +335,15 @@ ############################################################################## # The different types of values accepted for ``contraction_optimizer`` are determined by the ``optimize`` parameter in ``Quimb`` (see `docs `_) as this is the backend behind the :class:`~pennylane.devices.default_tensor.DefaultTensor` device. See `this tutorial `_ to learn more the use of this device in ``Pennylane``. # -# The size of (intermediate) tensors can grow exponential with the number of indices and dimensions, specially for large-scale tensor networks. Thus, we might run into memory problems when performing the contractions. A useful additional technique to split these tensors into more manageable pieces is known as **slicing**. +# The size of (intermediate) tensors can grow exponentially with the number of indices and dimensions, especially for large-scale tensor networks. Thus, we might run into memory problems when performing the contractions. A useful additional technique to split these tensors into more manageable pieces is known as **slicing**. # -# The idea is to change space for computation time, by temporarily fixing the values of some indices in the tensors, performing independently the contraction for each fixed value and summing the results [#Gray2021]_. +# The idea is to change space for computation time, by temporarily fixing the values of some indices in the tensors, performing independently the contraction for each fixed value, and summing the results [#Gray2021]_. ############################################################################## # Quantum circuits are tensor networks # --------------------------------- # -# Until now, we have looked at general tensor networks, while ✨sparkling✨ the discussions with examples related to quantum circuits. Here, we leverage the components we have learned to explore this relation more in depth. +# Until now, we have looked at general tensor networks, while ✨sparkling✨ the discussions with examples related to quantum circuits. Here, we leverage the components we have learned to explore this relation more in-depth. # # First, it is important to emphasize that quantum circuits don't just "look" or "behave" like tensor networks, but rather they **are** tensor networks! Quantum Circuits are a special class of tensor networks where each horizontal wire corresponds to the Hilbert space of a single qubit and the tensors acting on these subsystems are restricted to be unitary operators, denoting the time evolution of the state (from left to right). # @@ -353,13 +352,13 @@ # .. math:: # | \psi \rangle = U |\psi_0 \rangle, # -# which can also be depicted diagramatically as +# which can also be depicted diagrammatically as # # .. figure:: ../_static/demonstration_assets/tn_basics/11-circuit.png # :align: center # :width: 45% # -# In the right hand side of the equality we have assumed a specific form for the U tensor in terms of local 2-qubit gates, which is often the case when dealing with real quantum hardware. In addition, it is common for the initial state to be a product state such as :math:`|0\rangle^{\otimes N}`, hence the form of the tensor in the diagram. +# In the right-hand side of the equality we have assumed a specific form for the U tensor in terms of local 2-qubit gates, which is often the case when dealing with real quantum hardware. In addition, it is common for the initial state to be a product state such as :math:`|0\rangle^{\otimes N}`, hence the form of the tensor in the diagram. # # Now we can ask the important question: what quantities can we compute from this tensor network? # @@ -374,14 +373,14 @@ # If the observable is a linear combination of hermitian operators (e.g., a Hamiltonian) # # .. math:: -# O = \sum_i c_i O_i +# O = \sum_i c_i O_i. # -# we can calculate the total expectation value "naïvely" by computing the inner product for each component :math:`O_i` and summing up the weighted results. +# We can calculate the total expectation value "naïvely" by computing the inner product for each component :math:`O_i` and summing up the weighted results. # # .. math:: # O = \sum_i c_i \langle O_i \rangle = \sum_i c_i \langle \psi | O_i | \psi \rangle. # -# However, it is possible to perform this operation more efficiently using tensor networks by means of a structure called Matrix Product Operator (MPO) [#Pirvu2010]_. The idea is to construct an efficient representation of the observable :math:`O` which can be contracted with the tensor network from :math:`|\psi \rangle`. Constructing these networks efficiently for hamiltonians of arbitrary structure is an interesting task, which goes beyond the scope of this tutorial. +# However, it is possible to perform this operation more efficiently using tensor networks by means of a structure called Matrix Product Operator (MPO) [#Pirvu2010]_. The idea is to construct an efficient representation of the observable :math:`O` which can be contracted with the tensor network from :math:`|\psi \rangle`. Constructing these networks efficiently for Hamiltonians of arbitrary structure is an interesting task, which goes beyond the scope of this tutorial. # # When the observable of interest is *local*, i.e., it acts on a few neighbouring qubits, we can calculate the expectation value by considering only the section of the quantum circuit within the *reverse light cone* (causal cone) of the observable :math:`O_l`. # @@ -399,12 +398,12 @@ # Sampling # ~~~~~~~~ # -# In addition to calculating expectation values, we can also use the tensor network arising from a quantum circuit to sample bitstrings from the evolved probability distribution :math:`| \psi \rangle` - emulating what you would obtain from a real quantum computer. Since this is an ubiquitous task in quantum information, several algorithms have been proposed to generate samples from probability distributions represented as tensor networks. In particular, here we discuss the "Perfect Sampling Algorithm" applied to unitary tensor networks [#Ferris2012]_, as this generates uncorrelated samples, unlike markov-based approaches. +# In addition to calculating expectation values, we can also use the tensor network arising from a quantum circuit to sample bitstrings from the evolved probability distribution :math:`| \psi \rangle` - emulating what you would obtain from a real quantum computer. Since this is a ubiquitous task in quantum information, several algorithms have been proposed to generate samples from probability distributions represented as tensor networks. In particular, here we discuss the "Perfect Sampling Algorithm" applied to unitary tensor networks [#Ferris2012]_, as this generates uncorrelated samples, unlike Markov-based approaches. # # .. note:: # The method used in `Quimb `_ (the backend behind :class:`~pennylane.devices.default_tensor.DefaultTensor`) to generate samples from the quantum circuit is also based on this algorithm. # -# A cornerstone behind this algorithm is the well known `chain rule `_ which allows us to write the joint probability of an event using only conditional probabilities. Using this, we can express the probability of sampling the bitstring :math:`(x_1, x_2, x_3, \ldots, x_N)` from :math:`| \psi \rangle` as +# A cornerstone behind this algorithm is the well-known `chain rule `_ which allows us to write the joint probability of an event using only conditional probabilities. Using this, we can express the probability of sampling the bitstring :math:`(x_1, x_2, x_3, \ldots, x_N)` from :math:`| \psi \rangle` as # # .. math:: # p(x_1, x_2, x_3, \ldots, x_N) = p(x_1) p(x_2|x_1) p(x_3| x_1 x_2) \ldots p(x_N | x_1 x_2 x_3 \ldots x_{N-1}) @@ -425,11 +424,11 @@ # :align: center # :width: 45% # .. note:: -# In this diagram, we have extracted the diagonal of the reduced density matrix by contracting it with the COPY tensor introduced earlier on this tutorial! +# In this diagram, we have extracted the diagonal of the reduced density matrix by contracting it with the COPY tensor introduced earlier in this tutorial! # # Once we obtain the probability vector, we can generate a random sample weighted by these probabilities. To do so, we generate a random number :math:`r \in [0,1]` and choose :math:`x_1 = 0` if :math:`r < p(x_1=0)` and :math:`x_1 = 1` otherwise. We save this sample as :math:`\hat{x}_1`. # -# Next, we can calculate the following term :math:`p(x_2|\hat{x}_1)` conditioned on the sample we have just obtained. To accomplish this, we **project** the first qubit to be :math:`\hat{x}_1`. We can do this by contracting the computational basis state :math:`| \hat{x}_1 \rangle` with :math:`|\psi \rangle`, resulting in a smaller state :math:`|\psi_{x_1} \rangle`. Then, we can proceed exactly as we did in the previous step, calculating the reduced density matrix :math:`\rho_2` by tracing out the remaining qubits :math:`3,4,\ldots,N` and computing the probability vector from its diagonal +# Next, we can calculate the following term :math:`p(x_2|\hat{x}_1)` conditioned on the sample we have just obtained. To accomplish this, we **project** the first qubit to be :math:`\hat{x}_1`. We can do this by contracting the computational basis state :math:`| \hat{x}_1 \rangle` with :math:`|\psi \rangle`, resulting in a smaller state :math:`|\psi_{x_1} \rangle`. Then, we can proceed exactly as we did in the previous step, calculating the reduced density matrix :math:`\rho_2` by tracing out the remaining qubits :math:`3,4,\ldots, N` and computing the probability vector from its diagonal # # .. math:: # | p_{x_2 | \hat{x}_1} \rangle = \mathrm{diag} \left( \mathrm{Tr}_{3,4, \ldots,N}(| \psi_{x_1} \rangle \langle \psi_{x_1} |) \right). @@ -443,16 +442,16 @@ # Analogously as done with the expectation values, these contractions only involve the sections of the circuit within the light cone of **both** the projection with :math:`| \hat{x}_1 \rangle` and the contraction with the COPY tensor (diagonal computation). This procedure can be repeated recursively using the chain rule equation until we obtain the full bitstring :math:`(\hat{x}_1, \hat{x}_2, \hat{x}_3, \ldots, \hat{x}_N)`. To obtain more samples, we repeat the procedure from the beginning - this is what makes every sample memoryless or a perfect sample from the probability distribution. # # .. note:: -# We can reduce the computational cost of this algorithm by **caching** results from previous contractions. When we draw a new sample that partially matches a previously explored configuration (marginal probability), we can reuse the cached results and avoid contracting this part of the network over again. For example, let's assume we have performed the perfect sampling algorithm once and obtained the sample :math:`0110`. If the next sample we need to generate starts with the substring :math:`01`, we can reuse the marginal probabilies up to :math:`p(x_3|01)` and only calculate the new parts of the sequence. The same caching idea can be applied to other tensor network algorithms involving many contractions. +# We can reduce the computational cost of this algorithm by **caching** results from previous contractions. When we draw a new sample that partially matches a previously explored configuration (marginal probability), we can reuse the cached results and avoid contracting this part of the network over again. For example, let's assume we have performed the perfect sampling algorithm once and obtained the sample :math:`0110`. If the next sample we need to generate starts with the substring :math:`01`, we can reuse the marginal probabilities up to :math:`p(x_3|01)` and only calculate the new parts of the sequence. The same caching idea can be applied to other tensor network algorithms involving many contractions. ############################################################################## # # Conclusion # ---------- # -# Although the world of tensor networks and their relation to quantum computing is vastly wider than what we could ever cover in one tutorial, we hope that after this explanations you now feel equipped with the tools needed to dive deeper into this topic by yourself. 🔍 +# Although the world of tensor networks and their relation to quantum computing is vastly wider than what we could ever cover in one tutorial, we hope that after these explanations you now feel equipped with the tools needed to dive deeper into this topic by yourself. 🔍 # -# If you want to learn more about how to use tensor networks as a diagramatic tool, check out `these amazing lectures notes `_ by J.Biamonte. In addition, check out `this website `_ for great explanations on many important algorithms and tensor network structures by Flatiron Institute. +# If you want to learn more about how to use tensor networks as a diagrammatic tool, check out `these amazing lecture notes `_ by J.Biamonte. In addition, check out `this website `_ for great explanations on many important algorithms and tensor network structures by Flatiron Institute. # # And that is it for this tutorial! 🎉 From 83608d4ad199dadee2bcc918ef82a7fd71e8af58 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Sat, 26 Oct 2024 19:25:52 +0200 Subject: [PATCH 41/51] change size of images --- demonstrations/tutorial_tn_basics.py | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 77dc38d7c6..9c08d3119f 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -386,7 +386,7 @@ # # .. figure:: ../_static/demonstration_assets/tn_basics/12-expectation-local.png # :align: center -# :width: 45% +# :width: 60% # # Then, the sections outside of the light cone (grayed-out gates in the figure above) can be ignored since these are contractions resulting in the identity: :math:`G G^\dagger = I`. This helps us decrease the size of the tensor to be contracted, and consequently, the computational expense, by focusing on the section of the circuit with support inside the light cone of the observable - i.e., the gates that affect the calculation of the expectation value. # @@ -422,7 +422,7 @@ # # .. figure:: ../_static/demonstration_assets/tn_basics/13-sample.png # :align: center -# :width: 45% +# :width: 60% # .. note:: # In this diagram, we have extracted the diagonal of the reduced density matrix by contracting it with the COPY tensor introduced earlier in this tutorial! # @@ -437,7 +437,7 @@ # # .. figure:: ../_static/demonstration_assets/tn_basics/14-sample-cntd.png # :align: center -# :width: 45% +# :width: 60% # # Analogously as done with the expectation values, these contractions only involve the sections of the circuit within the light cone of **both** the projection with :math:`| \hat{x}_1 \rangle` and the contraction with the COPY tensor (diagonal computation). This procedure can be repeated recursively using the chain rule equation until we obtain the full bitstring :math:`(\hat{x}_1, \hat{x}_2, \hat{x}_3, \ldots, \hat{x}_N)`. To obtain more samples, we repeat the procedure from the beginning - this is what makes every sample memoryless or a perfect sample from the probability distribution. # From c8dec353541124d2266da6e389e262f333784bb8 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Mon, 28 Oct 2024 15:59:18 +0100 Subject: [PATCH 42/51] increase size of images --- demonstrations/tutorial_tn_basics.py | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 9c08d3119f..d538c82a3d 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -386,7 +386,7 @@ # # .. figure:: ../_static/demonstration_assets/tn_basics/12-expectation-local.png # :align: center -# :width: 60% +# :width: 65% # # Then, the sections outside of the light cone (grayed-out gates in the figure above) can be ignored since these are contractions resulting in the identity: :math:`G G^\dagger = I`. This helps us decrease the size of the tensor to be contracted, and consequently, the computational expense, by focusing on the section of the circuit with support inside the light cone of the observable - i.e., the gates that affect the calculation of the expectation value. # @@ -422,7 +422,7 @@ # # .. figure:: ../_static/demonstration_assets/tn_basics/13-sample.png # :align: center -# :width: 60% +# :width: 65% # .. note:: # In this diagram, we have extracted the diagonal of the reduced density matrix by contracting it with the COPY tensor introduced earlier in this tutorial! # @@ -437,7 +437,7 @@ # # .. figure:: ../_static/demonstration_assets/tn_basics/14-sample-cntd.png # :align: center -# :width: 60% +# :width: 65% # # Analogously as done with the expectation values, these contractions only involve the sections of the circuit within the light cone of **both** the projection with :math:`| \hat{x}_1 \rangle` and the contraction with the COPY tensor (diagonal computation). This procedure can be repeated recursively using the chain rule equation until we obtain the full bitstring :math:`(\hat{x}_1, \hat{x}_2, \hat{x}_3, \ldots, \hat{x}_N)`. To obtain more samples, we repeat the procedure from the beginning - this is what makes every sample memoryless or a perfect sample from the probability distribution. # From 71c367ea7e606d6ca3ebadb771d6bbbbe7d92a5e Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Mon, 28 Oct 2024 18:36:58 +0100 Subject: [PATCH 43/51] apply spelling corrections --- demonstrations/tutorial_tn_basics.py | 29 ++++++++++++++-------------- 1 file changed, 14 insertions(+), 15 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index d538c82a3d..ce98332c84 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -1,11 +1,11 @@ r"""Tutorial: ABC of tensor networks for quantum circuits ============================================================= -If you are in the domain of quantum computing or quantum information, chances are that you have heard (maybe a lot) about tensor networks being brought into the conversation. In fact, they are not only useful in these areas, but actually they are rather a widespread "tool" with applications ranging across physics, math, and computer science. +If you are in the domain of quantum computing or quantum information, chances are you have heard (maybe a lot) about tensor networks being brought into the conversation. They are not only useful in these areas, but actually, they are a rather widespread "tool" with applications ranging across physics, math, and computer science. Part of the excitement surrounding tensor networks is due to their ability to represent complex data efficiently, which allows for — among other things — fast classical simulations. In addition, the diagrammatic language accompanying tensor networks makes working with them intuitive and suitable for describing a vast range of mathematical concepts, including quantum circuits. -In this tutorial, we aim to provide an introduction to tensor networks with a focus on their applications in quantum computing. We choose to start by discussing the basic notions and definitions of tensors and tensor networks, and work our way up to more advanced topics such as contraction paths and algorithms used to simulate quantum computers using tensor networks. +In this tutorial, we aim to provide an introduction to tensor networks with a focus on their applications in quantum computing. We choose to start by discussing the basic notions and definitions of tensors and tensor networks and work our way up to more advanced topics such as contraction paths and algorithms used to simulate quantum computers using tensor networks. Without further ado, let’s dive right in! 🤓📚 @@ -73,7 +73,7 @@ print("Rank-3 tensor: \n", tensor_rank3) ############################################################################## -# We can create a tensor of arbitrary rank following a similar procedure. This recursive approach is instructive to understand a rank-:math:`r` tensor as made up of rank-:math:`(r-1)` tensors, which translates to the code as adding an additional level in the nested bracket structure ``[tensor_rank_r-1]``. +# We can create a tensor of arbitrary rank following a similar procedure. This recursive approach illustrates how a rank-:math:`r` tensor can be seen as consisting of nested rank-:math:`(r-1)` tensors, represented in code by adding another level to the nested bracket structure ``[tensor_rank_r-1]``. # ############################################################################## @@ -102,9 +102,9 @@ # :align: center # :width: 55% # -# Here, the resulting tensor has two dangling indices, :math:i and :math:k, defining a matrix, as expected! +# Here, the resulting tensor has two dangling indices, :math:`i` and :math:`k`, defining a matrix, as expected! # -# We can now generalize this concept to tensors, and consequently, to more than two legs being contracted. For example, let us look at three tensors :math:`A_{i,j,k}`, :math:`B_{j,l,m}` and :math:`C_{k,m,n}`. To contract them, all we need to do is to sum over repeated indices (:math:`j`, :math:`k`, :math:`m`), just as we would do in `Einstein convention `_. Thus, the :math:`(i,l,n)`-th element of the resulting tensor :math:`D` is +# We can now generalize this concept to tensors, and consequently, to more than two legs being contracted. For example, let us look at three tensors :math:`A_{i,j,k}`, :math:`B_{j,l,m}`, and :math:`C_{k,m,n}`. To contract them, all we need to do is to sum over repeated indices (:math:`j`, :math:`k`, :math:`m`), just as we would do in `Einstein convention `_. Thus, the :math:`(i,l,n)`-th element of the resulting tensor :math:`D` is # # .. math:: # (D)_{i,l,n} = \sum_{j,k,m} A_{i,j,k} B_{j,l,m} C_{k,m,n} . @@ -118,11 +118,10 @@ # With the above contraction, we have formed a network of tensors, i.e., a **Tensor Network**! # # .. tip:: -# A common question arising when drawing a tensor is "what is the correct order to draw the indices?". For instance, in the figure above, we have adopted the convention that a tensor :math:`A_{i,j,k}` corresponds to a diagram with the first leg (:math:`i`) pointing left, the second leg (:math:`j`) pointing upwards, and the third leg (:math:`k`) pointing right, and similarly for the other two tensors. However, this need not be the case. We could have defined the first leg to be the one pointing upwards, for example. Based on the use case, and the user, some conventions might seem more natural than others. The only important thing to keep in mind is to be consistent. In other words, once we choose a convention for the order, we should apply it to all the tensors to avoid contracting the wrong indices ❌. +# A common question arising when drawing a tensor is "What is the correct order to draw the indices?". For instance, in the figure above, we have adopted the convention that a tensor :math:`A_{i,j,k}` corresponds to a diagram with the first leg (:math:`i`) pointing left, the second leg (:math:`j`) pointing upwards, and the third leg (:math:`k`) pointing right, and similarly for the other two tensors. However, this need not be the case. We could have defined the first leg to be the one pointing upwards, for example. Based on the use case, and the user, some conventions might seem more natural than others. The only important thing to keep in mind is to be consistent. In other words, once we choose a convention for the order, we should apply it to all the tensors to avoid contracting the wrong indices ❌. # # Remember the similarity between tensor network contractions and Einstein notation? Then, it doesn't come as a surprise that we can perform a contraction using the function ``np.einsum``. To do so, we can start by creating the 3 tensors to be contracted by reshaping a 1D array (created using ``np.arange``) into rank-3 tensors of the correct dimensions. - # Create the individual rank-3 tensors A = np.arange(6).reshape(1, 2, 3) # ijk B = np.arange(6).reshape(2, 3, 1) # jlm @@ -164,7 +163,7 @@ # .. math:: # CNOT = \sum_j (T^1)_{i,j,k} \otimes (T^2)_{l,j,m}. # -# It turns out :math:`T^1` and :math:`T^2` are special tensors known as the COPY and XOR tensors, respectively, and therefore have special diagrammatic representations. +# It turns out : that math:`T^1` and :math:`T^2` are special tensors known as the COPY and XOR tensors, respectively, and therefore have special diagrammatic representations. # # .. figure:: ../_static/demonstration_assets/tn_basics/07-t1-t2.png # :align: center @@ -178,19 +177,19 @@ # :width: 45% # # .. note:: -# By looking at the elements of the COPY tensor, we can interpret it as being equal to unity when all the indices have the same value (0 or 1) and vanishing otherwise. On the other hand, the XOR tensor can be understood as being equal to unity when the values of the three indices contain an even number of 1's, and vanishing otherwise. We anticipate that the COPY tensor can be used to obtain the diagonal of a matrix. This will be useful in the last section of this tutorial. +# By looking at the elements of the COPY tensor, we can interpret it as being equal to unity when all the indices have the same value (0 or 1) and vanishing otherwise. On the other hand, the XOR tensor can be understood as being equal to unity when the values of the three indices contain an even number of 1's and vanishing otherwise. We anticipate that the COPY tensor can be used to obtain the diagonal of a matrix. This will be useful in the last section of this tutorial. # # This demonstrates the relation between the CNOT acting as a rank-4 tensor with dimensions :math:`2 \times 2 \times 2 \times 2` and its decomposition in terms of two rank-3 local tensors (:math:`T^1` and :math:`T^2`) of dimensions :math:`2 \times 2 \times 2`. # # .. note:: # -# More generally, we can find decompositions of multi-qubit gates into local tensors by means of the ubiquitous singular value decomposition (SVD). This method is explained in detail in our :doc:`MPS tutorial `. +# More generally, we can find decompositions of multi-qubit gates into local tensors employing the ubiquitous singular value decomposition (SVD). This method is explained in detail in our :doc:`MPS tutorial `. ############################################################################## # The cost of contracting a network # --------------------------------- # -# A common task when dealing with tensors is the contraction of large networks resulting in a single tensor (including scalars). To arrive at this final tensor, we can start with a single tensor and contract it with adjacent tensors one-at-a-time. The order in which this is carried out is known as the *contraction path* or *bubbling*. +# A common task when dealing with tensors is the contraction of large networks resulting in a single tensor (including scalars). To arrive at this final tensor, we can start with a single tensor and contract it with adjacent tensors one at a time. The order in which this is carried out is known as the *contraction path* or *bubbling*. # While the final tensor is independent of the order of the contraction, the number of operations performed can vary greatly depending on the order in which we contract the intermediate tensors. Moreover, in a general setup, finding the optimal order of indices to be contracted is not a trivial task. # # .. note:: @@ -386,7 +385,7 @@ # # .. figure:: ../_static/demonstration_assets/tn_basics/12-expectation-local.png # :align: center -# :width: 65% +# :width: 70% # # Then, the sections outside of the light cone (grayed-out gates in the figure above) can be ignored since these are contractions resulting in the identity: :math:`G G^\dagger = I`. This helps us decrease the size of the tensor to be contracted, and consequently, the computational expense, by focusing on the section of the circuit with support inside the light cone of the observable - i.e., the gates that affect the calculation of the expectation value. # @@ -422,7 +421,7 @@ # # .. figure:: ../_static/demonstration_assets/tn_basics/13-sample.png # :align: center -# :width: 65% +# :width: 70% # .. note:: # In this diagram, we have extracted the diagonal of the reduced density matrix by contracting it with the COPY tensor introduced earlier in this tutorial! # @@ -437,7 +436,7 @@ # # .. figure:: ../_static/demonstration_assets/tn_basics/14-sample-cntd.png # :align: center -# :width: 65% +# :width: 70% # # Analogously as done with the expectation values, these contractions only involve the sections of the circuit within the light cone of **both** the projection with :math:`| \hat{x}_1 \rangle` and the contraction with the COPY tensor (diagonal computation). This procedure can be repeated recursively using the chain rule equation until we obtain the full bitstring :math:`(\hat{x}_1, \hat{x}_2, \hat{x}_3, \ldots, \hat{x}_N)`. To obtain more samples, we repeat the procedure from the beginning - this is what makes every sample memoryless or a perfect sample from the probability distribution. # @@ -451,7 +450,7 @@ # # Although the world of tensor networks and their relation to quantum computing is vastly wider than what we could ever cover in one tutorial, we hope that after these explanations you now feel equipped with the tools needed to dive deeper into this topic by yourself. 🔍 # -# If you want to learn more about how to use tensor networks as a diagrammatic tool, check out `these amazing lecture notes `_ by J.Biamonte. In addition, check out `this website `_ for great explanations on many important algorithms and tensor network structures by Flatiron Institute. +# If you want to learn more about using tensor networks as a diagrammatic tool, check out `these amazing lecture notes `_ by J.Biamonte. In addition, check out `this website `_ for great explanations on many important algorithms and tensor network structures by Flatiron Institute. # # And that is it for this tutorial! 🎉 From e751ba829adf4d0ed5f74b34b515131f35849601 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Mon, 11 Nov 2024 01:12:04 +0100 Subject: [PATCH 44/51] apply changes from demo revision --- demonstrations/tutorial_tn_basics.py | 46 ++++++++++++++++------------ 1 file changed, 27 insertions(+), 19 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index ce98332c84..f349394c44 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -18,12 +18,12 @@ .. math:: T_{i_1, i_2, \ldots, i_r} \in \mathbb{C}^{d_1 \times d_2 \times \ldots \times d_r}, -where each :math:`i_n` is an *index* of dimension :math:`d_n`—it takes integer values such that :math:`i_n \in [1, d_n]`—and the number of indices :math:`r` is known as the *rank* of the tensor. We say :math:`T` is a rank-:math:`r` tensor. - -.. tip:: - Some authors refer to the indices of the tensors as their dimensions. In this tutorial, these two concepts will have different meanings, although related. +where each :math:`i_n` is an *index* of dimension :math:`d_n` and the number of indices :math:`r` is known as the *rank* of the tensor. We say :math:`T` is a rank-:math:`r` tensor. For example, a scalar :math:`s` is a rank-0 tensor, a vector :math:`v_i` is a rank-1 tensor, and a matrix :math:`G_{i,j}` is a rank-2 tensor. + +.. note:: + Some authors refer to the indices :math:`i_n` as the "dimensions of the tensor". In this tutorial, however, the term "dimensions" :math:`d_n` instead refers to the range of values that each index :math:`i_n` can take, specifically :math:`i_n \in [1, d_n]`, meaning :math:`i_n` takes integer values within this range. A beautiful and powerful tool accompanying tensors (networks) is their graphical language representation. The diagram of a tensor is simply a geometric shape with a leg sticking out of it for every index in the tensor. For example, @@ -73,7 +73,7 @@ print("Rank-3 tensor: \n", tensor_rank3) ############################################################################## -# We can create a tensor of arbitrary rank following a similar procedure. This recursive approach illustrates how a rank-:math:`r` tensor can be seen as consisting of nested rank-:math:`(r-1)` tensors, represented in code by adding another level to the nested bracket structure ``[tensor_rank_r-1]``. +# We can create a tensor of arbitrary rank following a similar procedure. This recursive approach illustrates how a rank-:math:`r` tensor can be seen as consisting of nested rank-:math:`(r-1)` tensors. This translates into code as adding another level to the nested bracket structure: ``[tensor_rank_r-1]``. # ############################################################################## @@ -104,12 +104,12 @@ # # Here, the resulting tensor has two dangling indices, :math:`i` and :math:`k`, defining a matrix, as expected! # -# We can now generalize this concept to tensors, and consequently, to more than two legs being contracted. For example, let us look at three tensors :math:`A_{i,j,k}`, :math:`B_{j,l,m}`, and :math:`C_{k,m,n}`. To contract them, all we need to do is to sum over repeated indices (:math:`j`, :math:`k`, :math:`m`), just as we would do in `Einstein convention `_. Thus, the :math:`(i,l,n)`-th element of the resulting tensor :math:`D` is +# We can now generalize this concept to tensors, and consequently, to more than two legs being contracted. For example, let us look at three tensors :math:`A_{i,j,k}`, :math:`B_{j,l,m}`, and :math:`C_{k,m,n}`. To contract them, all we need to do is to sum over repeated indices (:math:`j`, :math:`k`, :math:`m`), just as we would do in `Einstein convention `_. To obtain the :math:`(i,l,n)`-th element of the resulting tensor :math:`D`, we perform this contraction by using the tensor product :math:`\otimes` between the 3 tensors and summing over :math:`j, k, m` [#Bridgeman2017]_ # # .. math:: -# (D)_{i,l,n} = \sum_{j,k,m} A_{i,j,k} B_{j,l,m} C_{k,m,n} . +# (D)_{i,l,n} = \sum_{j,k,m} A_{i,j,k} \otimes B_{j,l,m} \otimes C_{k,m,n} = \sum_{j,k,m} A_{i,j,k} B_{j,l,m} C_{k,m,n} . # -# Notice how the resulting rank-3 tensor consists of the remaining open legs from the initial tensors :math:`(i,l,n)`. The diagrammatic representation of this equation is obtained by connecting all the legs with the same indices. +# Notice how the resulting rank-3 tensor consists of the remaining open legs from the initial tensors :math:`(i,l,n)`. It is customary to omit the tensor product from the contraction expression, just as we did in the last part of this equation. The diagrammatic representation of this equation is obtained by connecting all the legs with the same indices. # # .. figure:: ../_static/demonstration_assets/tn_basics/06-tensor-tensor.png # :align: center @@ -117,7 +117,7 @@ # # With the above contraction, we have formed a network of tensors, i.e., a **Tensor Network**! # -# .. tip:: +# .. note:: # A common question arising when drawing a tensor is "What is the correct order to draw the indices?". For instance, in the figure above, we have adopted the convention that a tensor :math:`A_{i,j,k}` corresponds to a diagram with the first leg (:math:`i`) pointing left, the second leg (:math:`j`) pointing upwards, and the third leg (:math:`k`) pointing right, and similarly for the other two tensors. However, this need not be the case. We could have defined the first leg to be the one pointing upwards, for example. Based on the use case, and the user, some conventions might seem more natural than others. The only important thing to keep in mind is to be consistent. In other words, once we choose a convention for the order, we should apply it to all the tensors to avoid contracting the wrong indices ❌. # # Remember the similarity between tensor network contractions and Einstein notation? Then, it doesn't come as a surprise that we can perform a contraction using the function ``np.einsum``. To do so, we can start by creating the 3 tensors to be contracted by reshaping a 1D array (created using ``np.arange``) into rank-3 tensors of the correct dimensions. @@ -158,12 +158,12 @@ # X # \end{pmatrix} # -# This means :math:`(T^1)_{i,j,k}` and :math:`(T^2)_{l,j,m}` are two rank-3 tensors, where the index :math:`j` *"picks"* the elements in the column vector while the first and last indices correspond to the indices of the internal tensors (matrices). For instance, the :math:`j = 0`-th element of :math:`T^1` and :math:`T^2` are :math:`|0\rangle \langle 0 |` and :math:`I`, respectively, and similarly for their :math:`j = 1`-st element. This means we can redefine the CNOT expression from above as +# This means :math:`(T^1)_{i,j,k}` and :math:`(T^2)_{l,j,m}` are two rank-3 tensors, where the index :math:`j` *"picks"* the elements in the column vector while the the other two indices correspond to the indices of the internal tensors (matrices). For instance, the :math:`j = 0`-th element of :math:`T^1` and :math:`T^2` are :math:`|0\rangle \langle 0 |` and :math:`I`, respectively, and similarly for their :math:`j = 1`-st element. This means we can redefine the CNOT expression from above as # # .. math:: -# CNOT = \sum_j (T^1)_{i,j,k} \otimes (T^2)_{l,j,m}. +# \mathrm{CNOT}_{i,l,k,m} = \sum_j (T^1)_{i,j,k} \otimes (T^2)_{l,j,m} = \sum_j T^1{i, j, k} T^2_{l, j, m}. # -# It turns out : that math:`T^1` and :math:`T^2` are special tensors known as the COPY and XOR tensors, respectively, and therefore have special diagrammatic representations. +# It turns out that :math:`T^1` and :math:`T^2` are special tensors known as the COPY and XOR tensors, respectively, and therefore have special diagrammatic representations. # # .. figure:: ../_static/demonstration_assets/tn_basics/07-t1-t2.png # :align: center @@ -206,13 +206,13 @@ # .. math:: # \mathcal{O}(d_i \times d_j \times d_k) # -# To illustrate the importance of choosing an efficient contraction path, let us look at a similar contraction between 3 rank-3 tensors, as shown in the previous section. +# To illustrate the importance of choosing an efficient contraction path, let us look at a similar contraction between 3 rank-3 tensors: :math:`A_{i,j,k} \in \mathbb{C}^{d_i \times d_j \times d_k}`, :math:`B_{j,l,m} \in \mathbb{C}^{d_j \times d_l \times d_m}`, and :math:`C_{k,m,n} \in \mathbb{C}^{d_k \times d_m \times d_n}`. # # .. figure:: ../_static/demonstration_assets/tn_basics/09-contraction.png # :align: center # :width: 50% # -# In this case, the tensors are such that :math:`A_{i,j,k} \in \mathcal{C}^{d_i \times d_j \times d_j}`, :math:`B_{j,l,m} \in \mathcal{C}^{d_j \times 1 \times d_m}`, and :math:`C_{k,m,n} \in \mathcal{C}^{d_j \times d_m \times d_i}`. First, we look at the complexity of contracting :math:`(AB)` and then :math:`C`. Following the procedure explained above, the first contraction results in a complexity of +# In particular, let us look at an example where :math:`d_k = d_j`, :math:`d_l = 1`, and :math:`d_n = d_i`. First, we look at the complexity of contracting :math:`(AB)` and then :math:`C`. Following the procedure explained above, the first contraction results in a complexity of # # .. math:: # \sum_{j} A_{i,j,k} B_{j,l,m} \implies \mathcal{O}(d_i \times d_m \times d_j^2 ) @@ -233,7 +233,8 @@ # \sum_{j, k} A_{i,j,k} (BC)_{j,l,k,n} \implies \mathcal{O}(d_i^2 \times d_j^2) . # # This means the second contraction path results in an asymptotic cost of :math:`\mathcal{O}(d_i^2 \times d_j^2)`—lower than the first contraction path. -# To see this in practice, let us perform the above contraction using ``np.einsum``. First, we create 3 tensors with the correct dimensions. +# +# To see this in practice, let us implement the above contractions using ``np.einsum``. First, we create the 3 tensors with dimensions as specified in the example above. import timeit @@ -241,9 +242,11 @@ dm = 100 dj = 10 -A = np.arange(di*dj*dj).reshape(di,dj,dj) # ijk -B = np.arange(dj*1*dm).reshape(dj,1,dm) # jlm -C = np.arange(dj*dm*di).reshape(dj,dm,di) # kmn +np.random.seed(20) + +A = np.random.rand(di, dj, dj) # ijk +B = np.random.rand(dj,1,dm) # jlm +C = np.random.rand(dj,dm,di) # kmn ############################################################################## # Then, we perform the individual contractions between pairs of tensors and time them using ``timeit``. We repeat the contraction ``20`` times and average the computation time to account for smaller fluctuations. First, we start by contracting :math:`A` and :math:`B`. @@ -359,7 +362,7 @@ # # In the right-hand side of the equality we have assumed a specific form for the U tensor in terms of local 2-qubit gates, which is often the case when dealing with real quantum hardware. In addition, it is common for the initial state to be a product state such as :math:`|0\rangle^{\otimes N}`, hence the form of the tensor in the diagram. # -# Now we can ask the important question: what quantities can we compute from this tensor network? +# Now we can ask the important question: what quantities can we compute from this tensor network? 🤔 # # Expectation values # ~~~~~~~~~~~~~~~~~~ @@ -468,6 +471,11 @@ # "Quantum computation and the evaluation of tensor networks", # ``__, 2010. # +# .. [#Bridgeman2017] +# J. C. Bridgeman and C. T. Chubb. +# "Hand-waving and interpretive dance: an introductory course on tensor networks," +# ``__, Journal of Physics A: Mathematical and Theoretical, vol. 50, no. 22, 2017. +# # .. [#Lam1997] # C.-C. Lam, P. Sadayappan, and R. Wenger. # "On Optimizing a Class of Multi-Dimensional Loops with Reduction for Parallel Execution", From a584f2671bc5fe762756f3535d1e69916719483a Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Thu, 14 Nov 2024 20:25:10 +0100 Subject: [PATCH 45/51] update username in metadata --- demonstrations/tutorial_tn_basics.metadata.json | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.metadata.json b/demonstrations/tutorial_tn_basics.metadata.json index cf773425c2..50c747976f 100644 --- a/demonstrations/tutorial_tn_basics.metadata.json +++ b/demonstrations/tutorial_tn_basics.metadata.json @@ -2,10 +2,10 @@ "title": "Tutorial: ABC of tensor networks", "authors": [ { - "id": "emiliano_godinez" + "username": "emiliano" }, { - "id": "ahmed_darwish" + "username": "ShiroRaven" } ], "dateOfPublication": "2024-08-06T00:00:00+00:00", From ec8ce1416e713638a993ad97ae302e050dfb930a Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Wed, 20 Nov 2024 12:20:05 +0100 Subject: [PATCH 46/51] add intro, transition, and finishing sentences to increase coherence --- demonstrations/tutorial_tn_basics.py | 54 +++++++++++++++++++--------- 1 file changed, 37 insertions(+), 17 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index f349394c44..5a9dc4d1ea 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -1,24 +1,27 @@ r"""Tutorial: ABC of tensor networks for quantum circuits ============================================================= -If you are in the domain of quantum computing or quantum information, chances are you have heard (maybe a lot) about tensor networks being brought into the conversation. They are not only useful in these areas, but actually, they are a rather widespread "tool" with applications ranging across physics, math, and computer science. +If you are in the domain of quantum computing or quantum information, chances are you have heard (maybe a lot) about tensor networks being brought into the conversation. Actually, they are a rather widespread "tool" with applications ranging across physics, math, and computer science. Part of the excitement surrounding tensor networks is due to their ability to represent complex data efficiently, which allows for — among other things — fast classical simulations. In addition, the diagrammatic language accompanying tensor networks makes working with them intuitive and suitable for describing a vast range of mathematical concepts, including quantum circuits. -In this tutorial, we aim to provide an introduction to tensor networks with a focus on their applications in quantum computing. We choose to start by discussing the basic notions and definitions of tensors and tensor networks and work our way up to more advanced topics such as contraction paths and algorithms used to simulate quantum computers using tensor networks. +The first sections aim to introduce the basic notions and definitions of tensor networks to those quantum practitioners familiar with quantum computing but who are new to the tensor networks world 🌎. -Without further ado, let’s dive right in! 🤓📚 +Then, building on this introduction, in the following sections we explore topics akin to more seasoned 🧂 readers who want to learn about concepts and algorithms used to connect tensor networks and quantum computing - like those used to simulate the output of a quantum circuit. +Without further ado, let’s dive right in! 🤓📚 From matrices to tensors ------------------------------- +First, we start by answering the question: what is a tensor❓ + A common and intuitive way of thinking about tensors is as generalizations of vectors and matrices. That is, we can think of them as multidimensional arrays—i.e., multidimensional maps that are linear with respect to every parameter. A tensor of dimensions :math:`d_1 \times d_2 \times \ldots \times d_r` can be expressed as .. math:: T_{i_1, i_2, \ldots, i_r} \in \mathbb{C}^{d_1 \times d_2 \times \ldots \times d_r}, -where each :math:`i_n` is an *index* of dimension :math:`d_n` and the number of indices :math:`r` is known as the *rank* of the tensor. We say :math:`T` is a rank-:math:`r` tensor. +where each :math:`i_n` is an **index** of dimension :math:`d_n` and the number of indices :math:`r` is known as the **rank** of the tensor. We say :math:`T` is a rank-:math:`r` tensor. For example, a scalar :math:`s` is a rank-0 tensor, a vector :math:`v_i` is a rank-1 tensor, and a matrix :math:`G_{i,j}` is a rank-2 tensor. @@ -75,6 +78,7 @@ ############################################################################## # We can create a tensor of arbitrary rank following a similar procedure. This recursive approach illustrates how a rank-:math:`r` tensor can be seen as consisting of nested rank-:math:`(r-1)` tensors. This translates into code as adding another level to the nested bracket structure: ``[tensor_rank_r-1]``. # +# Now that we know what a tensor is (and can even code it up 💻), let us look at how to combine them to create a tensor network. ############################################################################## # From matrix multiplication to tensor contractions @@ -115,7 +119,7 @@ # :align: center # :width: 55% # -# With the above contraction, we have formed a network of tensors, i.e., a **Tensor Network**! +# With the above contraction, we have formed a network of tensors, i.e., a **tensor network**! # # .. note:: # A common question arising when drawing a tensor is "What is the correct order to draw the indices?". For instance, in the figure above, we have adopted the convention that a tensor :math:`A_{i,j,k}` corresponds to a diagram with the first leg (:math:`i`) pointing left, the second leg (:math:`j`) pointing upwards, and the third leg (:math:`k`) pointing right, and similarly for the other two tensors. However, this need not be the case. We could have defined the first leg to be the one pointing upwards, for example. Based on the use case, and the user, some conventions might seem more natural than others. The only important thing to keep in mind is to be consistent. In other words, once we choose a convention for the order, we should apply it to all the tensors to avoid contracting the wrong indices ❌. @@ -158,10 +162,10 @@ # X # \end{pmatrix} # -# This means :math:`(T^1)_{i,j,k}` and :math:`(T^2)_{l,j,m}` are two rank-3 tensors, where the index :math:`j` *"picks"* the elements in the column vector while the the other two indices correspond to the indices of the internal tensors (matrices). For instance, the :math:`j = 0`-th element of :math:`T^1` and :math:`T^2` are :math:`|0\rangle \langle 0 |` and :math:`I`, respectively, and similarly for their :math:`j = 1`-st element. This means we can redefine the CNOT expression from above as +# This means :math:`(T^1)_{i,j,k}` and :math:`(T^2)_{l,j,m}` are two rank-3 tensors, where the index :math:`j` *picks* the elements in the column vector while the the other two indices correspond to the indices of the internal tensors (matrices). For instance, the :math:`j = 0`-th element of :math:`T^1` and :math:`T^2` are :math:`|0\rangle \langle 0 |` and :math:`I`, respectively, and similarly for their :math:`j = 1`-st element. This means we can redefine the CNOT expression from above as # # .. math:: -# \mathrm{CNOT}_{i,l,k,m} = \sum_j (T^1)_{i,j,k} \otimes (T^2)_{l,j,m} = \sum_j T^1{i, j, k} T^2_{l, j, m}. +# \mathrm{CNOT}_{i,l,k,m} = \sum_j (T^1)_{i,j,k} \otimes (T^2)_{l,j,m} = \sum_j (T^1)_{i,j,k} (T^2)_{l,j,m}. # # It turns out that :math:`T^1` and :math:`T^2` are special tensors known as the COPY and XOR tensors, respectively, and therefore have special diagrammatic representations. # @@ -189,14 +193,15 @@ # The cost of contracting a network # --------------------------------- # -# A common task when dealing with tensors is the contraction of large networks resulting in a single tensor (including scalars). To arrive at this final tensor, we can start with a single tensor and contract it with adjacent tensors one at a time. The order in which this is carried out is known as the *contraction path* or *bubbling*. +# A common task when dealing with tensors is the contraction of large networks resulting in a single tensor (including scalars). To arrive at this final tensor, we can start with a single tensor and contract it with adjacent tensors one at a time. The order in which this is carried out is known as the **contraction path** or **bubbling**. +# # While the final tensor is independent of the order of the contraction, the number of operations performed can vary greatly depending on the order in which we contract the intermediate tensors. Moreover, in a general setup, finding the optimal order of indices to be contracted is not a trivial task. # # .. note:: # # Actually, finding the optimal contraction path of a tensor network is an NP-complete problem [#Arad]_. # -# For this reason, it is useful to understand how to calculate the computational cost or the *complexity* of a tensor network contraction. First, we look at a simple matrix-matrix contraction. Given rank-2 tensors :math:`G^1_{i,j}` and :math:`G^2_{j,k}`, we have seen that the :math:`(i,k)`-th element of the resulting contraction along the :math:`j`-th index is +# For this reason, in this section we will look at how to calculate the computational cost or the **complexity** of a tensor network contraction. First, we look at a simple matrix-matrix contraction. Given rank-2 tensors :math:`G^1_{i,j}` and :math:`G^2_{j,k}`, we have seen that the :math:`(i,k)`-th element of the resulting contraction along the :math:`j`-th index is # # .. math:: # (G^3)_{i,k} = G^2 \cdot G^1 = \sum_{j=1}^{d_j} G^{2}_{j,k} G^{1}_{i,j} @@ -290,7 +295,17 @@ print(f"Computation cost for A(BC) contraction: {average_time_ms:.8f} ms") ############################################################################## -# From this, we see that the second contraction path results in a much lower complexity compared to the first one, just as we expected! +# From this, we see that the second contraction path results in a much lower complexity compared to the first one, just as we expected! 💪 + +############################################################################## +# Intermezzo +# ---------- +# +# So far, we have discussed the definition of a tensor, how to combine them to create a tensor network, and how to calculate the complexity of the contraction operations. Hopefully, after this brief introduction, you will feel more comfortable whenever tensor networks are brought into the conversation. +# +# Perhaps you even feel motivated to dive deeper into the vast world of tensor networks! 🧐 +# +# To help you with this endeavour, in the following sections we will summarize some ubiquitous algorithms used to connect tensor networks and quantum computers. These topics can become quite technical real fast, and thus, we can only scratch the surface in this demo. For this reason, we will reference the relevant sources when pertinent. Now, take a sip of coffee, brace yourself, and let's continue! ☕️ ############################################################################## # Contraction paths @@ -306,7 +321,7 @@ # :align: center # :width: 50% # -# Transforming a tensor network with an arbitrary structure into this binary tree can be achieved by a *tree embedding* of the tensor network graph [#Bienstock1990]_. Thus, optimization of the contraction path is equivalent to a search over tree embeddings of the network. +# Transforming a tensor network with an arbitrary structure into this binary tree can be achieved by a **tree embedding** of the tensor network graph [#Bienstock1990]_. Thus, optimization of the contraction path is equivalent to a search over tree embeddings of the network. # # .. note:: # @@ -326,7 +341,7 @@ # # Further approaches introduced in [#Gray2021]_ are based on alternative common graph-theoretic tasks, rather than searching over the contraction tree space, such as the `balanced bi-partitioning `_ and `community detection `_ algorithms. And even though these are only heuristics that do not guarantee an optimal contraction path, they can often achieve an arbitrarily close to optimal performance. # -# An extra level of optimization, known as *hyper-optimization* is introduced by the use of different algorithms to find the optimal contraction based on the specific tensor network, as some algorithms are better suited for certain network structures. For an in-depth exploration of these heuristics, please refer to [#Gray2021]_. +# An extra level of optimization, known as **hyper-optimization** is introduced by the use of different algorithms to find the optimal contraction based on the specific tensor network, as some algorithms are better suited for certain network structures. For an in-depth exploration of these heuristics, please refer to [#Gray2021]_. # # As we will explore in the next section, we can use tensor networks to simulate quantum circuits. In particular, the calculation of an expectation value corresponds to the contraction of the tensor network into a single tensor (scalar) as discussed in this section. In ``Pennylane``, this simulation can be performed using the :class:`~pennylane.devices.default_tensor.DefaultTensor` device, and the method used to find the contraction path can be chosen via the ``contraction_optimizer`` keyword argument. @@ -337,13 +352,18 @@ ############################################################################## # The different types of values accepted for ``contraction_optimizer`` are determined by the ``optimize`` parameter in ``Quimb`` (see `docs `_) as this is the backend behind the :class:`~pennylane.devices.default_tensor.DefaultTensor` device. See `this tutorial `_ to learn more the use of this device in ``Pennylane``. # +# Slicing +# ~~~~~~~ +# # The size of (intermediate) tensors can grow exponentially with the number of indices and dimensions, especially for large-scale tensor networks. Thus, we might run into memory problems when performing the contractions. A useful additional technique to split these tensors into more manageable pieces is known as **slicing**. # # The idea is to change space for computation time, by temporarily fixing the values of some indices in the tensors, performing independently the contraction for each fixed value, and summing the results [#Gray2021]_. +# +# To end this demo, let us answer the question: how can we use tensor networks to simulate the output of a quantum circuit ⁉️ ############################################################################## # Quantum circuits are tensor networks -# --------------------------------- +# ------------------------------------ # # Until now, we have looked at general tensor networks, while ✨sparkling✨ the discussions with examples related to quantum circuits. Here, we leverage the components we have learned to explore this relation more in-depth. # @@ -362,7 +382,7 @@ # # In the right-hand side of the equality we have assumed a specific form for the U tensor in terms of local 2-qubit gates, which is often the case when dealing with real quantum hardware. In addition, it is common for the initial state to be a product state such as :math:`|0\rangle^{\otimes N}`, hence the form of the tensor in the diagram. # -# Now we can ask the important question: what quantities can we compute from this tensor network? 🤔 +# Now we can ask ourselves: what quantities can we compute from this tensor network? 🤔 # # Expectation values # ~~~~~~~~~~~~~~~~~~ @@ -384,7 +404,7 @@ # # However, it is possible to perform this operation more efficiently using tensor networks by means of a structure called Matrix Product Operator (MPO) [#Pirvu2010]_. The idea is to construct an efficient representation of the observable :math:`O` which can be contracted with the tensor network from :math:`|\psi \rangle`. Constructing these networks efficiently for Hamiltonians of arbitrary structure is an interesting task, which goes beyond the scope of this tutorial. # -# When the observable of interest is *local*, i.e., it acts on a few neighbouring qubits, we can calculate the expectation value by considering only the section of the quantum circuit within the *reverse light cone* (causal cone) of the observable :math:`O_l`. +# When the observable of interest is *local*, i.e., it acts on a few neighbouring qubits, we can calculate the expectation value by considering only the section of the quantum circuit within the **reverse light cone** (causal cone) of the observable :math:`O_l`. # # .. figure:: ../_static/demonstration_assets/tn_basics/12-expectation-local.png # :align: center @@ -430,7 +450,7 @@ # # Once we obtain the probability vector, we can generate a random sample weighted by these probabilities. To do so, we generate a random number :math:`r \in [0,1]` and choose :math:`x_1 = 0` if :math:`r < p(x_1=0)` and :math:`x_1 = 1` otherwise. We save this sample as :math:`\hat{x}_1`. # -# Next, we can calculate the following term :math:`p(x_2|\hat{x}_1)` conditioned on the sample we have just obtained. To accomplish this, we **project** the first qubit to be :math:`\hat{x}_1`. We can do this by contracting the computational basis state :math:`| \hat{x}_1 \rangle` with :math:`|\psi \rangle`, resulting in a smaller state :math:`|\psi_{x_1} \rangle`. Then, we can proceed exactly as we did in the previous step, calculating the reduced density matrix :math:`\rho_2` by tracing out the remaining qubits :math:`3,4,\ldots, N` and computing the probability vector from its diagonal +# Next, we can calculate the following term :math:`p(x_2|\hat{x}_1)` conditioned on the sample we have just obtained. To accomplish this, we *project* the first qubit to be :math:`\hat{x}_1`. We can do this by contracting the computational basis state :math:`| \hat{x}_1 \rangle` with :math:`|\psi \rangle`, resulting in a smaller state :math:`|\psi_{x_1} \rangle`. Then, we can proceed exactly as we did in the previous step, calculating the reduced density matrix :math:`\rho_2` by tracing out the remaining qubits :math:`3,4,\ldots, N` and computing the probability vector from its diagonal # # .. math:: # | p_{x_2 | \hat{x}_1} \rangle = \mathrm{diag} \left( \mathrm{Tr}_{3,4, \ldots,N}(| \psi_{x_1} \rangle \langle \psi_{x_1} |) \right). @@ -441,7 +461,7 @@ # :align: center # :width: 70% # -# Analogously as done with the expectation values, these contractions only involve the sections of the circuit within the light cone of **both** the projection with :math:`| \hat{x}_1 \rangle` and the contraction with the COPY tensor (diagonal computation). This procedure can be repeated recursively using the chain rule equation until we obtain the full bitstring :math:`(\hat{x}_1, \hat{x}_2, \hat{x}_3, \ldots, \hat{x}_N)`. To obtain more samples, we repeat the procedure from the beginning - this is what makes every sample memoryless or a perfect sample from the probability distribution. +# Analogously as done with the expectation values, these contractions only involve the sections of the circuit within the light cone of *both* the projection with :math:`| \hat{x}_1 \rangle` and the contraction with the COPY tensor (diagonal computation). This procedure can be repeated recursively using the chain rule equation until we obtain the full bitstring :math:`(\hat{x}_1, \hat{x}_2, \hat{x}_3, \ldots, \hat{x}_N)`. To obtain more samples, we repeat the procedure from the beginning - this is what makes every sample memoryless or a perfect sample from the probability distribution. # # .. note:: # We can reduce the computational cost of this algorithm by **caching** results from previous contractions. When we draw a new sample that partially matches a previously explored configuration (marginal probability), we can reuse the cached results and avoid contracting this part of the network over again. For example, let's assume we have performed the perfect sampling algorithm once and obtained the sample :math:`0110`. If the next sample we need to generate starts with the substring :math:`01`, we can reuse the marginal probabilities up to :math:`p(x_3|01)` and only calculate the new parts of the sequence. The same caching idea can be applied to other tensor network algorithms involving many contractions. From a13a8975933a405b7f2060965ced89d0aca17361 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Sun, 24 Nov 2024 18:12:46 +0100 Subject: [PATCH 47/51] address comments from Korbinian --- demonstrations/tutorial_tn_basics.py | 22 ++++++++++++++++------ 1 file changed, 16 insertions(+), 6 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 5a9dc4d1ea..76a75e7922 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -380,7 +380,7 @@ # :align: center # :width: 45% # -# In the right-hand side of the equality we have assumed a specific form for the U tensor in terms of local 2-qubit gates, which is often the case when dealing with real quantum hardware. In addition, it is common for the initial state to be a product state such as :math:`|0\rangle^{\otimes N}`, hence the form of the tensor in the diagram. +# In the right-hand side of the equality we have assumed a specific form for the U tensor in terms of local 2-qubit gates, which is often the case when dealing with real quantum hardware. In addition, it is common for the initial state to be a product state such as :math:`|0\rangle^{\otimes N}`, hence the form of the tensor in the diagram as :math:`N` independent tensors of rank-1. However, an arbitrary input state is in general represented as one big rank-:math:`N` tensor. # # Now we can ask ourselves: what quantities can we compute from this tensor network? 🤔 # @@ -461,21 +461,25 @@ # :align: center # :width: 70% # -# Analogously as done with the expectation values, these contractions only involve the sections of the circuit within the light cone of *both* the projection with :math:`| \hat{x}_1 \rangle` and the contraction with the COPY tensor (diagonal computation). This procedure can be repeated recursively using the chain rule equation until we obtain the full bitstring :math:`(\hat{x}_1, \hat{x}_2, \hat{x}_3, \ldots, \hat{x}_N)`. To obtain more samples, we repeat the procedure from the beginning - this is what makes every sample memoryless or a perfect sample from the probability distribution. +# Analogously as done with the expectation values, these contractions only involve the sections of the circuit within the light cone of *both* the projection with :math:`| \hat{x}_1 \rangle` and the contraction with the COPY tensor (diagonal computation). This procedure can be repeated recursively using the chain rule equation until we obtain the full bitstring :math:`(\hat{x}_1, \hat{x}_2, \hat{x}_3, \ldots, \hat{x}_N)`. Then to obtain more samples, we just repeat the procedure from the beginning! # # .. note:: -# We can reduce the computational cost of this algorithm by **caching** results from previous contractions. When we draw a new sample that partially matches a previously explored configuration (marginal probability), we can reuse the cached results and avoid contracting this part of the network over again. For example, let's assume we have performed the perfect sampling algorithm once and obtained the sample :math:`0110`. If the next sample we need to generate starts with the substring :math:`01`, we can reuse the marginal probabilities up to :math:`p(x_3|01)` and only calculate the new parts of the sequence. The same caching idea can be applied to other tensor network algorithms involving many contractions. +# By generating each bitstring independently from each other, i.e., by restarting the sampling algorithm without knowledge of the previous samples, we ensure perfect sampling from the probability distribution, contrary to other Markov-based algorithms [#Schuch2008]_ . We then say the sample is *memoryless*. +# +# We can reduce the computational cost of the sampling algorithm by **caching** results from previous contractions. When we draw a new sample that partially matches a previously explored configuration (marginal probability), we can reuse the cached results and avoid contracting this part of the network over again. +# +# For example, let's assume we have performed the perfect sampling algorithm once and obtained the sample :math:`0110`. If the next sample we need to generate starts with the substring :math:`01`, we can reuse the marginal probabilities up to :math:`p(x_3|01)` and only calculate the new parts of the sequence. The same caching idea can be applied to other tensor network algorithms involving many contractions. ############################################################################## # # Conclusion # ---------- # -# Although the world of tensor networks and their relation to quantum computing is vastly wider than what we could ever cover in one tutorial, we hope that after these explanations you now feel equipped with the tools needed to dive deeper into this topic by yourself. 🔍 +# And that is it for this tutorial! 🎉 # -# If you want to learn more about using tensor networks as a diagrammatic tool, check out `these amazing lecture notes `_ by J.Biamonte. In addition, check out `this website `_ for great explanations on many important algorithms and tensor network structures by Flatiron Institute. +# Although the world of tensor networks and their relation to quantum computing is vastly wider than what we could ever cover in one tutorial, we hope that after these explanations you now feel equipped with the tools needed to dive deeper into this topic by yourself. 🔍 # -# And that is it for this tutorial! 🎉 +# If you want to learn more about using tensor networks as a diagrammatic tool, check out `these amazing lecture notes `_ by J.Biamonte. In addition, check out `this website `_ for great explanations on many important algorithms and tensor network structures by Flatiron Institute . ############################################################################## # References @@ -525,6 +529,12 @@ # A. J. Ferris and G. Vidal. # "Perfect sampling with unitary tensor networks," # ``__, Physical Review B, vol. 85, no. 16, 2012. +# +# .. [#Schuch2008] +# N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac. +# "Simulation of Quantum Many-Body Systems with Strings of Operators and Monte Carlo Tensor Contractions," +# ``__, Physical Review Letters, vol. 100, no. 4, Jan 2008. + ############################################################################## # About the author From 8609239d6627121b6c36968d789d46bd8dd5f14f Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Sun, 24 Nov 2024 18:28:22 +0100 Subject: [PATCH 48/51] address einstein-related comment --- demonstrations/tutorial_tn_basics.py | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 76a75e7922..0c73409545 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -108,7 +108,7 @@ # # Here, the resulting tensor has two dangling indices, :math:`i` and :math:`k`, defining a matrix, as expected! # -# We can now generalize this concept to tensors, and consequently, to more than two legs being contracted. For example, let us look at three tensors :math:`A_{i,j,k}`, :math:`B_{j,l,m}`, and :math:`C_{k,m,n}`. To contract them, all we need to do is to sum over repeated indices (:math:`j`, :math:`k`, :math:`m`), just as we would do in `Einstein convention `_. To obtain the :math:`(i,l,n)`-th element of the resulting tensor :math:`D`, we perform this contraction by using the tensor product :math:`\otimes` between the 3 tensors and summing over :math:`j, k, m` [#Bridgeman2017]_ +# We can now generalize this concept to tensors, and consequently, to more than two legs being contracted. For example, let us look at three tensors :math:`A_{i,j,k}`, :math:`B_{j,l,m}`, and :math:`C_{k,m,n}`. To contract them, all we need to do is to sum over repeated indices (:math:`j`, :math:`k`, :math:`m`). To obtain the :math:`(i,l,n)`-th element of the resulting tensor :math:`D`, we perform this contraction by using the tensor product :math:`\otimes` between the 3 tensors and summing over :math:`j, k, m` [#Bridgeman2017]_ # # .. math:: # (D)_{i,l,n} = \sum_{j,k,m} A_{i,j,k} \otimes B_{j,l,m} \otimes C_{k,m,n} = \sum_{j,k,m} A_{i,j,k} B_{j,l,m} C_{k,m,n} . @@ -124,7 +124,7 @@ # .. note:: # A common question arising when drawing a tensor is "What is the correct order to draw the indices?". For instance, in the figure above, we have adopted the convention that a tensor :math:`A_{i,j,k}` corresponds to a diagram with the first leg (:math:`i`) pointing left, the second leg (:math:`j`) pointing upwards, and the third leg (:math:`k`) pointing right, and similarly for the other two tensors. However, this need not be the case. We could have defined the first leg to be the one pointing upwards, for example. Based on the use case, and the user, some conventions might seem more natural than others. The only important thing to keep in mind is to be consistent. In other words, once we choose a convention for the order, we should apply it to all the tensors to avoid contracting the wrong indices ❌. # -# Remember the similarity between tensor network contractions and Einstein notation? Then, it doesn't come as a surprise that we can perform a contraction using the function ``np.einsum``. To do so, we can start by creating the 3 tensors to be contracted by reshaping a 1D array (created using ``np.arange``) into rank-3 tensors of the correct dimensions. +# In our code, we can perform a tensor contraction using the ``Numpy`` function ``np.einsum``. To do so, we can start by creating the 3 tensors to be contracted by reshaping a 1D array (created using ``np.arange``) into rank-3 tensors of the correct dimensions. # Create the individual rank-3 tensors A = np.arange(6).reshape(1, 2, 3) # ijk @@ -239,7 +239,7 @@ # # This means the second contraction path results in an asymptotic cost of :math:`\mathcal{O}(d_i^2 \times d_j^2)`—lower than the first contraction path. # -# To see this in practice, let us implement the above contractions using ``np.einsum``. First, we create the 3 tensors with dimensions as specified in the example above. +# To see this in practice, let us implement the above contractions using ``np.einsum``. First, we create the 3 tensors with the dimensions specified in the example above. We demonstrate a different form of creating tensors of the desired dimensions using the ``random`` module. import timeit From a1f5ea622ee068aef146db10654cd707c7d65a21 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Sun, 1 Dec 2024 22:17:13 +0100 Subject: [PATCH 49/51] add additional title level correct explanation on the definition of contraction --- demonstrations/tutorial_tn_basics.py | 63 ++++++++++++++++++---------- 1 file changed, 41 insertions(+), 22 deletions(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 0c73409545..960cb0b80f 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -1,18 +1,23 @@ r"""Tutorial: ABC of tensor networks for quantum circuits ============================================================= -If you are in the domain of quantum computing or quantum information, chances are you have heard (maybe a lot) about tensor networks being brought into the conversation. Actually, they are a rather widespread "tool" with applications ranging across physics, math, and computer science. +If you are in the domain of quantum computing or quantum information, chances are you have heard (maybe a lot) about tensor networks being brought into the conversation. In fact, tensor networks are a rather widespread "tool" with applications ranging across physics, math, and computer science. Part of the excitement surrounding tensor networks is due to their ability to represent complex data efficiently, which allows for — among other things — fast classical simulations. In addition, the diagrammatic language accompanying tensor networks makes working with them intuitive and suitable for describing a vast range of mathematical concepts, including quantum circuits. -The first sections aim to introduce the basic notions and definitions of tensor networks to those quantum practitioners familiar with quantum computing but who are new to the tensor networks world 🌎. +The :ref:`first part of this tutorial ` aims to introduce the basic notions and definitions of tensor networks to those quantum practitioners familiar with quantum computing but who are new to the tensor networks world 🌎. -Then, building on this introduction, in the following sections we explore topics akin to more seasoned 🧂 readers who want to learn about concepts and algorithms used to connect tensor networks and quantum computing - like those used to simulate the output of a quantum circuit. +Then, building on this introduction, in the :ref:`second part ` we explore topics akin to more seasoned 🧂 readers who want to learn about concepts and algorithms used to connect tensor networks and quantum computing - like those used to simulate the output of a quantum circuit. Without further ado, let’s dive right in! 🤓📚 +.. _part_one: + +Part 1: A first glimpse into the tensor networks world 🔍👀 +---------------------------------------------------------- + From matrices to tensors -------------------------------- +~~~~~~~~~~~~~~~~~~~~~~~~ First, we start by answering the question: what is a tensor❓ @@ -82,7 +87,7 @@ ############################################################################## # From matrix multiplication to tensor contractions -# ------------------------------------------------- +# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # # Matrix-matrix and matrix-vector multiplications are familiar operations within the context of quantum computing. We can now study these operations through the lens of the tensor notation introduced above. First, a matrix and a vector can be multiplied as # @@ -100,7 +105,7 @@ # We see that summing over the shared index :math:`i` is equivalent to **contracting** the corresponding legs from the matrix and vector diagrams. As expected, the result of this multiplication is another rank-1 tensor with dangling leg :math:`j`. Similarly, we can look at the matrix-matrix multiplication: # # .. math:: -# (G^3)_{i,k} = G^2 \cdot G^1 = \sum_j G^{2}_{j,k} G^{1}_{i,j} +# (G^3)_{i,k} = G^2 \cdot G^1 = \sum_j G^{1}_{i,j} G^{2}_{j,k} # # .. figure:: ../_static/demonstration_assets/tn_basics/05-matrix-matrix.png # :align: center @@ -108,12 +113,17 @@ # # Here, the resulting tensor has two dangling indices, :math:`i` and :math:`k`, defining a matrix, as expected! # -# We can now generalize this concept to tensors, and consequently, to more than two legs being contracted. For example, let us look at three tensors :math:`A_{i,j,k}`, :math:`B_{j,l,m}`, and :math:`C_{k,m,n}`. To contract them, all we need to do is to sum over repeated indices (:math:`j`, :math:`k`, :math:`m`). To obtain the :math:`(i,l,n)`-th element of the resulting tensor :math:`D`, we perform this contraction by using the tensor product :math:`\otimes` between the 3 tensors and summing over :math:`j, k, m` [#Bridgeman2017]_ +# We can now generalize this concept to tensors, and consequently, to more than two legs being contracted. For example, let us look at three tensors :math:`A_{i,j,k}`, :math:`B_{j,l,m}`, and :math:`C_{k,m,n}`. To contract them, all we need to do is to sum over repeated indices (:math:`j`, :math:`k`, :math:`m`). To obtain the :math:`(i,l,n)`-th element of the resulting tensor :math:`D`, we perform this contraction by summing over :math:`j, k, m` +# +# .. math:: +# (D)_{i,l,n} = \sum_{j,k,m} A_{i,j,k} B_{j,l,m} C_{k,m,n} . +# +# Using the tensor product :math:`\otimes` between the 3 tensors and summing over the repeated indices we can obtain a similar expression for the full tensor :math:`D` [#Bridgeman2017]_ # # .. math:: -# (D)_{i,l,n} = \sum_{j,k,m} A_{i,j,k} \otimes B_{j,l,m} \otimes C_{k,m,n} = \sum_{j,k,m} A_{i,j,k} B_{j,l,m} C_{k,m,n} . +# D_{i,l,n} = \sum_{j,k,m} A_{i,j,k} \otimes B_{j,l,m} \otimes C_{k,m,n}. # -# Notice how the resulting rank-3 tensor consists of the remaining open legs from the initial tensors :math:`(i,l,n)`. It is customary to omit the tensor product from the contraction expression, just as we did in the last part of this equation. The diagrammatic representation of this equation is obtained by connecting all the legs with the same indices. +# The resulting rank-3 tensor consists of the remaining open legs from the initial tensors :math:`(i,l,n)`. The first equation involving only scalars is the more widely used expression for the contraction operation, and thus, we will use it throughout this tutorial unless otherwise stated. The diagrammatic representation of this contraction is obtained by connecting all the legs with the same indices. # # .. figure:: ../_static/demonstration_assets/tn_basics/06-tensor-tensor.png # :align: center @@ -139,12 +149,12 @@ ############################################################################## # The CNOT gate -# ~~~~~~~~~~~~~ +# ^^^^^^^^^^^^^ # # To end this section, we want to discuss a common example of a tensor network contraction arising in quantum computing, namely the **CNOT** gate. The CNOT gate can be expressed in the computational basis as # # .. math:: -# CNOT = |0\rangle \langle 0 | \otimes I + |1\rangle \langle 1 | \otimes X. +# \mathrm{CNOT} = |0\rangle \langle 0 | \otimes I + |1\rangle \langle 1 | \otimes X. # # That is, if the control qubit is in the :math:`|1\rangle` state, we apply the :math:`X` gate on the target qubit; otherwise, we leave it untouched. However, we can also rewrite this equation as a contraction. To do so, we define two tensors: # @@ -162,10 +172,15 @@ # X # \end{pmatrix} # -# This means :math:`(T^1)_{i,j,k}` and :math:`(T^2)_{l,j,m}` are two rank-3 tensors, where the index :math:`j` *picks* the elements in the column vector while the the other two indices correspond to the indices of the internal tensors (matrices). For instance, the :math:`j = 0`-th element of :math:`T^1` and :math:`T^2` are :math:`|0\rangle \langle 0 |` and :math:`I`, respectively, and similarly for their :math:`j = 1`-st element. This means we can redefine the CNOT expression from above as +# This means :math:`T^1_{i,j,k}` and :math:`T^2_{l,j,m}` are two rank-3 tensors, where the index :math:`j` *picks* the elements in the column vector while the other two indices correspond to the indices of the internal tensors (matrices). For instance, the :math:`j = 0`-th element of :math:`T^1` and :math:`T^2` are :math:`|0\rangle \langle 0 |` and :math:`I`, respectively, and similarly for their :math:`j = 1`-st element. This means we can redefine the CNOT expression from above as # # .. math:: -# \mathrm{CNOT}_{i,l,k,m} = \sum_j (T^1)_{i,j,k} \otimes (T^2)_{l,j,m} = \sum_j (T^1)_{i,j,k} (T^2)_{l,j,m}. +# \mathrm{CNOT}_{i,l,k,m} = \sum_j T^1_{i,j,k} \otimes T^2_{l,j,m} . +# +# We recognize this expression as the second definition for the contraction operation we introduced earlier. Hence, the CNOT gate can be seen as the contraction between :math:`T^1` and :math:`T^2`. +# +# .. math:: +# (\mathrm{CNOT})_{i,l,k,m} = \sum_j (T^1)_{i,j,k} (T^2)_{l,j,m}. # # It turns out that :math:`T^1` and :math:`T^2` are special tensors known as the COPY and XOR tensors, respectively, and therefore have special diagrammatic representations. # @@ -175,7 +190,6 @@ # # Then, their contraction results in the well-known CNOT quantum circuit representation. # -# # .. figure:: ../_static/demonstration_assets/tn_basics/08-cnot.png # :align: center # :width: 45% @@ -187,11 +201,11 @@ # # .. note:: # -# More generally, we can find decompositions of multi-qubit gates into local tensors employing the ubiquitous singular value decomposition (SVD). This method is explained in detail in our :doc:`MPS tutorial `. +# More generally, we can find decompositions of multi-qubit gates into local tensors employing the ubiquitous singular value decomposition (SVD). This method is explained in detail in our :doc:`MPS tutorial `. This decomposition is helpful when contracting non-local tensors in, e.g., a quantum circuit. ############################################################################## # The cost of contracting a network -# --------------------------------- +# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # # A common task when dealing with tensors is the contraction of large networks resulting in a single tensor (including scalars). To arrive at this final tensor, we can start with a single tensor and contract it with adjacent tensors one at a time. The order in which this is carried out is known as the **contraction path** or **bubbling**. # @@ -308,8 +322,13 @@ # To help you with this endeavour, in the following sections we will summarize some ubiquitous algorithms used to connect tensor networks and quantum computers. These topics can become quite technical real fast, and thus, we can only scratch the surface in this demo. For this reason, we will reference the relevant sources when pertinent. Now, take a sip of coffee, brace yourself, and let's continue! ☕️ ############################################################################## +# .. _part_two: + +# Part 2: Connecting tensor networks and quantum circuits 🔌 +# ----------------------------------------------------------- +# # Contraction paths -# ----------------- +# ~~~~~~~~~~~~~~~~~ # # In the previous section, we explored how the choice of the contraction path affects the computational cost of the tensor network contraction through a toy example. As shown in [#Lam1997]_, finding an optimal contraction path is equivalent to solving the "multiplication problem," and thus, it is NP-hard. In this section, we provide a general description of the widespread techniques used to tackle this ubiquitous task. # @@ -353,7 +372,7 @@ # The different types of values accepted for ``contraction_optimizer`` are determined by the ``optimize`` parameter in ``Quimb`` (see `docs `_) as this is the backend behind the :class:`~pennylane.devices.default_tensor.DefaultTensor` device. See `this tutorial `_ to learn more the use of this device in ``Pennylane``. # # Slicing -# ~~~~~~~ +# ^^^^^^^ # # The size of (intermediate) tensors can grow exponentially with the number of indices and dimensions, especially for large-scale tensor networks. Thus, we might run into memory problems when performing the contractions. A useful additional technique to split these tensors into more manageable pieces is known as **slicing**. # @@ -363,7 +382,7 @@ ############################################################################## # Quantum circuits are tensor networks -# ------------------------------------ +# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # # Until now, we have looked at general tensor networks, while ✨sparkling✨ the discussions with examples related to quantum circuits. Here, we leverage the components we have learned to explore this relation more in-depth. # @@ -385,7 +404,7 @@ # Now we can ask ourselves: what quantities can we compute from this tensor network? 🤔 # # Expectation values -# ~~~~~~~~~~~~~~~~~~ +# ^^^^^^^^^^^^^^^^^^ # # As anticipated in the previous section, a natural quantity to compute using the tensor network arising from a quantum circuit is the expectation value of an observable :math:`O` evaluated at the quantum state :math:`|\psi \rangle` # @@ -418,7 +437,7 @@ # where :math:`| \psi_l \rangle` is the section of the evolved state within the light cone of :math:`O_l`. # # Sampling -# ~~~~~~~~ +# ^^^^^^^^ # # In addition to calculating expectation values, we can also use the tensor network arising from a quantum circuit to sample bitstrings from the evolved probability distribution :math:`| \psi \rangle` - emulating what you would obtain from a real quantum computer. Since this is a ubiquitous task in quantum information, several algorithms have been proposed to generate samples from probability distributions represented as tensor networks. In particular, here we discuss the "Perfect Sampling Algorithm" applied to unitary tensor networks [#Ferris2012]_, as this generates uncorrelated samples, unlike Markov-based approaches. # @@ -477,7 +496,7 @@ # # And that is it for this tutorial! 🎉 # -# Although the world of tensor networks and their relation to quantum computing is vastly wider than what we could ever cover in one tutorial, we hope that after these explanations you now feel equipped with the tools needed to dive deeper into this topic by yourself. 🔍 +# Although the world of tensor networks and their relation to quantum computing is vastly wider than what we could ever cover in one tutorial, we hope that after these explanations you now feel equipped with the tools needed to dive deeper into this topic by yourself. 🔍🛠️ # # If you want to learn more about using tensor networks as a diagrammatic tool, check out `these amazing lecture notes `_ by J.Biamonte. In addition, check out `this website `_ for great explanations on many important algorithms and tensor network structures by Flatiron Institute . From 8fb96ef48f23a9d52a62e87dc2495a8e1a32df45 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Mon, 2 Dec 2024 01:16:42 +0100 Subject: [PATCH 50/51] empty commit to retriger deployment --- demonstrations/tutorial_tn_basics.py | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 960cb0b80f..9485bb8a6c 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -558,4 +558,5 @@ ############################################################################## # About the author # ---------------- -# \ No newline at end of file +# +# \ No newline at end of file From 8a6f757df2d172124a4d7db375deda638038a5a7 Mon Sep 17 00:00:00 2001 From: EmilianoG-byte Date: Mon, 2 Dec 2024 15:21:20 +0100 Subject: [PATCH 51/51] correct empty space in part 2 --- demonstrations/tutorial_tn_basics.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_tn_basics.py b/demonstrations/tutorial_tn_basics.py index 9485bb8a6c..356272f88b 100644 --- a/demonstrations/tutorial_tn_basics.py +++ b/demonstrations/tutorial_tn_basics.py @@ -323,7 +323,7 @@ ############################################################################## # .. _part_two: - +# # Part 2: Connecting tensor networks and quantum circuits 🔌 # ----------------------------------------------------------- #