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a/demonstrations/tutorial_preparation_mps_mcm.metadata.json b/demonstrations/tutorial_preparation_mps_mcm.metadata.json new file mode 100644 index 0000000000..fe0471e3f5 --- /dev/null +++ b/demonstrations/tutorial_preparation_mps_mcm.metadata.json @@ -0,0 +1,53 @@ +{ + "title": "Introduction to mid-circuit measurements", + "authors": [ + { + "id": "david_wierichs" + } + ], + "dateOfPublication": "2024-05-10T00:00:00+00:00", + "dateOfLastModification": "2024-06-21T00:00:00+00:00", + "categories": [ + "Getting Started", + "Quantum Computing" + ], + "tags": [], + "previewImages": [ + { + "type": "thumbnail", + "uri": "/_static/demonstration_assets/regular_demo_thumbnails/thumbnail_mcm_introduction.png" + }, + { + "type": "large_thumbnail", + "uri": "/_static/large_demo_thumbnails/thumbnail_large_mcm_introduction.png" + } + ], + "seoDescription": "Learn the basics of mid-circuit measurements and how to use them in PennyLane.", + "doi": "", + "canonicalURL": "/qml/demos/tutorial_mcm_introduction", + "references": [], + "basedOnPapers": [], + "referencedByPapers": [], + "relatedContent": [ + { + "type": "demonstration", + "id": "tutorial_how_to_create_dynamic_mcm_circuits", + "weight": 1.0 + }, + { + "type": "demonstration", + "id": "tutorial_how_to_collect_mcm_stats", + "weight": 1.0 + }, + { + "type": "demonstration", + "id": "tutorial_mbqc", + "weight": 1.0 + }, + { + "type": "demonstration", + "id": "tutorial_teleportation", + "weight": 1.0 + } + ] +} diff --git a/demonstrations/tutorial_preparation_mps_mcm.py b/demonstrations/tutorial_preparation_mps_mcm.py new file mode 100644 index 0000000000..607d3f2b3d --- /dev/null +++ b/demonstrations/tutorial_preparation_mps_mcm.py @@ -0,0 +1,747 @@ +r"""Constant-depth preparation of matrix product states with dynamic circuits +============================================================================= + +Matrix product states (MPS) are an important class of quantum states that is +used in a plethora of applications on quantum many-body systems. +This makes the preparation of MPS on a quantum computer a useful subroutine. +For example, a quantum algorithm could prepare a classically pre-computed MPS +as initial state in order to then perform operations that are no longer +classically tractable. + +.. figure:: ../_static/todo + :align: center + :width: 50% + +In this demo you will learn about a new algorithm to prepare MPS with a dynamic +quantum circuit of constant depth. We will implement the circuit, which makes use +of mid-circuit measurements and conditionally applied operations, for a specific +MPS, and compare it to a (static) sequential circuit from the literature with +linear depth. + +This demo closely follows the eponymous article by Smith et al. [#smith]_. + +If you would like to first learn about the dynamic circuit tools used in this +algorithm, have a look at our :doc:`introduction to mid-circuit measurements +` and learn :doc:`how to collect statics of +mid-circuit measurements ` and +:doc:`how to create dynamic circuits with mid-circuit measurements +`. + +If you first want to familiarize yourself with tensor network states and MPS +in particular, take a look at our demo on +:doc:`tensor-network quantum circuits `. +Finally, our demo on :doc:`initial state preparation for quantum chemistry +` shows you how to +proceed to a concrete application, once an MPS is prepared. + +Outline +------- + +We will start with a brief (no, really!) mathematical overview of the used +objects and subroutines. Then we define them in code for a specific parametric MPS +and verify the conditions for the algorithm to apply. +One of the subroutines is the sequential MPS preparation circuit that we +compare to later. +Afterwards we combine the building blocks into the constant-depth algorithm +by Smith et al. We conclude by computing the correlation length of the prepared +MPS for different parameter values. + +Mathematical derivation +----------------------- + +We briefly introduce matrix product states and a sequential circuit with linear depth +that prepares them, as well as two tools from quantum information theory +that we will use below, namely fusion of MPS with mid-circuit measurements +and so-called operator pushing. + +Matrix product states +~~~~~~~~~~~~~~~~~~~~~ + +Matrix product states (MPS) are an important class of quantum states. +They can be described and manipulated conveniently on classical computers and +are able to approximate relevant states in quantum many-body systems. +In particular, MPS can accurately describe ground states of (gapped local) +one-dimensional Hamiltonians, which can be found efficiently using the density +matrix renormalization group (DMRG). +For reviews of MPS see [#todo]_, [#todo]_, and [#todo]_. + +Following [#smith]_, we will look at translation-invariant MPS +of a quantum :math:`N`-body system where each body, or site, has local (physical) +dimension :math:`d`. We will further restrict to periodic boundary conditions, +corresponding to translational invariance of the MPS on a closed loop. +A general MPS with this properties is given by + +.. math:: + + |\Psi\rangle = \sum_{\vec{m}} \operatorname{tr}[A^{m_1}A^{m_2}\cdots A^{m_N}]|\vec{m}\rangle, + +where :math:`\vec{m}` is a multi-index of :math:`N` indices ranging over :math:`d`, +i.e., :math:`\vec{m}\in\{0, 1 \dots, d-1\}^N`, and :math:`\{A^{m}\}_{m}` are :math:`d` +square matrices of equal dimension. This dimension :math:`D` is called the bond dimension, +and it is a crucial quantity for the expressivitiy and complexity of the MPS. +We see that :math:`|\Psi\rangle` is fully specified by the :math:`d\times D\times D=dD^2` +numbers in the rank-3 tensor :math:`A`. +However, this specification is not unique. We remove some of the redundancies by assuming +that :math:`A` is in so-called left-canonical form, meaning that it satisfies + +.. math:: + + \sum_m {A^m}^\dagger A^m = \mathbb{I}. + +We will not discuss additional redundancies here but refer to [#smith]_ and the +mentioned reviews for more details. + +**Example** + +As an example, consider an :math:`N`-site qubit (:math:`d=2`) chain and a :math:`D=2` MPS +:math:`|\Psi(g)\rangle` on this system defined by the matrices + +.. math:: + + A^0 = \eta \left(\begin{matrix} 1 & 0 \\ \sqrt{-g} & 0 \end{matrix}\right) + \quad + A^1 = \eta \left(\begin{matrix} 0 & -\sqrt{-g} \\ 0 & 1 \end{matrix}\right) + +where :math:`\eta = \sqrt{1-g}^{-1}` and :math:`g\in[-1, 0]` is a freely chosen parameter. + +This MPS is an important example because it can be tuned from the long-range correlated +GHZ state :math:`|\Psi(0)\rangle=(|0>^{\otimes N} + |1>^{\otimes N})/\sqrt(2)` +to the state :math:`|\Psi(-1)\rangle=|+\rangle^{\otimes N}` with vanishing correlation length. +In general, the correlation length is given by + +.. math:: + + \xi = \left|\ln\left(\frac{1+g}{1-g}\right)\right|^{-1}. + +Long-range correlated states require linear-depth unitary circuits in general. +Therefore, the constant-depth circuit below for :math:`|\Psi(0)\rangle` will demonstrate +that dynamic quantum circuits are more powerful than unitary operations alone. + +The MPS :math:`|\Psi(g)\rangle` will be our working example throughout this demo. Therefore +we warm up our coding by defining the tensor :math:`A` and testing some of its properties. + +""" +import numpy as np +import matplotlib.pyplot as plt +from scipy.optimize import curve_fit + +import pennylane as qml + +def A(g): + """Compute the tensor A in form of d=2 matrices with shape (D, D) = (2, 2) each.""" + eta = 1 / np.sqrt(1 - g) + A_0 = np.array([[1, 0], [np.sqrt(-g), 0]]) * eta + A_1 = np.array([[0, -np.sqrt(-g)], [0, 1]]) * eta + return (A_0, A_1) + +g = -0.6 +A_0, A_1 = A(g) +is_left_canonical = np.allclose(A_0.conj().T @ A_0 + A_1.conj().T @ A_1, np.eye(2)) +print(f"The matrices A^m are in left-canonical form: {is_left_canonical}") + +xi = 1 / abs(np.log((1+g) / (1-g))) +print(f"For {g=}, the theoretical correlation length is {xi=:.4f}") + +###################################################################### +# Sequential preparation circuit +# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +# +# An MPS like the one above can be prepared in linear depth using an established technique +# by Schön et al. [#schoen]_. We introduce this technique here because the new +# constant-depth construction uses it as a parallelized subroutine, and because we will +# later compare the new construction to the sequential circuit. +# +# The MPS :math:`|\Psi\rangle` above is given by the rank-3 tensor :math:`A`. +# In order to prepare the state in a circuit, we want to construct unitary operations +# from :math:`A`. For this, we build a new rank-4 tensor :math:`U` with an additional +# axis of dimension :math:`d`. If this new physical input axis is in the state +# :math:`|0\rangle`, we define :math:`U` to reproduce one :math:`A^m` for each physical +# output state :math:`|m\rangle`. We leave the remaining action undefined and only require +# it to make :math:`U` unitary overall. In short, this can be written as +# +# .. math:: +# +# U = \sum_m A^m \otimes |m\rangle\!\langle 0| + C_\perp. +# +# Let us denote :math:`U` acting on the :math:`i`-th physical site as :math:`U^{(i)}` +# for a fixed non-physical site. +# The sequential preparation circuit now consists of the following steps. +# +# #. Start in the state :math:`|\psi_0\rangle = |0_d\rangle^{\otimes N}\otimes |00\rangle_D`. +# The last two sites are non-physical bond sites, encoded by :math:`D`-dimensional qudits. +# +# #. Entangle the bond qudits into the state :math:`|I\rangle = 1/\sqrt{D}\sum_j |jj\rangle_D`. +# +# #. Apply the unitary :math:`U` once to each of the :math:`N` physical sites, with the +# :math:`D`-dimensional factor always acting on the first bond qudit. This produces the state +# +# .. math:: +# +# |\psi_1\rangle +# &= \frac{1}{\sqrt{D}} \sum_j \prod_{i=N-1}^{1} U^{(i)} +# \sum_{m_N=0}^{d-1} |0_d\rangle^{\otimes N-1}|m_N\rangle A^{m_N}|jj\rangle _D\\ +# &= \frac{1}{\sqrt{D}} \sum_j +# \sum_{\vec{m}} |\vec{m}\rangle A^{m_1} A^{m_2}\cdots A^{m_N}|j\rangle_D |j\rangle_D\\ +# &= \frac{1}{\sqrt{D}} \sum_{i,j} +# \sum_{\vec{m}} |\vec{m}\rangle \langle i|A^{m_1} A^{m_2}\cdots A^{m_N}|j\rangle_D |i\rangle_D|j\rangle_D +# +# #. Measure the two bond qudits in the (generalized) Bell basis and postselect on the outcome +# being :math:`|I\rangle`. Then discard the bond qudits. This collapses :math:`\psi_1\rangle` +# into the (renormalized) state +# +# .. math:: +# +# |\psi_2\rangle = \sum_{\vec{m}} \operatorname{tr}[A^{m_1}A^{m_2}\cdots A^{m_N}]|\vec{m}\rangle = |\Psi\rangle +# +# Note that this step is probabilistic and we only succeed to produce the state if we measured +# the state :math:`|I\rangle`. +# +# We see that we prepared :math:`|\Psi\rangle` with an entangling operation on the bond qudits, one unitary per +# physical qubit, and a final basis change to measure the Bell basis. Overall, this amounts to a linear +# operation count and circuit depth. +# +# **Example** +# +# For our example MPS :math:`|\Psi(g)\rangle`, let us first find the unitary :math:`U`. +# For this, consider the fixed part of :math:`U`. + +E_00 = np.array([[1, 0], [0, 0]]) # |0><0| +E_10 = np.array([[0, 0], [1, 0]]) # |1><0| +U_first_term = np.kron(A_0, E_00) + np.kron(A_1, E_10) + +print(np.round(U_first_term, 5)) +print(np.linalg.norm(U_first_term, axis=0)) + +###################################################################### +# We see that this fixed part has two norm-1 columns already, and we are +# able to complement this by :math:`C_\perp` with the same columns: + +E_01 = np.array([[0, 1], [0, 0]]) # |0><1| +E_11 = np.array([[0, 0], [0, 1]]) # |1><1| +C_perp = np.kron(A_1, E_01) + np.kron(A_0, E_11) + +U = U_first_term + C_perp + +print(np.round(U, 5)) +print(f"U is unitary: {np.allclose(U.conj().T @ U, np.eye(4))}") + + +###################################################################### +# Great! This means that we already found the unitary :math:`U` for this MPS. +# Combining the terms above on paper, we see that +# +# .. math:: +# +# U &= A^0 \otimes (|0\rangle\!\langle 0| + |1\rangle\!\langle 1|) + A^1\otimes (|1\rangle\!\langle 0| + |0\rangle\!\langle 1|)\\ +# &= A^0 \otimes \mathbb{I} + A^1\otimes X +# +# This looks a lot like a :class:`~.pennylane.CNOT` gate plays a part in :math:`U`. +# Removing it from :math:`U` we find +# +# .. math:: +# +# U \operatorname{CNOT} +# &= A^0 P_0 \otimes \mathbb{I} + A^0P_1 \otimes X + A^1 P_0 \otimes X + A^1P_1 \otimes \mathbb{I}\\ +# &= \left(\begin{matrix} \eta & -\eta\sqrt{-g} \\ \eta\sqrt{-g} & \eta \end{matrix}\right), +# +# where we denoted the computational basis state projectors as :math:`P_i = |i\rangle\!\langle i|`. +# The remaining operation is a Pauli-:math:`Y` rotation by the angle :math:`2\arccos(\eta)`, so +# that we can easily code up the unitary as a small circuit template: + + +def U_template(eta, bond_idx, phys_idx): + qml.CNOT([bond_idx, phys_idx]) + qml.RY(2 * np.arccos(eta), bond_idx) + +eta = 1 / np.sqrt(1 - g) +template_matrix = qml.matrix(U_template, wire_order=["bond", "phys"])(eta, "bond", "phys") +print(f"The template reproduces U: {np.allclose(template_matrix, U)}") + +###################################################################### +# Great, now that we have a circuit that realizes the unitary :math:`U`, let's code up the entire +# sequential preparation circuit. We will leave the measurement \& postselect step as a separate +# function for now. + +def prepare_bell(wire_0, wire_1): + """Prepare two qubits in the Bell state (|00>+|11>)/sqrt(2).""" + qml.Hadamard(wire_0) + qml.CNOT([wire_0, wire_1]) + +def project_measure(wire_0, wire_1): + """Measure two qubits in the Bell basis and + postselect on (|00>+|11>)/sqrt(2).""" + # Move bond qubits to Bell basis by undoing the preparation + qml.adjoint(prepare_bell)(wire_0, wire_1) + # Measure and postselect |00> + qml.measure(wire_0, postselect=0) + qml.measure(wire_1, postselect=0) + +def sequential_preparation(g, wires): + """Prepare the example MPS \Psi(g) on N qubits where N is the length + of the passed wires minus 2. The bond qubits are still entangled.""" + eta = 1 / np.sqrt(1 - g) + bond_wires = [wires[0], wires[-1]] # Bond qubits [0, N+1] + phys_wires = wires[1:-1] # Physical qubits [1, 2, ... N] + prepare_bell(*bond_wires) + for phys_wire in phys_wires: + U_template(eta, bond_wires[1], phys_wire) + +dev = qml.device("default.qubit") + +@qml.qnode(dev) +def sequential_circuit(N, g): + """Run the preparation circuit and projectively measure the bond qubits.""" + wires = list(range(N+2)) + sequential_preparation(g, wires) + project_measure(0, N + 1) + return qml.probs(wires=wires[1:N+1]) + +###################################################################### +# We will verify the prepared state below, when comparing to the constant-depth +# preparation circuit. Let's just draw it for now. + +N = 12 +qml.draw_mpl(sequential_circuit)(N, g) + +###################################################################### +# +# Fusion of MPS states with mid-circuit measurements +# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +# +# The next ingredient to the constant-depth MPS preparation circuit is to fuse +# MPS states on different qubits together into one MPS. Under the hood, this is +# an application of entanglement swapping. +# +# Consider a state :math:`|\Phi\rangle=|\Psi_1\rangle\otimes|\Phi_2\rangle`, where +# :math:`|\Psi_{1,2}\rangle` are MPS prepared with the sequential technique from +# above on :math:`K` and :math:`L` physical qudits, respectively, but with the bond +# qudits left unmeasured. In particular, the postselection step has not been performed +# yet. Using the form of :math:`|\psi_1\rangle` from the recipe above, we can write +# this state as +# +# .. math:: +# +# |\Phi\rangle = \frac{1}{D} \sum_{i,j,\ell,p} +# \sum_{\vec{m}} |\vec{m}\rangle |ij\ell p\rangle_D +# \langle i|A^{m_1} \cdots A^{m_K}|j\rangle_D +# \langle \ell|A^{m_{K+1}} \cdots A^{m_{K+L}}|p\rangle_D. +# +# Now we want to measure two bond qudits, one of :math:`|\Psi_1\rangle` and +# :math:`|\Psi_2\rangle` each, in an entangled basis. Assume this basis to be given +# in the form of (orthogonal) states +# +# .. math:: +# +# |B^k\rangle=\frac{1}{D} \sum_{j,\ell=1}^D \left(B_{j\ell}^k\right)^\ast |j\ell\rangle, +# +# which in turn are characterized by :math:`D\times D` matrices :math:`B^k`. +# We can change the basis on the bond qudits to be measured by applying the unitary +# :math:`V=\sum_k |k\rangle\!\langle B^k|`, which leads to the state +# +# .. math:: +# +# |\Phi'\rangle = \frac{1}{D} \sum_k \sum_{i,j,\ell,p} +# \sum_{\vec{m}} |\vec{m}\rangle |k\rangle_{D^2} |ip\rangle_D +# \langle i|A^{m_1} \cdots A^{m_K}B^kA^{m_{K+1}} \cdots A^{m_{K+L}}|p\rangle_D, +# +# where we used that :math:`\sum_{j,\ell} |j\rangle B^k_{j\ell}\langle\ell|=B^k`. +# We now can measure the two bond qudits and will know that for a given measurement +# outcome :math:`k`, we obtained the MPS state on :math:`K+L` qubits, except for +# a small "impurity", namely the matrix :math:`B^k` corresponding to the outcome. +# +# In this generality, the fusion strategy may seem complicated, but it can take +# a very simply form, as we will now see in our example: +# +# **Example** +# +# For our example we will use the Bell basis to perform the fusing mid-circuit +# measurement, i.e., +# +# .. math:: +# +# |B^0\rangle &= \frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)\quad +# |B^1\rangle = \frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)\\ +# |B^2\rangle &= \frac{1}{\sqrt{2}}(|00\rangle-|11\rangle)\quad +# |B^3\rangle = \frac{i}{\sqrt{2}}(|01\rangle-|10\rangle). +# +# This choice must seem arbitrary at this point, but will become clear +# later on. There, we will see that in general the matrices :math:`B^k` and the MPS +# tensor :math:`A` have to play well together. +# For the choice above, we find the matrices +# +# .. math:: +# +# B^0 = I \quad B^1 = X \quad B^2 = Z \quad B^3 = Y, +# +# i.e., the standard Pauli matrices. +# The measurement procedure can then be coded up in the following short function, +# which is similar to the ``project_measure`` function from the sequential preparation +# routine. Note, however, that instead of postselecting, we record the measurement +# outcome and will use it further below. + +def fuse(wire_0, wire_1): + """Measure two qubits in the Bell basis and return the outcome + encoded in two bits.""" + qml.CNOT([wire_0, wire_1]) + qml.Hadamard(wire_0) + return np.array([qml.measure(i) for i in [wire_0, wire_1]]) + +###################################################################### +# +# Fusing two MPS that have been prepared by the sequential routine now is really simple: + +def two_qubit_mps_by_fusion(g): + sequential_preparation(g, [0, 1, 2]) + sequential_preparation(g, [3, 4, 5]) + mcms = fuse(2, 3) + +###################################################################### +# However, the produced state is not quite the MPS state on two qubits, +# because of the impurities caused through the fusion measurement. We can check this +# by undoing the preparation with the two qubit sequential circuit and measuring an +# exemplary expectation value. +# If the two sequentially prepared MPS and the fusion step did prepare the correct +# MPS already, the test measurement would just be :math:`\langle 00 | Z_0| 11\rangle=1`. + +@qml.qnode(qml.device("default.qubit", wires=6)) +def prepare_and_unprepare(g): + two_qubit_mps_by_fusion(g) + # The bond qubits for a sequential preparation are just 0 and 5 + # The bond qubits 2 and 3 have been measured out in the fusion protocol + qml.adjoint(sequential_preparation)(g, [0, 1, 4, 5]) + return qml.expval(qml.PauliZ(1)) + +test = prepare_and_unprepare(g) +print(f"The test measurement of the fusion preparation + sequential unpreparation is {test:.2f}") + +###################################################################### +# This means we still need a way to remove the "impurity" matrices :math:`B^k` from the +# state that ended up in the prepared state when we performed the fusion measurement. +# Operator pushing will allow us to do just that! +# +# Operator pushing +# ~~~~~~~~~~~~~~~~ +# +# The last building block we need is so-called operator pushing. In fact, whether or not an MPS +# can be prepared with the constant-depth circuit depends on whether certain operator pushing +# relations are satisfied between the matrices :math:`B^k` from the fusion step and the MPS +# tensor :math:`A`. We will not go into detail about these conditions but assume here that +# we work with compatible MPS and measurements. +# +# Operator pushing then allows us to push an impurity operator :math:`B^k` from one bond +# axis through the tensor :math:`A` of the MPS to the other bond axis and to the physical axis. +# While doing so, the operator in general will change, i.e., we end up with different operators +# :math:`C^k` and :math:`D^k` on the bond and physical axes, respectively. +# Visually, we find: +# +# .. image:: ../_static/demonstration_assets/preparation_mps_mcm/operator_pushing.png +# +# For simplicity, we denote a push by :math:`B^k\mapsto (C^k, D^k)`. +# +# **Example** +# +# In the fusion step above we picked the Bell basis as measurement basis, without +# further specifying why. As we saw, this basis leads to Pauli matrices as impurities +# :math:`B^k`. This choice is such that they satisfy pushing relations together with +# the tensor :math:`A` of our example MPS :math:`|\Psi(g)\rangle`. +# In particular, the relations are +# +# .. image:: ../_static/demonstration_assets/preparation_mps_mcm/operator_pushing.png +# +# That is, in the notation above we have +# :math:`X\mapsto (Y, Y)`, +# :math:`Y\mapsto (Y, X)`, and +# :math:`Z\mapsto (I, Z)` as well as the trivial +# :math:`I\mapsto (I, I)`. +# Note that the bond axis operator either is trivial (the identity), or a Pauli-:math:`Y` operator. +# These pushing relations allow us to push the bond axis operator through to an open boundary +# bond axis of the fused MPS, and to remove the physical axis operators by applying the correct +# Pauli operation to the physical site, conditioned on the fusion measurement outcomes. +# If we have an MPS on :math:`L` sites, pushing through a Pauli will have the following effects: +# +# - If it is a Pauli-:math:`Y`, all physical sites need to be corrected by a Pauli-:math:`X`, and +# a Pauli-:math:`Y` is pushed to the opposite boundary. +# +# - If it is a Pauli-:math:`X`, a Pauli-:math:`Y` is applied to the first physical qubit, and afterwards +# the case above is recovered, leading to applying Pauli-:math:`X` and pushing through a Pauli-:math:`Y`. +# +# - If it is a Pauli-:math:`Z`, a Pauli-:math:`Z` is applied to the first physical qubit and we are done. +# No operator is pushed through. +# +# This pushing and correction step is captured by the following function, where we use some additional +# simplifications based on the used encoding with two bits of the pushed operator. + +def push_and_correct(op_id, phys_wires): + """Push an operator from left to right along the bond sites of an MPS and + conditionally apply correcting operations on the corresponding physical sites. + - The operator is given by two bits indicating the Pauli operator type. + - The physical MPS sites are given by phys_wires + """ + w = phys_wires[0] + # Apply Y if input is X or Y + qml.cond(op_id[1] & op_id[0], qml.X)(w) + # Apply Z if input is Z or Y + qml.cond(op_id[1] & ~op_id[0], qml.Y)(w) + qml.cond(~op_id[1] & op_id[0], qml.Z)(w) + # Apply X on other physical sites if input is X or Y + for i in phys_wires[1:]: + qml.cond(op_id[1], qml.X)(i) + # Push through Y if input is X or Y + return np.array([op_id[1], op_id[1]]) + +def correction(idx, q, op_id): + w = (idx + 1) * q + op_int = np.array([2, 1]) @ op_id + # If measurement syndrome is X, apply Y and push Y's along, leading to consecutive X + qml.cond(op_int == 1, qml.Y)(w) + for i in range(w + 1, w + q): + qml.cond(op_int == 1, qml.X)(i) + # If measurement syndrome is Z, apply Z and terminate + qml.cond(op_int == 2, qml.Z)(w) + # If measurement syndrome is Y, apply X and push Y's along, leading to consecutive X + for i in range(w, w + q): + qml.cond(op_int == 3, qml.X)(i) + return np.array([op_id[1], op_id[1]]) + +###################################################################### +# +# Piecing everything together +# --------------------------- +# +# Now that we discussed the sequential preparation algorithm, fusion with mid-circuit measurements, +# and operator pushing, we have all components for the constant-depth preparation algorithm. It works +# as follows (recall that showing existence and finding suitable operator pushing relations is a +# crucial step in general, which goes beyond the scope of the demo). +# +# #. For a block size :math:`q`, prepare :math:`\frac{N}{q}` MPS of size :math:`q` in parallel, using +# the sequential preparation algorithm (without the final projection step). +# +# #. Fuse together the blocks of MPS using mid-circuit measurements and store the measurement outcomes. +# +# #. Push the "impurity" operators resulting from the measurements to one of the outer bond sites and +# conditionally apply correction operators to the physical sites. +# +# #. Undo the operator that was pushed to the outer bond site by conditionally applying its inverse. +# +# #. Perform the same projection step as in the sequential preparation algorithm on the two remaining +# bond sites. +# +# Block size :math:`q` +# ~~~~~~~~~~~~~~~~~~~~ +# The size of the blocks that have to be prepared in the first step depends on multiple considerations. +# First, the block must be such that operator pushing relations are available. Usually this will +# determine a minimal block size, and multiples of that size are valid choices as well. +# Second, the depth of the (parallelized) sequential preparation circuits is proportional to :math:`q`, +# determining a key contribution to the overall depth of the algorithm. +# Third, the sequential algorithm requires two bond qudits for each block, leading to +# :math:`\frac{2N}{q}` auxiliary qudits overall. +# +# +# **Example** +# +# For our example MPS :math:`|\Psi(g)\rangle`, we already defined and coded up the sequential preparation +# circuit for a flexible number of qubits, the measurement-based fusion, and the operator pushing +# and correction step. This makes putting the algorithm together quite simple. We only define an XOR +# for our operator IDs and a function that applies a correcting Pauli operator to the final bond +# site after pushing through the operator. +# + +def xor(op_id_0, op_id_1): + """Express logical XOR as "SUM - 2 * AND" on integers.""" + return op_id_0 + op_id_1 - 2 * op_id_0 * op_id_1 + +def correct_end_bond(bond_idx, op_id): + op_int = np.array([2, 1]) @ op_id + qml.cond(op_int == 1, qml.X)(bond_idx) + qml.cond(op_int == 2, qml.Z)(bond_idx) + qml.cond(op_int == 3, qml.Y)(bond_idx) + + +def constant_depth(N, g, q): + """Prepare the MPS |\Psi(g)> in constant depth.""" + num_blocks = N // q + wires_per_block = q + 2 # 2 bond wires added + + # Step 1: Prepare small block MPS + for i in range(num_blocks): + wires = list(range(wires_per_block * i, wires_per_block * (i+1))) + sequential_preparation(g, wires) + + # Step 2: Fusion with mid-circuit measurements + mcms = [] + for i in range(1, num_blocks): + bond_wires = (wires_per_block * i - 1, wires_per_block * i) + mcms.append(fuse(*bond_wires)) + + # Step 3: Push operators through to highest-index wire and correct phys. sites + pushed_op_id = np.array([0, 0]) # Start with identity + for i in range(1, num_blocks): + phys_wires = list(range(wires_per_block * i + 1, wires_per_block * (i+1) - 1)) + pushed_op_id = push_and_correct(xor(mcms[i-1], pushed_op_id), phys_wires) + + # Step 4: Undo the pushed-through operator on the highest-index wire bond site. + correct_end_bond(num_blocks * wires_per_block - 1, pushed_op_id) + +def constant_depth_ansatz(N, g, q): + num_blocks = N // q + wires_per_block = q + 2 # 2 bond wires added + outer_bond_sites = [0, num_blocks * wires_per_block - 1] + # Steps 1-4 + constant_depth(N, g, q) + # Step 5: Perform projective measurement on outer-most bond sites. + project_measure(*outer_bond_sites) + phys_wires = sum( + (list(range(wires_per_block * i + 1, wires_per_block * (i+1) - 1)) + for i in range(num_blocks)), + start = [], + ) + return phys_wires + +@qml.qnode(dev) +def constant_depth_circuit(N, g, q): + phys_wires = constant_depth_ansatz(N, g, q) + return qml.probs(wires=phys_wires) + +###################################################################### +# +# Great, we know built the full constant-depth circuit to prepare the MPS +# :math:`|\Psi(g)\rangle`. +# +# Before we evaluate the states it produces, let's see how the circuit looks. +# We'll add some boxes for the individual subroutines. +# + +N = 12 +q = 3 +g = -0.8 +qml.draw_mpl(constant_depth_circuit)(N, g, q) +#plt.show() + +###################################################################### +# Let's check that the constant-depth circuit produces the same state, or probability +# distribution, as the sequential circuit. +# + +p_seq = sequential_circuit(N, g) +p_const = constant_depth_circuit(N, g, q) +print(f"The two circuits agree on the probabilities for {g=:.1f}: {np.allclose(p_seq, p_const)}") + + +###################################################################### +# Great, we now have a constant-depth circuit that reproduces the existing sequential +# preparation circuit, which has linear depth! +# +# Correlation length +# ~~~~~~~~~~~~~~~~~~ +# +# When introducing the example MPS :math:`|\Psi(g)\rangle`, we mentioned that the +# parameter :math:`g` controls the correlation length of the state. As extreme examples, +# we can produce a zero-correlation length state for :math:`g=-1` and a long-range +# correlated state, namely the GHZ state, for :math:`g\to 0`. +# + +N = 6 +g = -1 +uniform_probs = constant_depth_circuit(N, g, q) +g = -1e-8 +ghz_probs = constant_depth_circuit(N, g, q) +ints = list(range(2**N)) +fig, ax = plt.subplots() +ax.bar(ints, uniform_probs, width=1, label="zero correlation length ($|+\\rangle^N$)") +ax.bar(ints, ghz_probs, width=1, label="long correlation length (GHZ)") +ax.set_yscale("log") +ax.set_ylim([1e-3, 1]) +ax.legend() +#plt.show() + +###################################################################### +# In a more nuanced setting, let us produce the MPS in question for a series of values +# for :math:`g`, and measure the correlation length ourselves. +# For this, we measure the two-body correlators +# +# .. math:: +# +# C(j) = \langle Z_0 Z_j\rangle - \langle Z_0\rangle \langle Z_1\rangle +# +# for an increasing distance :math:`j`. + + +@qml.qnode(dev, interface="numpy") +def pre_correlations(N, g, q): + phys_wires = constant_depth_ansatz(N, g, q) + return [qml.expval(qml.Z(1) @ qml.Z(i)) for i in phys_wires] + [qml.expval(qml.Z(i)) for i in phys_wires] + +def correlations(N, g, q): + expvals = pre_correlations(N, g, q) + prods = expvals[:-N] + expvals = expvals[-N:] + correls = [prods[i] - expvals[1] * expvals[i] for i in range(N) if i!=1] + return correls + +N = 18 +q = 6 +gs = [-1/5, -2/5, -3/5, -4/5] +fig, ax = plt.subplots() +all_correls = [] +for g in gs: + correls = correlations(N, g, q) + ax.plot(list(range(N-1)), correls, label=f"$g={g}$") + all_correls.append(correls) +ax.set_yscale("log") +ax.set_ylim([5e-8, 1.5]) +ax.legend() +plt.show() + +###################################################################### +# Note that due to the periodic boundary conditions, we have :math:`C(N-j)=C(j)`. +# This is why the correlations increase again after :math:`j=\frac{N}{2}`. +# To obtain the correlation length, we fit an exponential to the measured correlators +# up until :math:`j=\frac{N/2}` and invert the decay rate. +# We can also compare this result to the analytic result for the correlation +# length :math:`xi` mentioned in the beginning: + +def exp_fn(x, a, b): + return a * np.exp(x * b) + +print(f"Correlation lengths xi\n===================") +for g, correls in zip(gs, all_correls): + popt, pcov = curve_fit(exp_fn, list(range(1, N//2+1)), correls[1:N//2+1]) + xi_numerical = -1/popt[1] + xi_predicted = 1 / abs(np.log((1+g) / (1-g))) + print(f"{g=:.1f}") + print(f"Predicted: {xi_predicted:.5f}") + print(f"Numerical: {xi_numerical:.5f}") + print() + +###################################################################### +# As we can see, the produced MPS states indeed have the predicted correlation lengths, +# up to some numerical imprecisions. +# +# Conclusion +# ---------- +# +###################################################################### +# +# +# References +# ---------- +# .. [#smith] +# +# Kevin C. Smith, Abid Khan, Bryan K. Clark, S.M. Girvin, Tzu-Chieh Wei +# "Constant-depth preparation of matrix product states with adaptive quantum circuits", +# `arXiv:2404.16083 `__, 2024. +# +# .. [#schoen] +# +# C. Schön, E. Solano, F. Verstraete, J. I. Cirac, M. M. Wolf +# "Sequential Generation of Entangled Multiqubit States", Physical Review Letters, **95**, 110503, +# `closed access `__, 2005. +# `arXiv:quant-ph/0501096 `__, 2005. +# +# .. [#todo] +# +# Todo +# +# About the author +# ---------------- +# From 87a487859020b58ca402a8208ae13f6079d9ba91 Mon Sep 17 00:00:00 2001 From: dwierichs Date: Tue, 13 Aug 2024 14:18:16 +0200 Subject: [PATCH 02/37] update --- .../tutorial_preparation_mps_mcm.py | 85 +++++++++++++------ 1 file changed, 59 insertions(+), 26 deletions(-) diff --git a/demonstrations/tutorial_preparation_mps_mcm.py b/demonstrations/tutorial_preparation_mps_mcm.py index 607d3f2b3d..7c90e8d5cb 100644 --- a/demonstrations/tutorial_preparation_mps_mcm.py +++ b/demonstrations/tutorial_preparation_mps_mcm.py @@ -119,14 +119,15 @@ The MPS :math:`|\Psi(g)\rangle` will be our working example throughout this demo. Therefore we warm up our coding by defining the tensor :math:`A` and testing some of its properties. - """ + import numpy as np import matplotlib.pyplot as plt from scipy.optimize import curve_fit import pennylane as qml + def A(g): """Compute the tensor A in form of d=2 matrices with shape (D, D) = (2, 2) each.""" eta = 1 / np.sqrt(1 - g) @@ -134,12 +135,13 @@ def A(g): A_1 = np.array([[0, -np.sqrt(-g)], [0, 1]]) * eta return (A_0, A_1) + g = -0.6 A_0, A_1 = A(g) is_left_canonical = np.allclose(A_0.conj().T @ A_0 + A_1.conj().T @ A_1, np.eye(2)) print(f"The matrices A^m are in left-canonical form: {is_left_canonical}") -xi = 1 / abs(np.log((1+g) / (1-g))) +xi = 1 / abs(np.log((1 + g) / (1 - g))) print(f"For {g=}, the theoretical correlation length is {xi=:.4f}") ###################################################################### @@ -205,8 +207,8 @@ def A(g): # For our example MPS :math:`|\Psi(g)\rangle`, let us first find the unitary :math:`U`. # For this, consider the fixed part of :math:`U`. -E_00 = np.array([[1, 0], [0, 0]]) # |0><0| -E_10 = np.array([[0, 0], [1, 0]]) # |1><0| +E_00 = np.array([[1, 0], [0, 0]]) # |0><0| +E_10 = np.array([[0, 0], [1, 0]]) # |1><0| U_first_term = np.kron(A_0, E_00) + np.kron(A_1, E_10) print(np.round(U_first_term, 5)) @@ -216,8 +218,8 @@ def A(g): # We see that this fixed part has two norm-1 columns already, and we are # able to complement this by :math:`C_\perp` with the same columns: -E_01 = np.array([[0, 1], [0, 0]]) # |0><1| -E_11 = np.array([[0, 0], [0, 1]]) # |1><1| +E_01 = np.array([[0, 1], [0, 0]]) # |0><1| +E_11 = np.array([[0, 0], [0, 1]]) # |1><1| C_perp = np.kron(A_1, E_01) + np.kron(A_0, E_11) U = U_first_term + C_perp @@ -253,6 +255,7 @@ def U_template(eta, bond_idx, phys_idx): qml.CNOT([bond_idx, phys_idx]) qml.RY(2 * np.arccos(eta), bond_idx) + eta = 1 / np.sqrt(1 - g) template_matrix = qml.matrix(U_template, wire_order=["bond", "phys"])(eta, "bond", "phys") print(f"The template reproduces U: {np.allclose(template_matrix, U)}") @@ -262,11 +265,13 @@ def U_template(eta, bond_idx, phys_idx): # sequential preparation circuit. We will leave the measurement \& postselect step as a separate # function for now. + def prepare_bell(wire_0, wire_1): """Prepare two qubits in the Bell state (|00>+|11>)/sqrt(2).""" qml.Hadamard(wire_0) qml.CNOT([wire_0, wire_1]) + def project_measure(wire_0, wire_1): """Measure two qubits in the Bell basis and postselect on (|00>+|11>)/sqrt(2).""" @@ -276,6 +281,7 @@ def project_measure(wire_0, wire_1): qml.measure(wire_0, postselect=0) qml.measure(wire_1, postselect=0) + def sequential_preparation(g, wires): """Prepare the example MPS \Psi(g) on N qubits where N is the length of the passed wires minus 2. The bond qubits are still entangled.""" @@ -286,15 +292,18 @@ def sequential_preparation(g, wires): for phys_wire in phys_wires: U_template(eta, bond_wires[1], phys_wire) + dev = qml.device("default.qubit") + @qml.qnode(dev) def sequential_circuit(N, g): """Run the preparation circuit and projectively measure the bond qubits.""" - wires = list(range(N+2)) + wires = list(range(N + 2)) sequential_preparation(g, wires) project_measure(0, N + 1) - return qml.probs(wires=wires[1:N+1]) + return qml.probs(wires=wires[1 : N + 1]) + ###################################################################### # We will verify the prepared state below, when comparing to the constant-depth @@ -379,6 +388,7 @@ def sequential_circuit(N, g): # routine. Note, however, that instead of postselecting, we record the measurement # outcome and will use it further below. + def fuse(wire_0, wire_1): """Measure two qubits in the Bell basis and return the outcome encoded in two bits.""" @@ -386,15 +396,18 @@ def fuse(wire_0, wire_1): qml.Hadamard(wire_0) return np.array([qml.measure(i) for i in [wire_0, wire_1]]) + ###################################################################### # # Fusing two MPS that have been prepared by the sequential routine now is really simple: + def two_qubit_mps_by_fusion(g): sequential_preparation(g, [0, 1, 2]) sequential_preparation(g, [3, 4, 5]) mcms = fuse(2, 3) + ###################################################################### # However, the produced state is not quite the MPS state on two qubits, # because of the impurities caused through the fusion measurement. We can check this @@ -403,6 +416,7 @@ def two_qubit_mps_by_fusion(g): # If the two sequentially prepared MPS and the fusion step did prepare the correct # MPS already, the test measurement would just be :math:`\langle 00 | Z_0| 11\rangle=1`. + @qml.qnode(qml.device("default.qubit", wires=6)) def prepare_and_unprepare(g): two_qubit_mps_by_fusion(g) @@ -411,6 +425,7 @@ def prepare_and_unprepare(g): qml.adjoint(sequential_preparation)(g, [0, 1, 4, 5]) return qml.expval(qml.PauliZ(1)) + test = prepare_and_unprepare(g) print(f"The test measurement of the fusion preparation + sequential unpreparation is {test:.2f}") @@ -471,6 +486,7 @@ def prepare_and_unprepare(g): # This pushing and correction step is captured by the following function, where we use some additional # simplifications based on the used encoding with two bits of the pushed operator. + def push_and_correct(op_id, phys_wires): """Push an operator from left to right along the bond sites of an MPS and conditionally apply correcting operations on the corresponding physical sites. @@ -489,6 +505,7 @@ def push_and_correct(op_id, phys_wires): # Push through Y if input is X or Y return np.array([op_id[1], op_id[1]]) + def correction(idx, q, op_id): w = (idx + 1) * q op_int = np.array([2, 1]) @ op_id @@ -503,6 +520,7 @@ def correction(idx, q, op_id): qml.cond(op_int == 3, qml.X)(i) return np.array([op_id[1], op_id[1]]) + ###################################################################### # # Piecing everything together @@ -546,11 +564,14 @@ def correction(idx, q, op_id): # site after pushing through the operator. # + def xor(op_id_0, op_id_1): """Express logical XOR as "SUM - 2 * AND" on integers.""" return op_id_0 + op_id_1 - 2 * op_id_0 * op_id_1 + def correct_end_bond(bond_idx, op_id): + """Perform a correction on the end bond site.""" op_int = np.array([2, 1]) @ op_id qml.cond(op_int == 1, qml.X)(bond_idx) qml.cond(op_int == 2, qml.Z)(bond_idx) @@ -560,11 +581,11 @@ def correct_end_bond(bond_idx, op_id): def constant_depth(N, g, q): """Prepare the MPS |\Psi(g)> in constant depth.""" num_blocks = N // q - wires_per_block = q + 2 # 2 bond wires added + wires_per_block = q + 2 # 2 bond wires added # Step 1: Prepare small block MPS for i in range(num_blocks): - wires = list(range(wires_per_block * i, wires_per_block * (i+1))) + wires = list(range(wires_per_block * i, wires_per_block * (i + 1))) sequential_preparation(g, wires) # Step 2: Fusion with mid-circuit measurements @@ -574,34 +595,40 @@ def constant_depth(N, g, q): mcms.append(fuse(*bond_wires)) # Step 3: Push operators through to highest-index wire and correct phys. sites - pushed_op_id = np.array([0, 0]) # Start with identity + pushed_op_id = np.array([0, 0]) # Start with identity for i in range(1, num_blocks): - phys_wires = list(range(wires_per_block * i + 1, wires_per_block * (i+1) - 1)) - pushed_op_id = push_and_correct(xor(mcms[i-1], pushed_op_id), phys_wires) + phys_wires = list(range(wires_per_block * i + 1, wires_per_block * (i + 1) - 1)) + pushed_op_id = push_and_correct(xor(mcms[i - 1], pushed_op_id), phys_wires) # Step 4: Undo the pushed-through operator on the highest-index wire bond site. correct_end_bond(num_blocks * wires_per_block - 1, pushed_op_id) + def constant_depth_ansatz(N, g, q): + """Circuit ansatz for constant-depth preparation routine.""" num_blocks = N // q - wires_per_block = q + 2 # 2 bond wires added + wires_per_block = q + 2 # 2 bond wires added outer_bond_sites = [0, num_blocks * wires_per_block - 1] # Steps 1-4 constant_depth(N, g, q) # Step 5: Perform projective measurement on outer-most bond sites. project_measure(*outer_bond_sites) phys_wires = sum( - (list(range(wires_per_block * i + 1, wires_per_block * (i+1) - 1)) - for i in range(num_blocks)), - start = [], + ( + list(range(wires_per_block * i + 1, wires_per_block * (i + 1) - 1)) + for i in range(num_blocks) + ), + start=[], ) return phys_wires + @qml.qnode(dev) def constant_depth_circuit(N, g, q): phys_wires = constant_depth_ansatz(N, g, q) return qml.probs(wires=phys_wires) + ###################################################################### # # Great, we know built the full constant-depth circuit to prepare the MPS @@ -615,7 +642,7 @@ def constant_depth_circuit(N, g, q): q = 3 g = -0.8 qml.draw_mpl(constant_depth_circuit)(N, g, q) -#plt.show() +# plt.show() ###################################################################### # Let's check that the constant-depth circuit produces the same state, or probability @@ -652,7 +679,7 @@ def constant_depth_circuit(N, g, q): ax.set_yscale("log") ax.set_ylim([1e-3, 1]) ax.legend() -#plt.show() +# plt.show() ###################################################################### # In a more nuanced setting, let us produce the MPS in question for a series of values @@ -669,23 +696,27 @@ def constant_depth_circuit(N, g, q): @qml.qnode(dev, interface="numpy") def pre_correlations(N, g, q): phys_wires = constant_depth_ansatz(N, g, q) - return [qml.expval(qml.Z(1) @ qml.Z(i)) for i in phys_wires] + [qml.expval(qml.Z(i)) for i in phys_wires] + return [qml.expval(qml.Z(1) @ qml.Z(i)) for i in phys_wires] + [ + qml.expval(qml.Z(i)) for i in phys_wires + ] + def correlations(N, g, q): expvals = pre_correlations(N, g, q) prods = expvals[:-N] expvals = expvals[-N:] - correls = [prods[i] - expvals[1] * expvals[i] for i in range(N) if i!=1] + correls = [prods[i] - expvals[1] * expvals[i] for i in range(N) if i != 1] return correls + N = 18 q = 6 -gs = [-1/5, -2/5, -3/5, -4/5] +gs = [-1 / 5, -2 / 5, -3 / 5, -4 / 5] fig, ax = plt.subplots() all_correls = [] for g in gs: correls = correlations(N, g, q) - ax.plot(list(range(N-1)), correls, label=f"$g={g}$") + ax.plot(list(range(N - 1)), correls, label=f"$g={g}$") all_correls.append(correls) ax.set_yscale("log") ax.set_ylim([5e-8, 1.5]) @@ -700,14 +731,16 @@ def correlations(N, g, q): # We can also compare this result to the analytic result for the correlation # length :math:`xi` mentioned in the beginning: + def exp_fn(x, a, b): return a * np.exp(x * b) + print(f"Correlation lengths xi\n===================") for g, correls in zip(gs, all_correls): - popt, pcov = curve_fit(exp_fn, list(range(1, N//2+1)), correls[1:N//2+1]) - xi_numerical = -1/popt[1] - xi_predicted = 1 / abs(np.log((1+g) / (1-g))) + popt, pcov = curve_fit(exp_fn, list(range(1, N // 2 + 1)), correls[1 : N // 2 + 1]) + xi_numerical = -1 / popt[1] + xi_predicted = 1 / abs(np.log((1 + g) / (1 - g))) print(f"{g=:.1f}") print(f"Predicted: {xi_predicted:.5f}") print(f"Numerical: {xi_numerical:.5f}") From 8578a47957130539c485c9f632e4f1d7d6496c66 Mon Sep 17 00:00:00 2001 From: dwierichs Date: Tue, 13 Aug 2024 17:28:07 +0200 Subject: [PATCH 03/37] change file name, copy placeholder images --- ...humbnail_large_constant_depth_mps_prep.png | Bin 0 -> 96717 bytes .../thumbnail_constant_depth_mps_prep.png | Bin 0 -> 19508 bytes .../operator_pushing.png | Bin .../operator_pushing_example.png | Bin 0 -> 36918 bytes ...ocialthumbnail_constant_depth_mps_prep.png | Bin 0 -> 36918 bytes ...ial_constant_depth_mps_prep.metadata.json} | 26 +++++++++--------- ...py => tutorial_constant_depth_mps_prep.py} | 6 ++-- 7 files changed, 16 insertions(+), 16 deletions(-) create mode 100644 _static/demo_thumbnails/large_demo_thumbnails/thumbnail_large_constant_depth_mps_prep.png create mode 100644 _static/demo_thumbnails/regular_demo_thumbnails/thumbnail_constant_depth_mps_prep.png rename _static/demonstration_assets/{preparation_mps_mcm => constant_depth_mps_prep}/operator_pushing.png (100%) create mode 100644 _static/demonstration_assets/constant_depth_mps_prep/operator_pushing_example.png create mode 100644 _static/demonstration_assets/constant_depth_mps_prep/socialthumbnail_constant_depth_mps_prep.png rename demonstrations/{tutorial_preparation_mps_mcm.metadata.json => tutorial_constant_depth_mps_prep.metadata.json} (57%) rename demonstrations/{tutorial_preparation_mps_mcm.py => 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a/demonstrations/tutorial_preparation_mps_mcm.metadata.json +++ b/demonstrations/tutorial_constant_depth_mps_prep.metadata.json @@ -1,52 +1,52 @@ { - "title": "Introduction to mid-circuit measurements", + "title": "Constant-depth preparation of matrix product states with dynamic circuits", "authors": [ { "id": "david_wierichs" } ], - "dateOfPublication": "2024-05-10T00:00:00+00:00", - "dateOfLastModification": "2024-06-21T00:00:00+00:00", + "dateOfPublication": "2024-09-10T00:00:00+00:00", + "dateOfLastModification": "2024-09-10T00:00:00+00:00", "categories": [ - "Getting Started", - "Quantum Computing" + "Quantum Computing", + "Algorithms" ], "tags": [], "previewImages": [ { "type": "thumbnail", - "uri": "/_static/demonstration_assets/regular_demo_thumbnails/thumbnail_mcm_introduction.png" + "uri": "/_static/demonstration_assets/regular_demo_thumbnails/thumbnail_constant_depth_mps_prep.png" }, { "type": "large_thumbnail", - "uri": "/_static/large_demo_thumbnails/thumbnail_large_mcm_introduction.png" + "uri": "/_static/large_demo_thumbnails/thumbnail_large_constant_depth_mps_prep.png" } ], - "seoDescription": "Learn the basics of mid-circuit measurements and how to use them in PennyLane.", + "seoDescription": "Prepare a matrix product state in constant depth with mid-circuit measurements and feedforward control.", "doi": "", - "canonicalURL": "/qml/demos/tutorial_mcm_introduction", + "canonicalURL": "/qml/demos/tutorial_constant_depth_mps_prep", "references": [], "basedOnPapers": [], "referencedByPapers": [], "relatedContent": [ { "type": "demonstration", - "id": "tutorial_how_to_create_dynamic_mcm_circuits", + "id": "tutorial_mcm_introduction", "weight": 1.0 }, { "type": "demonstration", - "id": "tutorial_how_to_collect_mcm_stats", + "id": "tutorial_how_to_create_dynamic_mcm_circuits", "weight": 1.0 }, { "type": "demonstration", - "id": "tutorial_mbqc", + "id": "tutorial_tn_circuits", "weight": 1.0 }, { "type": "demonstration", - "id": "tutorial_teleportation", + "id": "tutorial_initial_state_preparation", "weight": 1.0 } ] diff --git a/demonstrations/tutorial_preparation_mps_mcm.py b/demonstrations/tutorial_constant_depth_mps_prep.py similarity index 99% rename from demonstrations/tutorial_preparation_mps_mcm.py rename to demonstrations/tutorial_constant_depth_mps_prep.py index 7c90e8d5cb..87e77c00dc 100644 --- a/demonstrations/tutorial_preparation_mps_mcm.py +++ b/demonstrations/tutorial_constant_depth_mps_prep.py @@ -8,7 +8,7 @@ as initial state in order to then perform operations that are no longer classically tractable. -.. figure:: ../_static/todo +.. figure:: ../_static/demonstration_assets/preparation_mps_mcm/socialthumbnail_constant_depth_mps_prep.png :align: center :width: 50% @@ -449,7 +449,7 @@ def prepare_and_unprepare(g): # :math:`C^k` and :math:`D^k` on the bond and physical axes, respectively. # Visually, we find: # -# .. image:: ../_static/demonstration_assets/preparation_mps_mcm/operator_pushing.png +# .. image:: ../_static/demonstration_assets/constant_depth_mps_prep/operator_pushing.png # # For simplicity, we denote a push by :math:`B^k\mapsto (C^k, D^k)`. # @@ -461,7 +461,7 @@ def prepare_and_unprepare(g): # the tensor :math:`A` of our example MPS :math:`|\Psi(g)\rangle`. # In particular, the relations are # -# .. image:: ../_static/demonstration_assets/preparation_mps_mcm/operator_pushing.png +# .. image:: ../_static/demonstration_assets/constant_depth_mps_prep/operator_pushing_example.png # # That is, in the notation above we have # :math:`X\mapsto (Y, Y)`, From d7ffcd97253006541486c874975f601d7e5bbd1e Mon Sep 17 00:00:00 2001 From: dwierichs Date: Tue, 13 Aug 2024 17:36:32 +0200 Subject: [PATCH 04/37] update --- demonstrations/tutorial_constant_depth_mps_prep.py | 10 +++++++--- 1 file changed, 7 insertions(+), 3 deletions(-) diff --git a/demonstrations/tutorial_constant_depth_mps_prep.py b/demonstrations/tutorial_constant_depth_mps_prep.py index 87e77c00dc..8edd8969bb 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.py +++ b/demonstrations/tutorial_constant_depth_mps_prep.py @@ -8,7 +8,7 @@ as initial state in order to then perform operations that are no longer classically tractable. -.. figure:: ../_static/demonstration_assets/preparation_mps_mcm/socialthumbnail_constant_depth_mps_prep.png +.. figure:: ../_static/demonstration_assets/constant_depth_mps_prep/socialthumbnail_constant_depth_mps_prep.png :align: center :width: 50% @@ -63,7 +63,7 @@ In particular, MPS can accurately describe ground states of (gapped local) one-dimensional Hamiltonians, which can be found efficiently using the density matrix renormalization group (DMRG). -For reviews of MPS see [#todo]_, [#todo]_, and [#todo]_. +For reviews of MPS see [#todo]. Following [#smith]_, we will look at translation-invariant MPS of a quantum :math:`N`-body system where each body, or site, has local (physical) @@ -753,11 +753,15 @@ def exp_fn(x, a, b): # Conclusion # ---------- # -###################################################################### +# The circuit works and it has constant depth, yay! +# It needs additional qubits, nay! +# This can alleviate connectivity issues (?), yay! +# Entanglement swapping works, and we have MPS available in yet another way now, yay! # # # References # ---------- +# # .. [#smith] # # Kevin C. Smith, Abid Khan, Bryan K. Clark, S.M. Girvin, Tzu-Chieh Wei From 9e30e0e1a19ea2a29d70745becf3f1eeda1f41fd Mon Sep 17 00:00:00 2001 From: dwierichs Date: Thu, 15 Aug 2024 19:33:42 +0200 Subject: [PATCH 05/37] fix link --- demonstrations/tutorial_constant_depth_mps_prep.py | 2 +- demonstrations/tutorial_eqnn_force_field.py | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/demonstrations/tutorial_constant_depth_mps_prep.py b/demonstrations/tutorial_constant_depth_mps_prep.py index 8edd8969bb..be58eea4bd 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.py +++ b/demonstrations/tutorial_constant_depth_mps_prep.py @@ -773,7 +773,7 @@ def exp_fn(x, a, b): # C. Schön, E. Solano, F. Verstraete, J. I. Cirac, M. M. Wolf # "Sequential Generation of Entangled Multiqubit States", Physical Review Letters, **95**, 110503, # `closed access `__, 2005. -# `arXiv:quant-ph/0501096 `__, 2005. +# `arXiv:quant-ph/0501096 `__, 2005. # # .. [#todo] # diff --git a/demonstrations/tutorial_eqnn_force_field.py b/demonstrations/tutorial_eqnn_force_field.py index 66dffd7edd..ec05c8aa2c 100644 --- a/demonstrations/tutorial_eqnn_force_field.py +++ b/demonstrations/tutorial_eqnn_force_field.py @@ -45,7 +45,7 @@ Equivariant Quantum Machine learning ------------------------------------ +------------------------------------ In order to incorporate symmetries into machine learning models, we need a few concepts from group theory. A formal course on the subject is out of the scope of the present document, which is why we refer to two other demos, `equivariant graph embedding `_ From abec6f709aef97529dfa827e61c4424e0f48aa60 Mon Sep 17 00:00:00 2001 From: dwierichs Date: Mon, 19 Aug 2024 16:23:05 +0200 Subject: [PATCH 06/37] fix file path for regular thumbnail --- demonstrations/tutorial_constant_depth_mps_prep.metadata.json | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_constant_depth_mps_prep.metadata.json b/demonstrations/tutorial_constant_depth_mps_prep.metadata.json index 54b390799a..7cb03bad40 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.metadata.json +++ b/demonstrations/tutorial_constant_depth_mps_prep.metadata.json @@ -15,7 +15,7 @@ "previewImages": [ { "type": "thumbnail", - "uri": "/_static/demonstration_assets/regular_demo_thumbnails/thumbnail_constant_depth_mps_prep.png" + "uri": "/_static/regular_demo_thumbnails/thumbnail_constant_depth_mps_prep.png" }, { "type": "large_thumbnail", From 8dd1a57bae607ecb0105e3b7f09e10b0b6dc4710 Mon Sep 17 00:00:00 2001 From: dwierichs Date: Mon, 19 Aug 2024 21:14:13 +0200 Subject: [PATCH 07/37] fix file path for regular thumbnail --- demonstrations/tutorial_constant_depth_mps_prep.metadata.json | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_constant_depth_mps_prep.metadata.json b/demonstrations/tutorial_constant_depth_mps_prep.metadata.json index 7cb03bad40..85cc4c0ded 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.metadata.json +++ b/demonstrations/tutorial_constant_depth_mps_prep.metadata.json @@ -15,7 +15,7 @@ "previewImages": [ { "type": "thumbnail", - "uri": "/_static/regular_demo_thumbnails/thumbnail_constant_depth_mps_prep.png" + "uri": "/_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_constant_depth_mps_prep.png" }, { "type": "large_thumbnail", From 15fc897ef7b9e10ef5e420352a63df0ebb82b532 Mon Sep 17 00:00:00 2001 From: dwierichs Date: Mon, 19 Aug 2024 23:34:42 +0200 Subject: [PATCH 08/37] polish --- .../tutorial_constant_depth_mps_prep.py | 400 +++++++++--------- 1 file changed, 204 insertions(+), 196 deletions(-) diff --git a/demonstrations/tutorial_constant_depth_mps_prep.py b/demonstrations/tutorial_constant_depth_mps_prep.py index be58eea4bd..a32ff82ac8 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.py +++ b/demonstrations/tutorial_constant_depth_mps_prep.py @@ -1,58 +1,54 @@ r"""Constant-depth preparation of matrix product states with dynamic circuits ============================================================================= -Matrix product states (MPS) are an important class of quantum states that is -used in a plethora of applications on quantum many-body systems. -This makes the preparation of MPS on a quantum computer a useful subroutine. -For example, a quantum algorithm could prepare a classically pre-computed MPS -as initial state in order to then perform operations that are no longer -classically tractable. +Matrix product states (MPS) are used in a plethora of applications on quantum +many-body systems, both on classical and quantum computers. +This makes the preparation of MPS on a quantum computer an important subroutine. .. figure:: ../_static/demonstration_assets/constant_depth_mps_prep/socialthumbnail_constant_depth_mps_prep.png :align: center :width: 50% -In this demo you will learn about a new algorithm to prepare MPS with a dynamic +In this demo you will learn to prepare certain MPS with a dynamic quantum circuit of constant depth. We will implement the circuit, which makes use of mid-circuit measurements and conditionally applied operations, for a specific -MPS, and compare it to a (static) sequential circuit from the literature with -linear depth. +MPS, and compare it to a (static) sequential circuit with linear depth. +We will closely follow the eponymous article by Smith et al. [#smith]_. -This demo closely follows the eponymous article by Smith et al. [#smith]_. +.. note:: -If you would like to first learn about the dynamic circuit tools used in this -algorithm, have a look at our :doc:`introduction to mid-circuit measurements -` and learn :doc:`how to collect statics of -mid-circuit measurements ` and -:doc:`how to create dynamic circuits with mid-circuit measurements -`. + If you would like to first learn about the dynamic circuit tools used in this + algorithm, have a look at our :doc:`introduction to mid-circuit measurements + ` and learn :doc:`how to collect statics of + mid-circuit measurements ` and + :doc:`how to create dynamic circuits with mid-circuit measurements + `. -If you first want to familiarize yourself with tensor network states and MPS -in particular, take a look at our demo on -:doc:`tensor-network quantum circuits `. -Finally, our demo on :doc:`initial state preparation for quantum chemistry -` shows you how to -proceed to a concrete application, once an MPS is prepared. + If you first want to familiarize yourself with tensor network states and MPS + in particular, take a look at our demo on + :doc:`tensor-network quantum circuits `. + Finally, our demo on :doc:`initial state preparation for quantum chemistry + ` shows you how to + proceed to a concrete application, once an MPS is prepared. Outline ------- -We will start with a brief (no, really!) mathematical overview of the used -objects and subroutines. Then we define them in code for a specific parametric MPS -and verify the conditions for the algorithm to apply. -One of the subroutines is the sequential MPS preparation circuit that we -compare to later. -Afterwards we combine the building blocks into the constant-depth algorithm -by Smith et al. We conclude by computing the correlation length of the prepared -MPS for different parameter values. +We will start by briefly (no, really!) introducing the building blocks for +the algorithm from a mathematical perspective. One of these blocks +is the sequential MPS preparation circuit to which we will compare later. +Alongside this introduction we describe and code up each building block for a +specific MPS and verify the conditions for the algorithm to apply. +Finally, we combine the building blocks into the constant-depth algorithm by +Smith et al. We conclude by computing the correlation length of the prepared MPS. Mathematical derivation ----------------------- -We briefly introduce matrix product states and a sequential circuit with linear depth -that prepares them, as well as two tools from quantum information theory -that we will use below, namely fusion of MPS with mid-circuit measurements -and so-called operator pushing. +We briefly introduce MPS and a sequential circuit with linear +depth to prepare them, as well as two tools from quantum information theory +that we will use below: fusion of MPS with mid-circuit measurements +and *operator pushing*. Matrix product states ~~~~~~~~~~~~~~~~~~~~~ @@ -60,9 +56,9 @@ Matrix product states (MPS) are an important class of quantum states. They can be described and manipulated conveniently on classical computers and are able to approximate relevant states in quantum many-body systems. -In particular, MPS can accurately describe ground states of (gapped local) -one-dimensional Hamiltonians, which can be found efficiently using the density -matrix renormalization group (DMRG). +In particular, MPS can efficiently describe ground states of (gapped local) +one-dimensional Hamiltonians, which in addition can be found efficiently using +density matrix renormalization group (DMRG) algorithms. For reviews of MPS see [#todo]. Following [#smith]_, we will look at translation-invariant MPS @@ -78,7 +74,7 @@ where :math:`\vec{m}` is a multi-index of :math:`N` indices ranging over :math:`d`, i.e., :math:`\vec{m}\in\{0, 1 \dots, d-1\}^N`, and :math:`\{A^{m}\}_{m}` are :math:`d` square matrices of equal dimension. This dimension :math:`D` is called the bond dimension, -and it is a crucial quantity for the expressivitiy and complexity of the MPS. +which is a crucial quantity for the expressivity and complexity of the MPS. We see that :math:`|\Psi\rangle` is fully specified by the :math:`d\times D\times D=dD^2` numbers in the rank-3 tensor :math:`A`. However, this specification is not unique. We remove some of the redundancies by assuming @@ -93,8 +89,8 @@ **Example** -As an example, consider an :math:`N`-site qubit (:math:`d=2`) chain and a :math:`D=2` MPS -:math:`|\Psi(g)\rangle` on this system defined by the matrices +As an example, consider a chain of :math:`N` qubits (:math:`d=2`) and a :math:`D=2` MPS +:math:`|\Psi(g)\rangle` on this system, defined by the matrices .. math:: @@ -104,21 +100,21 @@ where :math:`\eta = \sqrt{1-g}^{-1}` and :math:`g\in[-1, 0]` is a freely chosen parameter. -This MPS is an important example because it can be tuned from the long-range correlated +This MPS is a simple yet important example because it can be tuned from the long-range correlated GHZ state :math:`|\Psi(0)\rangle=(|0>^{\otimes N} + |1>^{\otimes N})/\sqrt(2)` to the state :math:`|\Psi(-1)\rangle=|+\rangle^{\otimes N}` with vanishing correlation length. -In general, the correlation length is given by +In general, the correlation length of :math:`|\Psi(g)\rangle` is given by .. math:: - \xi = \left|\ln\left(\frac{1+g}{1-g}\right)\right|^{-1}. + \xi(g) = \left|\ln\left(\frac{1+g}{1-g}\right)\right|^{-1}. Long-range correlated states require linear-depth unitary circuits in general. Therefore, the constant-depth circuit below for :math:`|\Psi(0)\rangle` will demonstrate that dynamic quantum circuits are more powerful than unitary operations alone. The MPS :math:`|\Psi(g)\rangle` will be our working example throughout this demo. Therefore -we warm up our coding by defining the tensor :math:`A` and testing some of its properties. +we warm up our coding by defining the tensor :math:`A` and testing its properties. """ import numpy as np @@ -151,22 +147,21 @@ def A(g): # An MPS like the one above can be prepared in linear depth using an established technique # by Schön et al. [#schoen]_. We introduce this technique here because the new # constant-depth construction uses it as a parallelized subroutine, and because we will -# later compare the new construction to the sequential circuit. +# later compare the two approaches. # # The MPS :math:`|\Psi\rangle` above is given by the rank-3 tensor :math:`A`. # In order to prepare the state in a circuit, we want to construct unitary operations # from :math:`A`. For this, we build a new rank-4 tensor :math:`U` with an additional # axis of dimension :math:`d`. If this new physical input axis is in the state -# :math:`|0\rangle`, we define :math:`U` to reproduce one :math:`A^m` for each physical +# :math:`|0\rangle`, we demand :math:`U` to reproduce one :math:`A^m` for each physical # output state :math:`|m\rangle`. We leave the remaining action undefined and only require # it to make :math:`U` unitary overall. In short, this can be written as # # .. math:: # -# U = \sum_m A^m \otimes |m\rangle\!\langle 0| + C_\perp. +# U = \sum_m A^m \otimes |m\rangle\!\langle 0| + C_\perp, # -# Let us denote :math:`U` acting on the :math:`i`-th physical site as :math:`U^{(i)}` -# for a fixed non-physical site. +# where :math:`C_\perp` can be any operator making :math:`U` unitary. # The sequential preparation circuit now consists of the following steps. # # #. Start in the state :math:`|\psi_0\rangle = |0_d\rangle^{\otimes N}\otimes |00\rangle_D`. @@ -175,7 +170,8 @@ def A(g): # #. Entangle the bond qudits into the state :math:`|I\rangle = 1/\sqrt{D}\sum_j |jj\rangle_D`. # # #. Apply the unitary :math:`U` once to each of the :math:`N` physical sites, with the -# :math:`D`-dimensional factor always acting on the first bond qudit. This produces the state +# :math:`D`-dimensional factor always acting on the first bond qudit, denoted by :math:`U^{(i)}`. +# This produces the state # # .. math:: # @@ -188,14 +184,14 @@ def A(g): # \sum_{\vec{m}} |\vec{m}\rangle \langle i|A^{m_1} A^{m_2}\cdots A^{m_N}|j\rangle_D |i\rangle_D|j\rangle_D # # #. Measure the two bond qudits in the (generalized) Bell basis and postselect on the outcome -# being :math:`|I\rangle`. Then discard the bond qudits. This collapses :math:`\psi_1\rangle` -# into the (renormalized) state +# being :math:`|I\rangle`. Then discard the bond qudits, which collapses :math:`\psi_1\rangle` +# into the state # # .. math:: # # |\psi_2\rangle = \sum_{\vec{m}} \operatorname{tr}[A^{m_1}A^{m_2}\cdots A^{m_N}]|\vec{m}\rangle = |\Psi\rangle # -# Note that this step is probabilistic and we only succeed to produce the state if we measured +# Note that this step is *probabilistic* and we only succeed to produce the state if we measure # the state :math:`|I\rangle`. # # We see that we prepared :math:`|\Psi\rangle` with an entangling operation on the bond qudits, one unitary per @@ -234,7 +230,8 @@ def A(g): # # .. math:: # -# U &= A^0 \otimes (|0\rangle\!\langle 0| + |1\rangle\!\langle 1|) + A^1\otimes (|1\rangle\!\langle 0| + |0\rangle\!\langle 1|)\\ +# U &= A^0 \otimes (|0\rangle\!\langle 0| + |1\rangle\!\langle 1|) +# + A^1\otimes (|1\rangle\!\langle 0| + |0\rangle\!\langle 1|)\\ # &= A^0 \otimes \mathbb{I} + A^1\otimes X # # This looks a lot like a :class:`~.pennylane.CNOT` gate plays a part in :math:`U`. @@ -246,8 +243,9 @@ def A(g): # &= A^0 P_0 \otimes \mathbb{I} + A^0P_1 \otimes X + A^1 P_0 \otimes X + A^1P_1 \otimes \mathbb{I}\\ # &= \left(\begin{matrix} \eta & -\eta\sqrt{-g} \\ \eta\sqrt{-g} & \eta \end{matrix}\right), # -# where we denoted the computational basis state projectors as :math:`P_i = |i\rangle\!\langle i|`. -# The remaining operation is a Pauli-:math:`Y` rotation by the angle :math:`2\arccos(\eta)`, so +# where we denoted the computational basis state projectors as +# :math:`P_i = |i\rangle\!\langle i|`. The remaining operation is an +# :class:`~.pennylane.RY` rotation by the angle :math:`2\arccos(\eta)`, so # that we can easily code up the unitary as a small circuit template: @@ -262,8 +260,8 @@ def U_template(eta, bond_idx, phys_idx): ###################################################################### # Great, now that we have a circuit that realizes the unitary :math:`U`, let's code up the entire -# sequential preparation circuit. We will leave the measurement \& postselect step as a separate -# function for now. +# sequential preparation circuit. We implement the measurement \& postselect step as a separate +# function ``project_measure``. def prepare_bell(wire_0, wire_1): @@ -272,16 +270,6 @@ def prepare_bell(wire_0, wire_1): qml.CNOT([wire_0, wire_1]) -def project_measure(wire_0, wire_1): - """Measure two qubits in the Bell basis and - postselect on (|00>+|11>)/sqrt(2).""" - # Move bond qubits to Bell basis by undoing the preparation - qml.adjoint(prepare_bell)(wire_0, wire_1) - # Measure and postselect |00> - qml.measure(wire_0, postselect=0) - qml.measure(wire_1, postselect=0) - - def sequential_preparation(g, wires): """Prepare the example MPS \Psi(g) on N qubits where N is the length of the passed wires minus 2. The bond qubits are still entangled.""" @@ -293,6 +281,15 @@ def sequential_preparation(g, wires): U_template(eta, bond_wires[1], phys_wire) +def project_measure(wire_0, wire_1): + """Measure two qubits in the Bell basis and postselect on (|00>+|11>)/sqrt(2).""" + # Move bond qubits to Bell basis by undoing the preparation + qml.adjoint(prepare_bell)(wire_0, wire_1) + # Measure and postselect |00> + qml.measure(wire_0, postselect=0) + qml.measure(wire_1, postselect=0) + + dev = qml.device("default.qubit") @@ -307,22 +304,25 @@ def sequential_circuit(N, g): ###################################################################### # We will verify the prepared state below, when comparing to the constant-depth -# preparation circuit. Let's just draw it for now. +# preparation circuit. Let's just draw the circuit for now. -N = 12 +N = 10 qml.draw_mpl(sequential_circuit)(N, g) ###################################################################### +# As we can see, the sequential preparation circuit already uses mid-circuit +# measurements. However, there is no feed forward control that modifies the circuit +# dynamically based on measured values. # # Fusion of MPS states with mid-circuit measurements # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # -# The next ingredient to the constant-depth MPS preparation circuit is to fuse -# MPS states on different qubits together into one MPS. Under the hood, this is +# The next ingredient for the constant-depth MPS preparation circuit is to fuse +# two MPS states together into one MPS. Under the hood, this is # an application of entanglement swapping. # -# Consider a state :math:`|\Phi\rangle=|\Psi_1\rangle\otimes|\Phi_2\rangle`, where -# :math:`|\Psi_{1,2}\rangle` are MPS prepared with the sequential technique from +# Consider a state :math:`|\Phi\rangle=|\Phi_1\rangle\otimes|\Phi_2\rangle`, where +# :math:`|\Phi_{1,2}\rangle` are MPS prepared with the sequential technique from # above on :math:`K` and :math:`L` physical qudits, respectively, but with the bond # qudits left unmeasured. In particular, the postselection step has not been performed # yet. Using the form of :math:`|\psi_1\rangle` from the recipe above, we can write @@ -335,9 +335,9 @@ def sequential_circuit(N, g): # \langle i|A^{m_1} \cdots A^{m_K}|j\rangle_D # \langle \ell|A^{m_{K+1}} \cdots A^{m_{K+L}}|p\rangle_D. # -# Now we want to measure two bond qudits, one of :math:`|\Psi_1\rangle` and -# :math:`|\Psi_2\rangle` each, in an entangled basis. Assume this basis to be given -# in the form of (orthogonal) states +# Now we want to measure two bond qudits in an entangled basis, one of +# :math:`|\Phi_1\rangle` and :math:`|\Phi_2\rangle` each. Assume this basis +# to be given in the form of (orthogonal) states # # .. math:: # @@ -355,16 +355,15 @@ def sequential_circuit(N, g): # # where we used that :math:`\sum_{j,\ell} |j\rangle B^k_{j\ell}\langle\ell|=B^k`. # We now can measure the two bond qudits and will know that for a given measurement -# outcome :math:`k`, we obtained the MPS state on :math:`K+L` qubits, except for -# a small "impurity", namely the matrix :math:`B^k` corresponding to the outcome. +# outcome :math:`k`, we obtained the MPS state on :math:`K+L` qubits, up to +# a small "impurity", namely the matrix :math:`B^k` that depends to the outcome. # -# In this generality, the fusion strategy may seem complicated, but it can take +# In full generality, the fusion strategy may seem complicated, but it can take # a very simply form, as we will now see in our example: # # **Example** # -# For our example we will use the Bell basis to perform the fusing mid-circuit -# measurement, i.e., +# We will use the Bell basis to perform the mid-circuit measurement, i.e., # # .. math:: # @@ -375,8 +374,7 @@ def sequential_circuit(N, g): # # This choice must seem arbitrary at this point, but will become clear # later on. There, we will see that in general the matrices :math:`B^k` and the MPS -# tensor :math:`A` have to play well together. -# For the choice above, we find the matrices +# tensor :math:`A` have to "play well" together. For the choice above, we find the matrices # # .. math:: # @@ -386,7 +384,7 @@ def sequential_circuit(N, g): # The measurement procedure can then be coded up in the following short function, # which is similar to the ``project_measure`` function from the sequential preparation # routine. Note, however, that instead of postselecting, we record the measurement -# outcome and will use it further below. +# outcome in the form of two bits, to be used further below. def fuse(wire_0, wire_1): @@ -394,12 +392,12 @@ def fuse(wire_0, wire_1): encoded in two bits.""" qml.CNOT([wire_0, wire_1]) qml.Hadamard(wire_0) - return np.array([qml.measure(i) for i in [wire_0, wire_1]]) + return np.array([qml.measure(w) for w in [wire_0, wire_1]]) ###################################################################### -# -# Fusing two MPS that have been prepared by the sequential routine now is really simple: +# Fusing two MPS together that have been prepared by the sequential routine +# now is really simple: def two_qubit_mps_by_fusion(g): @@ -409,12 +407,12 @@ def two_qubit_mps_by_fusion(g): ###################################################################### -# However, the produced state is not quite the MPS state on two qubits, -# because of the impurities caused through the fusion measurement. We can check this -# by undoing the preparation with the two qubit sequential circuit and measuring an -# exemplary expectation value. -# If the two sequentially prepared MPS and the fusion step did prepare the correct -# MPS already, the test measurement would just be :math:`\langle 00 | Z_0| 11\rangle=1`. +# However, as mentioned above, the produced state is not quite the MPS state on two qubits, +# because of the impurities caused by the fusion measurement. We can check this +# by undoing the fusion-based preparation with the two-qubit sequential circuit +# and measuring an exemplary expectation value. +# If the two separately prepared MPS and the fusion step did prepare the correct +# MPS already, the test measurement would just be :math:`\langle 00 | Z_0| 00\rangle=1`. @qml.qnode(qml.device("default.qubit", wires=6)) @@ -430,9 +428,9 @@ def prepare_and_unprepare(g): print(f"The test measurement of the fusion preparation + sequential unpreparation is {test:.2f}") ###################################################################### -# This means we still need a way to remove the "impurity" matrices :math:`B^k` from the +# This means we still need a way to remove the impurity matrices :math:`B^k` from the # state that ended up in the prepared state when we performed the fusion measurement. -# Operator pushing will allow us to do just that! +# *Operator pushing* will allow us to do exactly that! # # Operator pushing # ~~~~~~~~~~~~~~~~ @@ -457,9 +455,9 @@ def prepare_and_unprepare(g): # # In the fusion step above we picked the Bell basis as measurement basis, without # further specifying why. As we saw, this basis leads to Pauli matrices as impurities -# :math:`B^k`. This choice is such that they satisfy pushing relations together with -# the tensor :math:`A` of our example MPS :math:`|\Psi(g)\rangle`. -# In particular, the relations are +# :math:`B^k`. It turns out that those matrices satisfy simple pushing relations together with +# the tensor :math:`A` of our example MPS :math:`|\Psi(g)\rangle`, explaining this choice of +# measurement basis. In particular, the relations are # # .. image:: ../_static/demonstration_assets/constant_depth_mps_prep/operator_pushing_example.png # @@ -470,22 +468,28 @@ def prepare_and_unprepare(g): # :math:`I\mapsto (I, I)`. # Note that the bond axis operator either is trivial (the identity), or a Pauli-:math:`Y` operator. # These pushing relations allow us to push the bond axis operator through to an open boundary -# bond axis of the fused MPS, and to remove the physical axis operators by applying the correct -# Pauli operation to the physical site, conditioned on the fusion measurement outcomes. -# If we have an MPS on :math:`L` sites, pushing through a Pauli will have the following effects: +# bond axis of the fused MPS. In addition, we can remove the operators on the physical axes by +# applying the correct Pauli operation to the physical site, conditioned on the fusion measurement +# outcomes. If we have an MPS on :math:`L` sites, pushing through a Pauli will have the +# following effect: # # - If it is a Pauli-:math:`Y`, all physical sites need to be corrected by a Pauli-:math:`X`, and -# a Pauli-:math:`Y` is pushed to the opposite boundary. +# a Pauli-:math:`Y` is pushed to the opposite bond site. # # - If it is a Pauli-:math:`X`, a Pauli-:math:`Y` is applied to the first physical qubit, and afterwards # the case above is recovered, leading to applying Pauli-:math:`X` and pushing through a Pauli-:math:`Y`. # # - If it is a Pauli-:math:`Z`, a Pauli-:math:`Z` is applied to the first physical qubit and we are done. -# No operator is pushed through. +# No operator is pushed through on the bond axis. # -# This pushing and correction step is captured by the following function, where we use some additional -# simplifications based on the used encoding with two bits of the pushed operator. - +# We can recast this condition into the following dynamic circuit instructions: +# +# - If the first (second) bit of the measurement outcome bitstring is active, apply a Pauli-:math:`Z` +# (Pauli-:math:`Y`) to the first physical qubit +# +# - If the second bit is active, apply a Pauli-:math:`X` to all other physical qubits. +# +# This pushing and correction step is captured by the following function. def push_and_correct(op_id, phys_wires): """Push an operator from left to right along the bond sites of an MPS and @@ -494,75 +498,66 @@ def push_and_correct(op_id, phys_wires): - The physical MPS sites are given by phys_wires """ w = phys_wires[0] - # Apply Y if input is X or Y - qml.cond(op_id[1] & op_id[0], qml.X)(w) + # Apply Z if input is Z or Y - qml.cond(op_id[1] & ~op_id[0], qml.Y)(w) - qml.cond(~op_id[1] & op_id[0], qml.Z)(w) + qml.cond(op_id[0], qml.Z)(w) + # Apply Y if input is X or Y + qml.cond(op_id[1], qml.Y)(w) # Apply X on other physical sites if input is X or Y for i in phys_wires[1:]: qml.cond(op_id[1], qml.X)(i) # Push through Y if input is X or Y return np.array([op_id[1], op_id[1]]) - -def correction(idx, q, op_id): - w = (idx + 1) * q - op_int = np.array([2, 1]) @ op_id - # If measurement syndrome is X, apply Y and push Y's along, leading to consecutive X - qml.cond(op_int == 1, qml.Y)(w) - for i in range(w + 1, w + q): - qml.cond(op_int == 1, qml.X)(i) - # If measurement syndrome is Z, apply Z and terminate - qml.cond(op_int == 2, qml.Z)(w) - # If measurement syndrome is Y, apply X and push Y's along, leading to consecutive X - for i in range(w, w + q): - qml.cond(op_int == 3, qml.X)(i) - return np.array([op_id[1], op_id[1]]) - - ###################################################################### -# # Piecing everything together # --------------------------- # -# Now that we discussed the sequential preparation algorithm, fusion with mid-circuit measurements, -# and operator pushing, we have all components for the constant-depth preparation algorithm. It works -# as follows (recall that showing existence and finding suitable operator pushing relations is a -# crucial step in general, which goes beyond the scope of the demo). +# Now that we discussed the sequential preparation algorithm, fusion with mid-circuit +# measurements, and operator pushing, we have all components for the constant-depth +# preparation algorithm. # -# #. For a block size :math:`q`, prepare :math:`\frac{N}{q}` MPS of size :math:`q` in parallel, using -# the sequential preparation algorithm (without the final projection step). +# #. For a block size :math:`q`, prepare :math:`\frac{N}{q}` MPS of size :math:`q` +# in parallel, using the sequential preparation algorithm (without the final +# projection step). # -# #. Fuse together the blocks of MPS using mid-circuit measurements and store the measurement outcomes. +# #. Fuse together the blocks of MPS using mid-circuit measurements and store the +# measurement outcomes. # -# #. Push the "impurity" operators resulting from the measurements to one of the outer bond sites and +# #. Push the resulting impurity operators to one of the outer bond sites and # conditionally apply correction operators to the physical sites. # -# #. Undo the operator that was pushed to the outer bond site by conditionally applying its inverse. +# #. Undo the operator that was pushed to the outer bond site by conditionally applying +# its inverse. # -# #. Perform the same projection step as in the sequential preparation algorithm on the two remaining -# bond sites. +# #. Perform the same projection step as in the sequential preparation algorithm on +# the two remaining bond sites. +# +# It is important to remember that showing the existence of suitable operator pushing +# relations (and finding them explicitly) is a crucial step in general, which goes +# beyond the scope of the demo. # # Block size :math:`q` # ~~~~~~~~~~~~~~~~~~~~ -# The size of the blocks that have to be prepared in the first step depends on multiple considerations. -# First, the block must be such that operator pushing relations are available. Usually this will -# determine a minimal block size, and multiples of that size are valid choices as well. -# Second, the depth of the (parallelized) sequential preparation circuits is proportional to :math:`q`, -# determining a key contribution to the overall depth of the algorithm. -# Third, the sequential algorithm requires two bond qudits for each block, leading to -# :math:`\frac{2N}{q}` auxiliary qudits overall. # +# The size of the blocks that have to be prepared in the first step depends on multiple +# considerations. First, the block must be such that operator pushing relations are +# available. Often this will determine a minimal block size, and multiples of that +# size are valid choices as well. Second, the depth of the (parallelized) sequential +# preparation circuits is proportional to :math:`q`, determining a key contribution +# to the overall depth of the algorithm. Third, the sequential algorithm requires +# two bond qudits for each block, leading to :math:`\frac{2N}{q}` auxiliary qudits +# overall. Note how the product of the depth and the number of auxiliary qubits is +# linear in :math:`N`. # # **Example** # -# For our example MPS :math:`|\Psi(g)\rangle`, we already defined and coded up the sequential preparation -# circuit for a flexible number of qubits, the measurement-based fusion, and the operator pushing -# and correction step. This makes putting the algorithm together quite simple. We only define an XOR -# for our operator IDs and a function that applies a correcting Pauli operator to the final bond -# site after pushing through the operator. -# +# For our example MPS :math:`|\Psi(g)\rangle`, we already defined and coded up the +# sequential preparation circuit for a flexible number of qubits (Step 1), +# the measurement-based fusion (Step 2), and the operator pushing and correction step +# (Step 3). This makes putting the algorithm together quite simple. We only need to +# define an XOR for our operator bit strings and a function that applies a correcting +# Pauli operator to the final bond site after pushing through the operator (Step 4). def xor(op_id_0, op_id_1): @@ -572,56 +567,57 @@ def xor(op_id_0, op_id_1): def correct_end_bond(bond_idx, op_id): """Perform a correction on the end bond site.""" - op_int = np.array([2, 1]) @ op_id - qml.cond(op_int == 1, qml.X)(bond_idx) - qml.cond(op_int == 2, qml.Z)(bond_idx) - qml.cond(op_int == 3, qml.Y)(bond_idx) + # Apply Z if correction op is Z or Y + qml.cond(op_id[0], qml.Z)(bond_idx) + # Apply X if correction op is X or Y + qml.cond(op_id[1], qml.X)(bond_idx) + #op_int = np.array([2, 1]) @ op_id + #qml.cond(op_int == 1, qml.X)(bond_idx) + #qml.cond(op_int == 2, qml.Z)(bond_idx) + #qml.cond(op_int == 3, qml.Y)(bond_idx) def constant_depth(N, g, q): """Prepare the MPS |\Psi(g)> in constant depth.""" num_blocks = N // q - wires_per_block = q + 2 # 2 bond wires added + block_wires = q + 2 # 2 bond wires added # Step 1: Prepare small block MPS for i in range(num_blocks): - wires = list(range(wires_per_block * i, wires_per_block * (i + 1))) + wires = list(range(block_wires * i, block_wires * (i + 1))) sequential_preparation(g, wires) # Step 2: Fusion with mid-circuit measurements mcms = [] for i in range(1, num_blocks): - bond_wires = (wires_per_block * i - 1, wires_per_block * i) + bond_wires = (block_wires * i - 1, block_wires * i) mcms.append(fuse(*bond_wires)) # Step 3: Push operators through to highest-index wire and correct phys. sites pushed_op_id = np.array([0, 0]) # Start with identity for i in range(1, num_blocks): - phys_wires = list(range(wires_per_block * i + 1, wires_per_block * (i + 1) - 1)) + phys_wires = list(range(block_wires * i + 1, block_wires * (i + 1) - 1)) pushed_op_id = push_and_correct(xor(mcms[i - 1], pushed_op_id), phys_wires) # Step 4: Undo the pushed-through operator on the highest-index wire bond site. - correct_end_bond(num_blocks * wires_per_block - 1, pushed_op_id) + correct_end_bond(num_blocks * block_wires - 1, pushed_op_id) def constant_depth_ansatz(N, g, q): """Circuit ansatz for constant-depth preparation routine.""" num_blocks = N // q - wires_per_block = q + 2 # 2 bond wires added - outer_bond_sites = [0, num_blocks * wires_per_block - 1] + block_wires = q + 2 + outer_bond_sites = [0, num_blocks * block_wires - 1] # Steps 1-4 constant_depth(N, g, q) # Step 5: Perform projective measurement on outer-most bond sites. project_measure(*outer_bond_sites) - phys_wires = sum( - ( - list(range(wires_per_block * i + 1, wires_per_block * (i + 1) - 1)) - for i in range(num_blocks) - ), - start=[], + # Collect wire ranges for physical wires, skipping bond wires + phys_wires = ( + range(block_wires * i + 1, block_wires * (i + 1) - 1) for i in range(num_blocks) ) - return phys_wires - + # Turn ranges to lists and sum them together + return sum(map(list, phys_wires), start=[]) @qml.qnode(dev) def constant_depth_circuit(N, g, q): @@ -631,9 +627,8 @@ def constant_depth_circuit(N, g, q): ###################################################################### # -# Great, we know built the full constant-depth circuit to prepare the MPS +# Great, we now built the full constant-depth circuit to prepare the MPS # :math:`|\Psi(g)\rangle`. -# # Before we evaluate the states it produces, let's see how the circuit looks. # We'll add some boxes for the individual subroutines. # @@ -642,20 +637,20 @@ def constant_depth_circuit(N, g, q): q = 3 g = -0.8 qml.draw_mpl(constant_depth_circuit)(N, g, q) -# plt.show() +# TODO: BOXES ###################################################################### -# Let's check that the constant-depth circuit produces the same state, or probability -# distribution, as the sequential circuit. +# Let's check that the constant-depth circuit produces the same probability +# distribution as the sequential circuit. # p_seq = sequential_circuit(N, g) p_const = constant_depth_circuit(N, g, q) -print(f"The two circuits agree on the probabilities for {g=:.1f}: {np.allclose(p_seq, p_const)}") +print(f"The two probabilities agree for {g=:.1f}: {np.allclose(p_seq, p_const)}") ###################################################################### -# Great, we now have a constant-depth circuit that reproduces the existing sequential +# Great, we have a constant-depth circuit that reproduces the existing sequential # preparation circuit, which has linear depth! # # Correlation length @@ -664,7 +659,8 @@ def constant_depth_circuit(N, g, q): # When introducing the example MPS :math:`|\Psi(g)\rangle`, we mentioned that the # parameter :math:`g` controls the correlation length of the state. As extreme examples, # we can produce a zero-correlation length state for :math:`g=-1` and a long-range -# correlated state, namely the GHZ state, for :math:`g\to 0`. +# correlated state, namely the GHZ state, for :math:`g\to 0`. We can visualize +# these states via their probability distributions (note the logarithmic axis). # N = 6 @@ -679,16 +675,15 @@ def constant_depth_circuit(N, g, q): ax.set_yscale("log") ax.set_ylim([1e-3, 1]) ax.legend() -# plt.show() ###################################################################### -# In a more nuanced setting, let us produce the MPS in question for a series of values -# for :math:`g`, and measure the correlation length ourselves. +# In a more nuanced setting, let us produce the MPS in question for a series of +# values for :math:`g`, and measure the correlation length ourselves. # For this, we measure the two-body correlators # # .. math:: # -# C(j) = \langle Z_0 Z_j\rangle - \langle Z_0\rangle \langle Z_1\rangle +# C(j) = \langle Z_1 Z_j\rangle - \langle Z_1\rangle \langle Z_j\rangle # # for an increasing distance :math:`j`. @@ -696,20 +691,23 @@ def constant_depth_circuit(N, g, q): @qml.qnode(dev, interface="numpy") def pre_correlations(N, g, q): phys_wires = constant_depth_ansatz(N, g, q) - return [qml.expval(qml.Z(1) @ qml.Z(i)) for i in phys_wires] + [ - qml.expval(qml.Z(i)) for i in phys_wires - ] + return ( + [qml.expval(qml.Z(1) @ qml.Z(i)) for i in phys_wires] + + [qml.expval(qml.Z(i)) for i in phys_wires] + ) def correlations(N, g, q): + # Compute expectation values for correlators expvals = pre_correlations(N, g, q) prods = expvals[:-N] expvals = expvals[-N:] - correls = [prods[i] - expvals[1] * expvals[i] for i in range(N) if i != 1] + # Exclude the self-correlation C(1) + correls = [prods[i] - expvals[0] * expvals[i] for i in range(N) if i != 0] return correls -N = 18 +N = 12 q = 6 gs = [-1 / 5, -2 / 5, -3 / 5, -4 / 5] fig, ax = plt.subplots() @@ -736,14 +734,16 @@ def exp_fn(x, a, b): return a * np.exp(x * b) -print(f"Correlation lengths xi\n===================") +mid_point = N // 2 + 1 +distances = list(range(1, mid_point)) +print(f"Correlation lengths xi\n======================") for g, correls in zip(gs, all_correls): - popt, pcov = curve_fit(exp_fn, list(range(1, N // 2 + 1)), correls[1 : N // 2 + 1]) + popt, pcov = curve_fit(exp_fn, distances, correls[1 : mid_point]) xi_numerical = -1 / popt[1] xi_predicted = 1 / abs(np.log((1 + g) / (1 - g))) print(f"{g=:.1f}") - print(f"Predicted: {xi_predicted:.5f}") print(f"Numerical: {xi_numerical:.5f}") + print(f"Predicted: {xi_predicted:.5f}") print() ###################################################################### @@ -753,11 +753,19 @@ def exp_fn(x, a, b): # Conclusion # ---------- # -# The circuit works and it has constant depth, yay! -# It needs additional qubits, nay! -# This can alleviate connectivity issues (?), yay! -# Entanglement swapping works, and we have MPS available in yet another way now, yay! -# +# We successfully implemented a constand-depth dynamic quantum circuit that +# prepares an MPS with a parametrized correlation length. In particular, +# we saw that a dynamic quantum circuit can reach quantum states in constant depth +# that require linear depth with purely unitary circuits. +# The cost for this advantage are the auxiliary qubits we had to add as intermediate +# bond sites. Nonetheless, this constant-depth algorithm is an exciting advance in +# state preparation techniques, alleviating requirements of coherence times and +# connectivity on hardware. +# +# For more information, consider the original article, the mentioned reviews, as well +# as our demos on dynamic circuits and tensor network states. +# +# Happy preparing! # # References # ---------- From 87f2d906cc55ef18012fe4d5a2b3a6eb19d07bc7 Mon Sep 17 00:00:00 2001 From: dwierichs Date: Tue, 20 Aug 2024 09:31:21 +0200 Subject: [PATCH 09/37] plot fancy --- .../tutorial_constant_depth_mps_prep.py | 85 ++++++++++++++----- 1 file changed, 63 insertions(+), 22 deletions(-) diff --git a/demonstrations/tutorial_constant_depth_mps_prep.py b/demonstrations/tutorial_constant_depth_mps_prep.py index a32ff82ac8..b3c73be905 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.py +++ b/demonstrations/tutorial_constant_depth_mps_prep.py @@ -119,6 +119,7 @@ import numpy as np import matplotlib.pyplot as plt +from matplotlib.patches import FancyBboxPatch from scipy.optimize import curve_fit import pennylane as qml @@ -284,7 +285,8 @@ def sequential_preparation(g, wires): def project_measure(wire_0, wire_1): """Measure two qubits in the Bell basis and postselect on (|00>+|11>)/sqrt(2).""" # Move bond qubits to Bell basis by undoing the preparation - qml.adjoint(prepare_bell)(wire_0, wire_1) + qml.CNOT([wire_0, wire_1]) + qml.Hadamard(wire_0) # Measure and postselect |00> qml.measure(wire_0, postselect=0) qml.measure(wire_1, postselect=0) @@ -491,6 +493,7 @@ def prepare_and_unprepare(g): # # This pushing and correction step is captured by the following function. + def push_and_correct(op_id, phys_wires): """Push an operator from left to right along the bond sites of an MPS and conditionally apply correcting operations on the corresponding physical sites. @@ -509,6 +512,7 @@ def push_and_correct(op_id, phys_wires): # Push through Y if input is X or Y return np.array([op_id[1], op_id[1]]) + ###################################################################### # Piecing everything together # --------------------------- @@ -532,7 +536,7 @@ def push_and_correct(op_id, phys_wires): # # #. Perform the same projection step as in the sequential preparation algorithm on # the two remaining bond sites. -# +# # It is important to remember that showing the existence of suitable operator pushing # relations (and finding them explicitly) is a crucial step in general, which goes # beyond the scope of the demo. @@ -571,10 +575,10 @@ def correct_end_bond(bond_idx, op_id): qml.cond(op_id[0], qml.Z)(bond_idx) # Apply X if correction op is X or Y qml.cond(op_id[1], qml.X)(bond_idx) - #op_int = np.array([2, 1]) @ op_id - #qml.cond(op_int == 1, qml.X)(bond_idx) - #qml.cond(op_int == 2, qml.Z)(bond_idx) - #qml.cond(op_int == 3, qml.Y)(bond_idx) + # op_int = np.array([2, 1]) @ op_id + # qml.cond(op_int == 1, qml.X)(bond_idx) + # qml.cond(op_int == 2, qml.Z)(bond_idx) + # qml.cond(op_int == 3, qml.Y)(bond_idx) def constant_depth(N, g, q): @@ -613,12 +617,11 @@ def constant_depth_ansatz(N, g, q): # Step 5: Perform projective measurement on outer-most bond sites. project_measure(*outer_bond_sites) # Collect wire ranges for physical wires, skipping bond wires - phys_wires = ( - range(block_wires * i + 1, block_wires * (i + 1) - 1) for i in range(num_blocks) - ) + phys_wires = (range(block_wires * i + 1, block_wires * (i + 1) - 1) for i in range(num_blocks)) # Turn ranges to lists and sum them together return sum(map(list, phys_wires), start=[]) + @qml.qnode(dev) def constant_depth_circuit(N, g, q): phys_wires = constant_depth_ansatz(N, g, q) @@ -633,13 +636,52 @@ def constant_depth_circuit(N, g, q): # We'll add some boxes for the individual subroutines. # -N = 12 +N = 9 q = 3 g = -0.8 -qml.draw_mpl(constant_depth_circuit)(N, g, q) -# TODO: BOXES +fig, ax = qml.draw_mpl(constant_depth_circuit)(N, g, q) + +# Cosmetics +options = { + "facecolor": "white", + "linewidth": 2, + "zorder": -1, + "boxstyle": "round, pad=0.1", +} +text_options = {"fontsize": 15, "ha": "center", "va": "top"} +box_data = [ + ((-0.45, -0.35), 1.7, 4.7, "#FF87EB"), # Bond entangling 1 + ((-0.45, 4.65), 1.7, 4.7, "#FF87EB"), # Bond entangling 2 + ((-0.45, 9.65), 1.7, 4.7, "#FF87EB"), # Bond entangling 3 + ((1.65, 0.55), 5.7, 3.8, "#FFE096"), # Sequential prep 1 + ((1.65, 5.55), 5.7, 3.8, "#FFE096"), # Sequential prep 2 + ((1.65, 10.55), 5.7, 3.8, "#FFE096"), # Sequential prep 3 + ((7.65, 3.6), 1.7, 1.65, "#D7A2F6"), # Fuse 1 and 2 + ((11.65, 8.65), 1.7, 1.65, "#D7A2F6"), # Fuse (1,2) and 3 + ((9.65, 3.6), 9.7, 4.75, "#70CEFF"), # Push and correct b/w 1 and 2 + ((13.65, 8.65), 9.7, 4.75, "#70CEFF"), # Push and correct b/w (1, 2) and 3 + ((23.6, 13.6), 1.75, 0.8, "#C756B2"), # Correct pushed bond operator + ((25.7, -0.35), 2.7, 14.75, "#B5F2ED"), # Projective measurement +] + +labels = [ + ("Step 1a", 0.5, 14.75), + ("Step 1b", 4.5, 14.75), + ("Step 2", 8.5, 5.5), + ("Step 2", 12.5, 10.5), + ("Step 3", 20.5, 7.5), + ("Step 4", 24.5, 13), + ("Step 5", 27, 14.75), +] + +for xy, width, height, color in box_data: + ax.add_patch(FancyBboxPatch(xy, width, height, edgecolor=color, **options)) +for label, x, y in labels: + t = ax.text(x, y, label, **text_options) + t.set_bbox({"facecolor": "white", "alpha": 0.85, "edgecolor": "white"}) ###################################################################### +# # Let's check that the constant-depth circuit produces the same probability # distribution as the sequential circuit. # @@ -691,20 +733,19 @@ def constant_depth_circuit(N, g, q): @qml.qnode(dev, interface="numpy") def pre_correlations(N, g, q): phys_wires = constant_depth_ansatz(N, g, q) - return ( - [qml.expval(qml.Z(1) @ qml.Z(i)) for i in phys_wires] - + [qml.expval(qml.Z(i)) for i in phys_wires] - ) + single_obs = [qml.expval(qml.Z(i)) for i in phys_wires] + # Exclude the product Z(1) @ Z(1) = I + prod_obs = [qml.expval(qml.Z(1) @ qml.Z(i)) for i in phys_wires[1:]] + return single_obs + prod_obs def correlations(N, g, q): # Compute expectation values for correlators expvals = pre_correlations(N, g, q) - prods = expvals[:-N] - expvals = expvals[-N:] + single_expvals = expvals[:N] + prod_expvals = expvals[N:] # Exclude the self-correlation C(1) - correls = [prods[i] - expvals[0] * expvals[i] for i in range(N) if i != 0] - return correls + return [p - single_expvals[0] * s for p, s in zip(prod_expvals, single_expvals[1:])] N = 12 @@ -738,7 +779,7 @@ def exp_fn(x, a, b): distances = list(range(1, mid_point)) print(f"Correlation lengths xi\n======================") for g, correls in zip(gs, all_correls): - popt, pcov = curve_fit(exp_fn, distances, correls[1 : mid_point]) + popt, pcov = curve_fit(exp_fn, distances, correls[1:mid_point]) xi_numerical = -1 / popt[1] xi_predicted = 1 / abs(np.log((1 + g) / (1 - g))) print(f"{g=:.1f}") @@ -761,7 +802,7 @@ def exp_fn(x, a, b): # bond sites. Nonetheless, this constant-depth algorithm is an exciting advance in # state preparation techniques, alleviating requirements of coherence times and # connectivity on hardware. -# +# # For more information, consider the original article, the mentioned reviews, as well # as our demos on dynamic circuits and tensor network states. # From 401a70e3bc0d674f6063c22a525802a5ac422e92 Mon Sep 17 00:00:00 2001 From: dwierichs Date: Tue, 20 Aug 2024 09:32:25 +0200 Subject: [PATCH 10/37] large thumbnail file path --- demonstrations/tutorial_constant_depth_mps_prep.metadata.json | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_constant_depth_mps_prep.metadata.json b/demonstrations/tutorial_constant_depth_mps_prep.metadata.json index 85cc4c0ded..76f589c5f6 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.metadata.json +++ b/demonstrations/tutorial_constant_depth_mps_prep.metadata.json @@ -19,7 +19,7 @@ }, { "type": "large_thumbnail", - "uri": "/_static/large_demo_thumbnails/thumbnail_large_constant_depth_mps_prep.png" + "uri": "/_static/demo_thumbnails/large_demo_thumbnails/thumbnail_large_constant_depth_mps_prep.png" } ], "seoDescription": "Prepare a matrix product state in constant depth with mid-circuit measurements and feedforward control.", From 8e5e91273e26a3f12fdd79c8005271fb66e52e84 Mon Sep 17 00:00:00 2001 From: dwierichs Date: Tue, 20 Aug 2024 09:47:14 +0200 Subject: [PATCH 11/37] undo eqnn change --- demonstrations/tutorial_eqnn_force_field.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_eqnn_force_field.py b/demonstrations/tutorial_eqnn_force_field.py index ec05c8aa2c..66dffd7edd 100644 --- a/demonstrations/tutorial_eqnn_force_field.py +++ b/demonstrations/tutorial_eqnn_force_field.py @@ -45,7 +45,7 @@ Equivariant Quantum Machine learning ------------------------------------- +----------------------------------- In order to incorporate symmetries into machine learning models, we need a few concepts from group theory. A formal course on the subject is out of the scope of the present document, which is why we refer to two other demos, `equivariant graph embedding `_ From e4df05ec8407cb5ae320b1ee48d1903740c2edb6 Mon Sep 17 00:00:00 2001 From: dwierichs Date: Tue, 20 Aug 2024 14:28:55 +0200 Subject: [PATCH 12/37] more polishing --- .../tutorial_constant_depth_mps_prep.py | 53 +++++++++++-------- 1 file changed, 30 insertions(+), 23 deletions(-) diff --git a/demonstrations/tutorial_constant_depth_mps_prep.py b/demonstrations/tutorial_constant_depth_mps_prep.py index b3c73be905..179867492d 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.py +++ b/demonstrations/tutorial_constant_depth_mps_prep.py @@ -98,10 +98,10 @@ \quad A^1 = \eta \left(\begin{matrix} 0 & -\sqrt{-g} \\ 0 & 1 \end{matrix}\right) -where :math:`\eta = \sqrt{1-g}^{-1}` and :math:`g\in[-1, 0]` is a freely chosen parameter. +where :math:`\eta = \frac{1}{\sqrt{1-g}}` and :math:`g\in[-1, 0]` is a freely chosen parameter. This MPS is a simple yet important example because it can be tuned from the long-range correlated -GHZ state :math:`|\Psi(0)\rangle=(|0>^{\otimes N} + |1>^{\otimes N})/\sqrt(2)` +GHZ state :math:`|\Psi(0)\rangle=\frac{1}{\sqrt{2}}(|0\rangle^{\otimes N} + |1\rangle^{\otimes N})` to the state :math:`|\Psi(-1)\rangle=|+\rangle^{\otimes N}` with vanishing correlation length. In general, the correlation length of :math:`|\Psi(g)\rangle` is given by @@ -110,7 +110,7 @@ \xi(g) = \left|\ln\left(\frac{1+g}{1-g}\right)\right|^{-1}. Long-range correlated states require linear-depth unitary circuits in general. -Therefore, the constant-depth circuit below for :math:`|\Psi(0)\rangle` will demonstrate +Therefore, the constant-depth circuit for :math:`|\Psi(0)\rangle` will demonstrate that dynamic quantum circuits are more powerful than unitary operations alone. The MPS :math:`|\Psi(g)\rangle` will be our working example throughout this demo. Therefore @@ -165,27 +165,28 @@ def A(g): # where :math:`C_\perp` can be any operator making :math:`U` unitary. # The sequential preparation circuit now consists of the following steps. # -# #. Start in the state :math:`|\psi_0\rangle = |0_d\rangle^{\otimes N}\otimes |00\rangle_D`. +# #. Start in the state :math:`|\psi_0\rangle = |0\rangle^{\otimes N}\otimes |00\rangle_D`. # The last two sites are non-physical bond sites, encoded by :math:`D`-dimensional qudits. # -# #. Entangle the bond qudits into the state :math:`|I\rangle = 1/\sqrt{D}\sum_j |jj\rangle_D`. +# #. Entangle the bond qudits into the state :math:`|I\rangle = \frac{1}{\sqrt{D}}\sum_j |jj\rangle_D`. # # #. Apply the unitary :math:`U` once to each of the :math:`N` physical sites, with the -# :math:`D`-dimensional factor always acting on the first bond qudit, denoted by :math:`U^{(i)}`. +# :math:`D`-dimensional tensor factor always acting on the first bond qudit, denoted by :math:`U^{(i)}`. # This produces the state # # .. math:: # # |\psi_1\rangle +# &= \prod_{i=N}^{1} U^{(i)} |0\rangle^{\otimes N} \frac{1}{\sqrt{D}} \sum_j|jj\rangle _D\\ # &= \frac{1}{\sqrt{D}} \sum_j \prod_{i=N-1}^{1} U^{(i)} -# \sum_{m_N=0}^{d-1} |0_d\rangle^{\otimes N-1}|m_N\rangle A^{m_N}|jj\rangle _D\\ +# \sum_{m_N=0}^{d-1} |0\rangle^{\otimes N-1}|m_N\rangle A^{m_N}|jj\rangle _D\\ # &= \frac{1}{\sqrt{D}} \sum_j -# \sum_{\vec{m}} |\vec{m}\rangle A^{m_1} A^{m_2}\cdots A^{m_N}|j\rangle_D |j\rangle_D\\ +# \sum_{\vec{m}} |\vec{m}\rangle A^{m_1} A^{m_2}\cdots A^{m_N}|jj\rangle_D\\ # &= \frac{1}{\sqrt{D}} \sum_{i,j} -# \sum_{\vec{m}} |\vec{m}\rangle \langle i|A^{m_1} A^{m_2}\cdots A^{m_N}|j\rangle_D |i\rangle_D|j\rangle_D +# \sum_{\vec{m}} |\vec{m}\rangle \langle i|A^{m_1} A^{m_2}\cdots A^{m_N}|j\rangle |ij\rangle_D # # #. Measure the two bond qudits in the (generalized) Bell basis and postselect on the outcome -# being :math:`|I\rangle`. Then discard the bond qudits, which collapses :math:`\psi_1\rangle` +# being :math:`|I\rangle`. Then discard the bond qudits, which collapses :math:`|\psi_1\rangle` # into the state # # .. math:: @@ -213,7 +214,7 @@ def A(g): ###################################################################### # We see that this fixed part has two norm-1 columns already, and we are -# able to complement this by :math:`C_\perp` with the same columns: +# able to complement this by :math:`C_\perp` using the same columns, but with shifted rows: E_01 = np.array([[0, 1], [0, 0]]) # |0><1| E_11 = np.array([[0, 0], [0, 1]]) # |1><1| @@ -222,7 +223,7 @@ def A(g): U = U_first_term + C_perp print(np.round(U, 5)) -print(f"U is unitary: {np.allclose(U.conj().T @ U, np.eye(4))}") +print(f"\nU is unitary: {np.allclose(U.conj().T @ U, np.eye(4))}") ###################################################################### @@ -241,8 +242,8 @@ def A(g): # .. math:: # # U \operatorname{CNOT} -# &= A^0 P_0 \otimes \mathbb{I} + A^0P_1 \otimes X + A^1 P_0 \otimes X + A^1P_1 \otimes \mathbb{I}\\ -# &= \left(\begin{matrix} \eta & -\eta\sqrt{-g} \\ \eta\sqrt{-g} & \eta \end{matrix}\right), +# &= (A^0 P_0) \otimes \mathbb{I} + (A^0P_1) \otimes X + (A^1 P_0) \otimes X + (A^1P_1) \otimes \mathbb{I}\\ +# &= \left(\begin{matrix} \eta & -\eta\sqrt{-g} \\ \eta\sqrt{-g} & \eta \end{matrix}\right)\otimes \mathbb{I}, # # where we denoted the computational basis state projectors as # :math:`P_i = |i\rangle\!\langle i|`. The remaining operation is an @@ -260,7 +261,7 @@ def U_template(eta, bond_idx, phys_idx): print(f"The template reproduces U: {np.allclose(template_matrix, U)}") ###################################################################### -# Great, now that we have a circuit that realizes the unitary :math:`U`, let's code up the entire +# Excellent, now that we have a circuit that realizes the unitary :math:`U`, let's code up the entire # sequential preparation circuit. We implement the measurement \& postselect step as a separate # function ``project_measure``. @@ -308,8 +309,8 @@ def sequential_circuit(N, g): # We will verify the prepared state below, when comparing to the constant-depth # preparation circuit. Let's just draw the circuit for now. -N = 10 -qml.draw_mpl(sequential_circuit)(N, g) +N = 7 +qml.draw_mpl(sequential_circuit)(N, g); ###################################################################### # As we can see, the sequential preparation circuit already uses mid-circuit @@ -386,7 +387,8 @@ def sequential_circuit(N, g): # The measurement procedure can then be coded up in the following short function, # which is similar to the ``project_measure`` function from the sequential preparation # routine. Note, however, that instead of postselecting, we record the measurement -# outcome in the form of two bits, to be used further below. +# outcome in the form of two bits representing the index :math:`k` of the operator +# :math:`B^k`, which we will use further below. def fuse(wire_0, wire_1): @@ -617,7 +619,9 @@ def constant_depth_ansatz(N, g, q): # Step 5: Perform projective measurement on outer-most bond sites. project_measure(*outer_bond_sites) # Collect wire ranges for physical wires, skipping bond wires - phys_wires = (range(block_wires * i + 1, block_wires * (i + 1) - 1) for i in range(num_blocks)) + phys_wires = ( + range(block_wires * i + 1, block_wires * (i + 1) - 1) for i in range(num_blocks) + ) # Turn ranges to lists and sum them together return sum(map(list, phys_wires), start=[]) @@ -630,7 +634,7 @@ def constant_depth_circuit(N, g, q): ###################################################################### # -# Great, we now built the full constant-depth circuit to prepare the MPS +# We built the full constant-depth circuit to prepare the MPS # :math:`|\Psi(g)\rangle`. # Before we evaluate the states it produces, let's see how the circuit looks. # We'll add some boxes for the individual subroutines. @@ -692,8 +696,11 @@ def constant_depth_circuit(N, g, q): ###################################################################### -# Great, we have a constant-depth circuit that reproduces the existing sequential +# Nice, we have a constant-depth circuit that reproduces the existing sequential # preparation circuit, which has linear depth! +# Not that while the dynamic operations of step 3 seem to have a linear depth, +# they can be applied in parallel because they are only controlled by classical +# registers. # # Correlation length # ~~~~~~~~~~~~~~~~~~ @@ -716,7 +723,7 @@ def constant_depth_circuit(N, g, q): ax.bar(ints, ghz_probs, width=1, label="long correlation length (GHZ)") ax.set_yscale("log") ax.set_ylim([1e-3, 1]) -ax.legend() +ax.legend(); ###################################################################### # In a more nuanced setting, let us produce the MPS in question for a series of @@ -766,7 +773,7 @@ def correlations(N, g, q): # Note that due to the periodic boundary conditions, we have :math:`C(N-j)=C(j)`. # This is why the correlations increase again after :math:`j=\frac{N}{2}`. # To obtain the correlation length, we fit an exponential to the measured correlators -# up until :math:`j=\frac{N/2}` and invert the decay rate. +# up until :math:`j=\frac{N}{2}` and invert the decay rate. # We can also compare this result to the analytic result for the correlation # length :math:`xi` mentioned in the beginning: From 344c93876a40eabed2f10c7fdea97cf1f2c306ce Mon Sep 17 00:00:00 2001 From: dwierichs Date: Tue, 20 Aug 2024 14:45:51 +0200 Subject: [PATCH 13/37] review reference --- .../tutorial_constant_depth_mps_prep.metadata.json | 2 +- demonstrations/tutorial_constant_depth_mps_prep.py | 11 ++++++++--- 2 files changed, 9 insertions(+), 4 deletions(-) diff --git a/demonstrations/tutorial_constant_depth_mps_prep.metadata.json b/demonstrations/tutorial_constant_depth_mps_prep.metadata.json index 76f589c5f6..4c405f4b97 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.metadata.json +++ b/demonstrations/tutorial_constant_depth_mps_prep.metadata.json @@ -26,7 +26,7 @@ "doi": "", "canonicalURL": "/qml/demos/tutorial_constant_depth_mps_prep", "references": [], - "basedOnPapers": [], + "basedOnPapers": ["10.48550/arXiv.2404.16083"], "referencedByPapers": [], "relatedContent": [ { diff --git a/demonstrations/tutorial_constant_depth_mps_prep.py b/demonstrations/tutorial_constant_depth_mps_prep.py index 179867492d..4fff5a99b6 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.py +++ b/demonstrations/tutorial_constant_depth_mps_prep.py @@ -59,7 +59,7 @@ In particular, MPS can efficiently describe ground states of (gapped local) one-dimensional Hamiltonians, which in addition can be found efficiently using density matrix renormalization group (DMRG) algorithms. -For reviews of MPS see [#todo]. +For a review of MPS see [#cirac]_. Following [#smith]_, we will look at translation-invariant MPS of a quantum :math:`N`-body system where each body, or site, has local (physical) @@ -831,9 +831,14 @@ def exp_fn(x, a, b): # `closed access `__, 2005. # `arXiv:quant-ph/0501096 `__, 2005. # -# .. [#todo] +# .. [#cirac] +# +# J. I. Cirac, D. Pérez-García, N. Schuch, F. Verstraete +# "Matrix product states and projected entangled pair states: Concepts, symmetries, theorems", +# Reviews of Modern Physics **93**, 045003, +# `closed access `__, 2021. +# `arXiv:2011.12127 `__, 2020. # -# Todo # # About the author # ---------------- From 97e065f6bf0bab96bc4199e07ba94cab4f03b035 Mon Sep 17 00:00:00 2001 From: dwierichs Date: Tue, 20 Aug 2024 20:59:57 +0200 Subject: [PATCH 14/37] polish --- .../constant_depth_mps_prep/algorithm.png | Bin 0 -> 215350 bytes .../tutorial_constant_depth_mps_prep.py | 33 ++++++++++-------- 2 files changed, 19 insertions(+), 14 deletions(-) create mode 100644 _static/demonstration_assets/constant_depth_mps_prep/algorithm.png diff --git a/_static/demonstration_assets/constant_depth_mps_prep/algorithm.png b/_static/demonstration_assets/constant_depth_mps_prep/algorithm.png new file mode 100644 index 0000000000000000000000000000000000000000..fd9ce2966f5417a510e5865e553891fe5b13b655 GIT binary patch literal 215350 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zu3xZ)Zl;=oJU@448?+=s^zo3b_)907UX1><^^r=jIlAuYYRT?t52JC-+=u-LLZh-2 zk_%|=bg7z2-S03qxda<^|twrYRQs$^$OOBnKc-4ljr;Vo8#7NvC)H?L~e#F zVO-`|hC5-Tg02|SH{`M>9uu16nELUR&z zP_w|`e;Gge2n0pxfB7E*#{Uo2y0t@yI)g;DVT1FaQ_+W-In literal 0 HcmV?d00001 diff --git a/demonstrations/tutorial_constant_depth_mps_prep.py b/demonstrations/tutorial_constant_depth_mps_prep.py index 4fff5a99b6..14af8be0e8 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.py +++ b/demonstrations/tutorial_constant_depth_mps_prep.py @@ -38,12 +38,13 @@ the algorithm from a mathematical perspective. One of these blocks is the sequential MPS preparation circuit to which we will compare later. Alongside this introduction we describe and code up each building block for a -specific MPS and verify the conditions for the algorithm to apply. -Finally, we combine the building blocks into the constant-depth algorithm by -Smith et al. We conclude by computing the correlation length of the prepared MPS. +specific MPS. +Then we combine the building blocks into the constant-depth algorithm by +Smith et al. We conclude by computing the correlation length of the prepared +example MPS. -Mathematical derivation ------------------------ +Building blocks +--------------- We briefly introduce MPS and a sequential circuit with linear depth to prepare them, as well as two tools from quantum information theory @@ -89,8 +90,8 @@ **Example** -As an example, consider a chain of :math:`N` qubits (:math:`d=2`) and a :math:`D=2` MPS -:math:`|\Psi(g)\rangle` on this system, defined by the matrices +As an example, consider a chain of :math:`N` qubits (:math:`d=2`) and an MPS +:math:`|\Psi(g)\rangle` on this system with :math:`D=2`, defined by the matrices .. math:: @@ -103,6 +104,7 @@ This MPS is a simple yet important example because it can be tuned from the long-range correlated GHZ state :math:`|\Psi(0)\rangle=\frac{1}{\sqrt{2}}(|0\rangle^{\otimes N} + |1\rangle^{\otimes N})` to the state :math:`|\Psi(-1)\rangle=|+\rangle^{\otimes N}` with vanishing correlation length. +This MPS also is discussed in Sec. III C 1. of [#smith]_. In general, the correlation length of :math:`|\Psi(g)\rangle` is given by .. math:: @@ -236,17 +238,16 @@ def A(g): # + A^1\otimes (|1\rangle\!\langle 0| + |0\rangle\!\langle 1|)\\ # &= A^0 \otimes \mathbb{I} + A^1\otimes X # -# This looks a lot like a :class:`~.pennylane.CNOT` gate plays a part in :math:`U`. +# This looks a lot like a :class:`~.pennylane.CNOT` gate. # Removing it from :math:`U` we find # # .. math:: # # U \operatorname{CNOT} -# &= (A^0 P_0) \otimes \mathbb{I} + (A^0P_1) \otimes X + (A^1 P_0) \otimes X + (A^1P_1) \otimes \mathbb{I}\\ +# &= (A^0 E_{00}) \otimes \mathbb{I} + (A^0E_{11}) \otimes X + (A^1 E_{00}) \otimes X + (A^1E_{11}) \otimes \mathbb{I}\\ # &= \left(\begin{matrix} \eta & -\eta\sqrt{-g} \\ \eta\sqrt{-g} & \eta \end{matrix}\right)\otimes \mathbb{I}, # -# where we denoted the computational basis state projectors as -# :math:`P_i = |i\rangle\!\langle i|`. The remaining operation is an +# The remaining operation is an # :class:`~.pennylane.RY` rotation by the angle :math:`2\arccos(\eta)`, so # that we can easily code up the unitary as a small circuit template: @@ -262,7 +263,7 @@ def U_template(eta, bond_idx, phys_idx): ###################################################################### # Excellent, now that we have a circuit that realizes the unitary :math:`U`, let's code up the entire -# sequential preparation circuit. We implement the measurement \& postselect step as a separate +# sequential preparation circuit. We implement the measurement \& postselection step as a separate # function ``project_measure``. @@ -310,7 +311,7 @@ def sequential_circuit(N, g): # preparation circuit. Let's just draw the circuit for now. N = 7 -qml.draw_mpl(sequential_circuit)(N, g); +_ = qml.draw_mpl(sequential_circuit)(N, g) ###################################################################### # As we can see, the sequential preparation circuit already uses mid-circuit @@ -359,7 +360,7 @@ def sequential_circuit(N, g): # where we used that :math:`\sum_{j,\ell} |j\rangle B^k_{j\ell}\langle\ell|=B^k`. # We now can measure the two bond qudits and will know that for a given measurement # outcome :math:`k`, we obtained the MPS state on :math:`K+L` qubits, up to -# a small "impurity", namely the matrix :math:`B^k` that depends to the outcome. +# an "impurity", namely the matrix :math:`B^k` that depends to the outcome. # # In full generality, the fusion strategy may seem complicated, but it can take # a very simply form, as we will now see in our example: @@ -539,6 +540,10 @@ def push_and_correct(op_id, phys_wires): # #. Perform the same projection step as in the sequential preparation algorithm on # the two remaining bond sites. # +# Smith et al. summarize their algorithm in the following chart: +# +# .. image:: ../_static/demonstration_assets/constant_depth_mps_prep/algorithm.png +# # It is important to remember that showing the existence of suitable operator pushing # relations (and finding them explicitly) is a crucial step in general, which goes # beyond the scope of the demo. From a4d140cfb33cdbe8555e02e7fd5e7f1def8ea56d Mon Sep 17 00:00:00 2001 From: dwierichs Date: Tue, 20 Aug 2024 21:09:42 +0200 Subject: [PATCH 15/37] tab recom --- demonstrations/tutorial_constant_depth_mps_prep.py | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/demonstrations/tutorial_constant_depth_mps_prep.py b/demonstrations/tutorial_constant_depth_mps_prep.py index 14af8be0e8..cd2f9598c9 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.py +++ b/demonstrations/tutorial_constant_depth_mps_prep.py @@ -642,7 +642,8 @@ def constant_depth_circuit(N, g, q): # We built the full constant-depth circuit to prepare the MPS # :math:`|\Psi(g)\rangle`. # Before we evaluate the states it produces, let's see how the circuit looks. -# We'll add some boxes for the individual subroutines. +# We'll add some boxes for the individual subroutines, and recommend opening +# the figure in a separate tab. # N = 9 From 506c3d713cb763308653a9c04340f04ab871b740 Mon Sep 17 00:00:00 2001 From: dwierichs Date: Tue, 20 Aug 2024 21:10:45 +0200 Subject: [PATCH 16/37] proper system size --- demonstrations/tutorial_constant_depth_mps_prep.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_constant_depth_mps_prep.py b/demonstrations/tutorial_constant_depth_mps_prep.py index cd2f9598c9..e4405da274 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.py +++ b/demonstrations/tutorial_constant_depth_mps_prep.py @@ -761,7 +761,7 @@ def correlations(N, g, q): return [p - single_expvals[0] * s for p, s in zip(prod_expvals, single_expvals[1:])] -N = 12 +N = 18 q = 6 gs = [-1 / 5, -2 / 5, -3 / 5, -4 / 5] fig, ax = plt.subplots() From 454a7d1e6bd367b91c238547d374d05e3ed4290e Mon Sep 17 00:00:00 2001 From: dwierichs Date: Thu, 22 Aug 2024 11:45:19 +0200 Subject: [PATCH 17/37] details --- .../tutorial_constant_depth_mps_prep.py | 22 +++++++++++++------ 1 file changed, 15 insertions(+), 7 deletions(-) diff --git a/demonstrations/tutorial_constant_depth_mps_prep.py b/demonstrations/tutorial_constant_depth_mps_prep.py index e4405da274..29e4740c86 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.py +++ b/demonstrations/tutorial_constant_depth_mps_prep.py @@ -7,7 +7,9 @@ .. figure:: ../_static/demonstration_assets/constant_depth_mps_prep/socialthumbnail_constant_depth_mps_prep.png :align: center - :width: 50% + :width: 70% + :alt: Illustration of two matrix product states merged together by a tree growing out of and back into them + :target: javascript:void(0); In this demo you will learn to prepare certain MPS with a dynamic quantum circuit of constant depth. We will implement the circuit, which makes use @@ -115,8 +117,8 @@ Therefore, the constant-depth circuit for :math:`|\Psi(0)\rangle` will demonstrate that dynamic quantum circuits are more powerful than unitary operations alone. -The MPS :math:`|\Psi(g)\rangle` will be our working example throughout this demo. Therefore -we warm up our coding by defining the tensor :math:`A` and testing its properties. +The MPS :math:`|\Psi(g)\rangle` will be our running example throughout this demo. +To warm up our coding, let's define the tensor :math:`A` and test that it is in left-canonical form. """ import numpy as np @@ -360,7 +362,7 @@ def sequential_circuit(N, g): # where we used that :math:`\sum_{j,\ell} |j\rangle B^k_{j\ell}\langle\ell|=B^k`. # We now can measure the two bond qudits and will know that for a given measurement # outcome :math:`k`, we obtained the MPS state on :math:`K+L` qubits, up to -# an "impurity", namely the matrix :math:`B^k` that depends to the outcome. +# a defect, namely the matrix :math:`B^k` that depends to the outcome. # # In full generality, the fusion strategy may seem complicated, but it can take # a very simply form, as we will now see in our example: @@ -433,7 +435,7 @@ def prepare_and_unprepare(g): print(f"The test measurement of the fusion preparation + sequential unpreparation is {test:.2f}") ###################################################################### -# This means we still need a way to remove the impurity matrices :math:`B^k` from the +# This means we still need a way to remove the defect matrices :math:`B^k` from the # state that ended up in the prepared state when we performed the fusion measurement. # *Operator pushing* will allow us to do exactly that! # @@ -446,13 +448,15 @@ def prepare_and_unprepare(g): # tensor :math:`A`. We will not go into detail about these conditions but assume here that # we work with compatible MPS and measurements. # -# Operator pushing then allows us to push an impurity operator :math:`B^k` from one bond +# Operator pushing then allows us to push a defect operator :math:`B^k` from one bond # axis through the tensor :math:`A` of the MPS to the other bond axis and to the physical axis. # While doing so, the operator in general will change, i.e., we end up with different operators # :math:`C^k` and :math:`D^k` on the bond and physical axes, respectively. # Visually, we find: # # .. image:: ../_static/demonstration_assets/constant_depth_mps_prep/operator_pushing.png +# :width: 35% +# :align: center # # For simplicity, we denote a push by :math:`B^k\mapsto (C^k, D^k)`. # @@ -465,6 +469,8 @@ def prepare_and_unprepare(g): # measurement basis. In particular, the relations are # # .. image:: ../_static/demonstration_assets/constant_depth_mps_prep/operator_pushing_example.png +# :width: 75% +# :align: center # # That is, in the notation above we have # :math:`X\mapsto (Y, Y)`, @@ -531,7 +537,7 @@ def push_and_correct(op_id, phys_wires): # #. Fuse together the blocks of MPS using mid-circuit measurements and store the # measurement outcomes. # -# #. Push the resulting impurity operators to one of the outer bond sites and +# #. Push the resulting defect operators to one of the outer bond sites and # conditionally apply correction operators to the physical sites. # # #. Undo the operator that was pushed to the outer bond site by conditionally applying @@ -543,6 +549,8 @@ def push_and_correct(op_id, phys_wires): # Smith et al. summarize their algorithm in the following chart: # # .. image:: ../_static/demonstration_assets/constant_depth_mps_prep/algorithm.png +# :width: 75% +# :align: center # # It is important to remember that showing the existence of suitable operator pushing # relations (and finding them explicitly) is a crucial step in general, which goes From 4c5328f3ad75ea4adddefb6d62bf8502c6cc49d4 Mon Sep 17 00:00:00 2001 From: dwierichs Date: Thu, 22 Aug 2024 16:44:04 +0200 Subject: [PATCH 18/37] polish --- .../tutorial_constant_depth_mps_prep.py | 18 ++++++++++++------ 1 file changed, 12 insertions(+), 6 deletions(-) diff --git a/demonstrations/tutorial_constant_depth_mps_prep.py b/demonstrations/tutorial_constant_depth_mps_prep.py index 29e4740c86..46a5135253 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.py +++ b/demonstrations/tutorial_constant_depth_mps_prep.py @@ -733,11 +733,13 @@ def constant_depth_circuit(N, g, q): ghz_probs = constant_depth_circuit(N, g, q) ints = list(range(2**N)) fig, ax = plt.subplots() -ax.bar(ints, uniform_probs, width=1, label="zero correlation length ($|+\\rangle^N$)") -ax.bar(ints, ghz_probs, width=1, label="long correlation length (GHZ)") +label_plus = "zero correlation length ($|+\\rangle^N$)" +label_ghz = "long correlation length (GHZ)" +ax.bar(ints, ghz_probs, width=1, color="#ff87eb", label=label_ghz, edgecolor="k") +ax.bar(ints, uniform_probs, width=1, color="#70ceff", label=label_plus, edgecolor="0.5") ax.set_yscale("log") ax.set_ylim([1e-3, 1]) -ax.legend(); +_ = ax.legend() ###################################################################### # In a more nuanced setting, let us produce the MPS in question for a series of @@ -771,14 +773,18 @@ def correlations(N, g, q): N = 18 q = 6 -gs = [-1 / 5, -2 / 5, -3 / 5, -4 / 5] +gs = [-0.2, -0.4, -0.6, -0.8] fig, ax = plt.subplots() +colors = ["#70ceff", "#ffe096", "#c756b2", "#e1eba8"] +plot_kwargs = dict(ls="", marker="o", markersize=7, markeredgecolor="k") all_correls = [] -for g in gs: +for g, c in zip(gs, colors): correls = correlations(N, g, q) - ax.plot(list(range(N - 1)), correls, label=f"$g={g}$") + ax.plot(list(range(N - 1)), correls, color=c, label=f"$g={g}$", **plot_kwargs) all_correls.append(correls) ax.set_yscale("log") +ax.set_xlabel("$j$") +ax.set_ylabel("$C(j)$") ax.set_ylim([5e-8, 1.5]) ax.legend() plt.show() From 64796f8202f0492d4384ead0c2776d4c75cd875a Mon Sep 17 00:00:00 2001 From: David Wierichs Date: Tue, 27 Aug 2024 09:10:29 +0200 Subject: [PATCH 19/37] Apply suggestions from code review Co-authored-by: Korbinian Kottmann <43949391+Qottmann@users.noreply.github.com> --- demonstrations/tutorial_constant_depth_mps_prep.py | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/demonstrations/tutorial_constant_depth_mps_prep.py b/demonstrations/tutorial_constant_depth_mps_prep.py index 46a5135253..995e9784c7 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.py +++ b/demonstrations/tutorial_constant_depth_mps_prep.py @@ -68,7 +68,7 @@ of a quantum :math:`N`-body system where each body, or site, has local (physical) dimension :math:`d`. We will further restrict to periodic boundary conditions, corresponding to translational invariance of the MPS on a closed loop. -A general MPS with this properties is given by +A general MPS with these properties is given by .. math:: @@ -76,7 +76,7 @@ where :math:`\vec{m}` is a multi-index of :math:`N` indices ranging over :math:`d`, i.e., :math:`\vec{m}\in\{0, 1 \dots, d-1\}^N`, and :math:`\{A^{m}\}_{m}` are :math:`d` -square matrices of equal dimension. This dimension :math:`D` is called the bond dimension, +square matrices of equal dimension :math:`D`. This dimension :math:`D` is called the bond dimension, which is a crucial quantity for the expressivity and complexity of the MPS. We see that :math:`|\Psi\rangle` is fully specified by the :math:`d\times D\times D=dD^2` numbers in the rank-3 tensor :math:`A`. @@ -795,7 +795,7 @@ def correlations(N, g, q): # To obtain the correlation length, we fit an exponential to the measured correlators # up until :math:`j=\frac{N}{2}` and invert the decay rate. # We can also compare this result to the analytic result for the correlation -# length :math:`xi` mentioned in the beginning: +# length :math:`\xi` mentioned in the beginning: def exp_fn(x, a, b): From 84fd872ac390f293b6907a5b9ef555c26b3ce166 Mon Sep 17 00:00:00 2001 From: dwierichs Date: Tue, 27 Aug 2024 09:58:58 +0200 Subject: [PATCH 20/37] code review --- .../tutorial_constant_depth_mps_prep.py | 135 +++++++++++------- 1 file changed, 83 insertions(+), 52 deletions(-) diff --git a/demonstrations/tutorial_constant_depth_mps_prep.py b/demonstrations/tutorial_constant_depth_mps_prep.py index 995e9784c7..919cc0bb06 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.py +++ b/demonstrations/tutorial_constant_depth_mps_prep.py @@ -3,7 +3,9 @@ Matrix product states (MPS) are used in a plethora of applications on quantum many-body systems, both on classical and quantum computers. -This makes the preparation of MPS on a quantum computer an important subroutine. +This makes the preparation of MPS on a quantum computer an important subroutine, +for example when warm-starting a quantum algorithm with a classically optimized +MPS or preparing initial states with well-studied physical properties. .. figure:: ../_static/demonstration_assets/constant_depth_mps_prep/socialthumbnail_constant_depth_mps_prep.png :align: center @@ -14,24 +16,24 @@ In this demo you will learn to prepare certain MPS with a dynamic quantum circuit of constant depth. We will implement the circuit, which makes use of mid-circuit measurements and conditionally applied operations, for a specific -MPS, and compare it to a (static) sequential circuit with linear depth. -We will closely follow the eponymous article by Smith et al. [#smith]_. +MPS. Then we will compare it to a (static) sequential circuit with linear depth. +Throughout, we closely follow the eponymous article by Smith et al. [#smith]_. .. note:: - If you would like to first learn about the dynamic circuit tools used in this - algorithm, have a look at our :doc:`introduction to mid-circuit measurements - ` and learn :doc:`how to collect statics of - mid-circuit measurements ` and + If you are new to dynamic circuit tools (mid-circuit measurements and classically + controlled operations) used in this algorithm, we recommend to check out + our :doc:`introduction to mid-circuit measurements + ` and learn :doc:`how to create dynamic circuits with mid-circuit measurements `. - If you first want to familiarize yourself with tensor network states and MPS - in particular, take a look at our demo on - :doc:`tensor-network quantum circuits `. - Finally, our demo on :doc:`initial state preparation for quantum chemistry - ` shows you how to - proceed to a concrete application, once an MPS is prepared. + Optionally, you may want to familiarize yourself with tensor network states + and MPS in particular. For this, take a look at our demos on + :doc:`tensor-network quantum circuits ` and + :doc:`initial state preparation for quantum chemistry + `, which uses a prepared MPS in + an application. Outline ------- @@ -62,7 +64,7 @@ In particular, MPS can efficiently describe ground states of (gapped local) one-dimensional Hamiltonians, which in addition can be found efficiently using density matrix renormalization group (DMRG) algorithms. -For a review of MPS see [#cirac]_. +For a review of MPS see [#schollwoeck]_. Following [#smith]_, we will look at translation-invariant MPS of a quantum :math:`N`-body system where each body, or site, has local (physical) @@ -106,16 +108,17 @@ This MPS is a simple yet important example because it can be tuned from the long-range correlated GHZ state :math:`|\Psi(0)\rangle=\frac{1}{\sqrt{2}}(|0\rangle^{\otimes N} + |1\rangle^{\otimes N})` to the state :math:`|\Psi(-1)\rangle=|+\rangle^{\otimes N}` with vanishing correlation length. -This MPS also is discussed in Sec. III C 1. of [#smith]_. +This example also is discussed in Sec. III C 1. of [#smith]_. In general, the correlation length of :math:`|\Psi(g)\rangle` is given by .. math:: \xi(g) = \left|\ln\left(\frac{1+g}{1-g}\right)\right|^{-1}. -Long-range correlated states require linear-depth unitary circuits in general. -Therefore, the constant-depth circuit for :math:`|\Psi(0)\rangle` will demonstrate -that dynamic quantum circuits are more powerful than unitary operations alone. +Long-range correlated states require linear-depth circuits if we are restricted to unitary +operations. Therefore, the constant-depth circuit for :math:`|\Psi(0)\rangle` will demonstrate +that dynamic quantum circuits, which are not unitary, are more powerful than unitary +operations alone. The MPS :math:`|\Psi(g)\rangle` will be our running example throughout this demo. To warm up our coding, let's define the tensor :math:`A` and test that it is in left-canonical form. @@ -151,32 +154,48 @@ def A(g): # # An MPS like the one above can be prepared in linear depth using an established technique # by Schön et al. [#schoen]_. We introduce this technique here because the new -# constant-depth construction uses it as a parallelized subroutine, and because we will +# constant-depth construction uses it as a (parallelized) subroutine, and because we will # later compare the two approaches. # # The MPS :math:`|\Psi\rangle` above is given by the rank-3 tensor :math:`A`. -# In order to prepare the state in a circuit, we want to construct unitary operations -# from :math:`A`. For this, we build a new rank-4 tensor :math:`U` with an additional -# axis of dimension :math:`d`. If this new physical input axis is in the state -# :math:`|0\rangle`, we demand :math:`U` to reproduce one :math:`A^m` for each physical -# output state :math:`|m\rangle`. We leave the remaining action undefined and only require -# it to make :math:`U` unitary overall. In short, this can be written as +# We can think of this as an operator that acts on a bond site and simultaneously *creates* +# a physical site. In order to prepare the state in a circuit, which usually is not allowed +# to create qudits out of nowhere, we want to construct a unitary operation +# from :math:`A` that already takes a physical site as input. +# This means that we build a new rank-4 tensor :math:`U` with an additional +# axis of dimension :math:`d`. Then we demand that :math:`U` acts like :math:`A` if this +# new physical input axis is in the state :math:`|0\rangle`. That is, after applying :math:`U` +# to a physical site in :math:`|0\rangle` and a bond site, we obtain the same result as +# applying :math:`A` to the bond site alone. +# +# We leave the action of :math:`U` on other physical input states undefined, and only require +# it to make :math:`U` unitary overall. +# In short, our construction can be written as # # .. math:: # # U = \sum_m A^m \otimes |m\rangle\!\langle 0| + C_\perp, # # where :math:`C_\perp` can be any operator making :math:`U` unitary. +# +# In the definition of the MPS, the bond site is passed on between copies of :math:`A`, +# sequentially creating physical sites. Likewise, we will apply our unitary :math:`U` to +# a sequence of physical sites (each in the initial state :math:`|0\rangle`) while passing +# on a single bond site. This way, we chain up the unitaries on the bond site like beads +# on a string. +# # The sequential preparation circuit now consists of the following steps. # # #. Start in the state :math:`|\psi_0\rangle = |0\rangle^{\otimes N}\otimes |00\rangle_D`. # The last two sites are non-physical bond sites, encoded by :math:`D`-dimensional qudits. # -# #. Entangle the bond qudits into the state :math:`|I\rangle = \frac{1}{\sqrt{D}}\sum_j |jj\rangle_D`. +# #. Entangle the bond qudits into the state +# :math:`|I\rangle = \frac{1}{\sqrt{D}}\sum_j |jj\rangle_D`. # -# #. Apply the unitary :math:`U` once to each of the :math:`N` physical sites, with the -# :math:`D`-dimensional tensor factor always acting on the first bond qudit, denoted by :math:`U^{(i)}`. -# This produces the state +# #. Apply the unitary :math:`U` to each of the :math:`N` physical sites, with the +# :math:`D`-dimensional tensor factor always acting on the first bond qudit. +# Denoting :math:`U` acting on the :math:`i`\ th physical site and the first bond site +# by :math:`U^{(i)}`, this produces the state # # .. math:: # @@ -186,8 +205,12 @@ def A(g): # \sum_{m_N=0}^{d-1} |0\rangle^{\otimes N-1}|m_N\rangle A^{m_N}|jj\rangle _D\\ # &= \frac{1}{\sqrt{D}} \sum_j # \sum_{\vec{m}} |\vec{m}\rangle A^{m_1} A^{m_2}\cdots A^{m_N}|jj\rangle_D\\ -# &= \frac{1}{\sqrt{D}} \sum_{i,j} -# \sum_{\vec{m}} |\vec{m}\rangle \langle i|A^{m_1} A^{m_2}\cdots A^{m_N}|j\rangle |ij\rangle_D +# &= \frac{1}{\sqrt{D}} \sum_{j,k} +# \sum_{\vec{m}} |\vec{m}\rangle \langle j|A^{m_1} A^{m_2}\cdots A^{m_N}|k\rangle |jk\rangle_D +# +# We can see how each :math:`U^{(i)}` contributes a factor of :math:`A` and a corresponding +# state :math:`|m_i\rangle` on the :math:`i`\ th physical site. The same would have +# come out when applying the 3-tensor :math:`A` to the first bond site :math:`N` times. # # #. Measure the two bond qudits in the (generalized) Bell basis and postselect on the outcome # being :math:`|I\rangle`. Then discard the bond qudits, which collapses :math:`|\psi_1\rangle` @@ -195,7 +218,9 @@ def A(g): # # .. math:: # -# |\psi_2\rangle = \sum_{\vec{m}} \operatorname{tr}[A^{m_1}A^{m_2}\cdots A^{m_N}]|\vec{m}\rangle = |\Psi\rangle +# |\psi_2\rangle +# = \sum_{\vec{m}} \operatorname{tr}[A^{m_1}A^{m_2}\cdots A^{m_N}]|\vec{m}\rangle +# = |\Psi\rangle # # Note that this step is *probabilistic* and we only succeed to produce the state if we measure # the state :math:`|I\rangle`. @@ -316,16 +341,22 @@ def sequential_circuit(N, g): _ = qml.draw_mpl(sequential_circuit)(N, g) ###################################################################### -# As we can see, the sequential preparation circuit already uses mid-circuit -# measurements. However, there is no feed forward control that modifies the circuit -# dynamically based on measured values. +# The sequential preparation circuit already uses mid-circuit measurements as is visible +# from the measurement steps on the first and last qubit, which are set to postselect the +# outcome :math:`|0\rangle`. However, there is no feed-forward control that modifies the circuit +# dynamically based on measured values. This will change in the following sections. # # Fusion of MPS states with mid-circuit measurements # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # -# The next ingredient for the constant-depth MPS preparation circuit is to fuse -# two MPS states together into one MPS. Under the hood, this is -# an application of entanglement swapping. +# The next ingredient for the constant-depth MPS preparation circuit is to fuse the product +# state of two MPS states together into one (entangled) MPS. Recall that a single sequential +# creation of this final MPS would pass a bond site between the first and second group of physical +# sites, chaining them like beads on the same string. The fusion step allows us to replace +# this connection with dynamic circuit components after creating two independent MPS. +# This is like knotting together two strings on which we chained beads independently before. +# As a matter of fact, chaining beads on smaller strings and the fact that we can knot them +# together simultaneously is at the heart of going from linear to constant circuit depth. # # Consider a state :math:`|\Phi\rangle=|\Phi_1\rangle\otimes|\Phi_2\rangle`, where # :math:`|\Phi_{1,2}\rangle` are MPS prepared with the sequential technique from @@ -528,9 +559,11 @@ def push_and_correct(op_id, phys_wires): # # Now that we discussed the sequential preparation algorithm, fusion with mid-circuit # measurements, and operator pushing, we have all components for the constant-depth -# preparation algorithm. +# preparation algorithm. As inputs it requires the tensor :math:`A` (or the unitary +# :math:`U` to reproduce :math:`A`), the total number of physical sites :math:`N`, +# and a block size :math:`q`. # -# #. For a block size :math:`q`, prepare :math:`\frac{N}{q}` MPS of size :math:`q` +# #. Prepare :math:`\frac{N}{q}` MPS of size :math:`q` # in parallel, using the sequential preparation algorithm (without the final # projection step). # @@ -546,7 +579,7 @@ def push_and_correct(op_id, phys_wires): # #. Perform the same projection step as in the sequential preparation algorithm on # the two remaining bond sites. # -# Smith et al. summarize their algorithm in the following chart: +# We summarize the algorithm in the following sketch, which again follows [#smith]_. # # .. image:: ../_static/demonstration_assets/constant_depth_mps_prep/algorithm.png # :width: 75% @@ -632,9 +665,7 @@ def constant_depth_ansatz(N, g, q): # Step 5: Perform projective measurement on outer-most bond sites. project_measure(*outer_bond_sites) # Collect wire ranges for physical wires, skipping bond wires - phys_wires = ( - range(block_wires * i + 1, block_wires * (i + 1) - 1) for i in range(num_blocks) - ) + phys_wires = (range(block_wires * i + 1, block_wires * (i + 1) - 1) for i in range(num_blocks)) # Turn ranges to lists and sum them together return sum(map(list, phys_wires), start=[]) @@ -771,7 +802,7 @@ def correlations(N, g, q): return [p - single_expvals[0] * s for p, s in zip(prod_expvals, single_expvals[1:])] -N = 18 +N = 12 q = 6 gs = [-0.2, -0.4, -0.6, -0.8] fig, ax = plt.subplots() @@ -851,13 +882,13 @@ def exp_fn(x, a, b): # `closed access `__, 2005. # `arXiv:quant-ph/0501096 `__, 2005. # -# .. [#cirac] -# -# J. I. Cirac, D. Pérez-García, N. Schuch, F. Verstraete -# "Matrix product states and projected entangled pair states: Concepts, symmetries, theorems", -# Reviews of Modern Physics **93**, 045003, -# `closed access `__, 2021. -# `arXiv:2011.12127 `__, 2020. +# .. [#schollwoeck] +# +# U. Schollwoeck +# "The density-matrix renormalization group in the age of matrix product states", +# Annals of Physics **326**, 96, +# `closed access `__, 2011. +# `arXiv:1008.3477 `__, 2010. # # # About the author From 4b92ac669ded01f481029d52784609112555f720 Mon Sep 17 00:00:00 2001 From: dwierichs Date: Tue, 27 Aug 2024 10:09:15 +0200 Subject: [PATCH 21/37] sketches --- .../thumbnail_constant_depth_mps_prep.png | Bin 19508 -> 62696 bytes .../constant_depth_mps_prep/algorithm.png | Bin 215350 -> 444775 bytes .../operator_pushing.png | Bin 36918 -> 61327 bytes .../operator_pushing_example.png | Bin 36918 -> 139606 bytes 4 files changed, 0 insertions(+), 0 deletions(-) diff --git a/_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_constant_depth_mps_prep.png b/_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_constant_depth_mps_prep.png index 4ea9d7c48f55c19031ac8d6b355ce738f730f91e..babc9ebf873e20cdcbcc9706cc3addaf6ccee603 100644 GIT binary patch literal 62696 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2001 From: dwierichs Date: Wed, 28 Aug 2024 14:34:23 +0200 Subject: [PATCH 22/37] code review; shorten; skip decomposition --- .../tutorial_constant_depth_mps_prep.py | 300 ++++++------------ 1 file changed, 105 insertions(+), 195 deletions(-) diff --git a/demonstrations/tutorial_constant_depth_mps_prep.py b/demonstrations/tutorial_constant_depth_mps_prep.py index 919cc0bb06..ea772972a6 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.py +++ b/demonstrations/tutorial_constant_depth_mps_prep.py @@ -42,10 +42,10 @@ the algorithm from a mathematical perspective. One of these blocks is the sequential MPS preparation circuit to which we will compare later. Alongside this introduction we describe and code up each building block for a -specific MPS. -Then we combine the building blocks into the constant-depth algorithm by -Smith et al. We conclude by computing the correlation length of the prepared -example MPS. +specific MPS. Then we combine the building blocks into the constant-depth +algorithm by Smith et al. and run it for the example MPS. + +TODO: INCLUDE ALGORITHM OVERVIEW IMAGE HERE ALREADY Building blocks --------------- @@ -78,7 +78,9 @@ where :math:`\vec{m}` is a multi-index of :math:`N` indices ranging over :math:`d`, i.e., :math:`\vec{m}\in\{0, 1 \dots, d-1\}^N`, and :math:`\{A^{m}\}_{m}` are :math:`d` -square matrices of equal dimension :math:`D`. This dimension :math:`D` is called the bond dimension, +square matrices of equal dimension :math:`D` (Note that we are using the same +:math:`A^m` for each physical site, due to the translation invariance). +This dimension :math:`D` is called the bond dimension, which is a crucial quantity for the expressivity and complexity of the MPS. We see that :math:`|\Psi\rangle` is fully specified by the :math:`d\times D\times D=dD^2` numbers in the rank-3 tensor :math:`A`. @@ -105,20 +107,31 @@ where :math:`\eta = \frac{1}{\sqrt{1-g}}` and :math:`g\in[-1, 0]` is a freely chosen parameter. -This MPS is a simple yet important example because it can be tuned from the long-range correlated -GHZ state :math:`|\Psi(0)\rangle=\frac{1}{\sqrt{2}}(|0\rangle^{\otimes N} + |1\rangle^{\otimes N})` -to the state :math:`|\Psi(-1)\rangle=|+\rangle^{\otimes N}` with vanishing correlation length. -This example also is discussed in Sec. III C 1. of [#smith]_. -In general, the correlation length of :math:`|\Psi(g)\rangle` is given by +This MPS is a simple yet important example (also discussed in Sec. III C 1. of [#smith]_) +because it can be tuned from the GHZ state +:math:`|\Psi(0)\rangle=\frac{1}{\sqrt{2}}(|0\rangle^{\otimes N} + |1\rangle^{\otimes N})` +to the product state :math:`|\Psi(-1)\rangle=|+\rangle^{\otimes N}`, two states that differ +greatly in their physical properties. + +One such property is the correlation length. +For the GHZ state, a correlator between sites :math:`0` and :math:`j` is + +.. math:: + + C_{\text{GHZ}}(j) = \langle Z(0) Z(j) \rangle_{\text{GHZ}} - \lange Z(0)\rangle_{\text{GHZ}}\lange Z(j)\rangle_{\text{GHZ}} = 1. + +That is, the correlation does not decay with the distance :math:`j`, but the correlation length is +infinite. For the product state, on the other hand, we get .. math:: - \xi(g) = \left|\ln\left(\frac{1+g}{1-g}\right)\right|^{-1}. + C_{+}(j) = \langle Z(0) Z(j) \rangle_{+} - \lange Z(0)\rangle_{+}\lange Z(j)\rangle_{+} = 0, +so that correlations decay "instantaneously". The correlation length vanishes. Long-range correlated states require linear-depth circuits if we are restricted to unitary operations. Therefore, the constant-depth circuit for :math:`|\Psi(0)\rangle` will demonstrate -that dynamic quantum circuits, which are not unitary, are more powerful than unitary -operations alone. +that dynamic quantum circuits, which are not purely unitary, are more powerful than unitary +operations alone! The MPS :math:`|\Psi(g)\rangle` will be our running example throughout this demo. To warm up our coding, let's define the tensor :math:`A` and test that it is in left-canonical form. @@ -127,7 +140,6 @@ import numpy as np import matplotlib.pyplot as plt from matplotlib.patches import FancyBboxPatch -from scipy.optimize import curve_fit import pennylane as qml @@ -152,6 +164,8 @@ def A(g): # Sequential preparation circuit # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # +# TODO IMAGE +# # An MPS like the one above can be prepared in linear depth using an established technique # by Schön et al. [#schoen]_. We introduce this technique here because the new # constant-depth construction uses it as a (parallelized) subroutine, and because we will @@ -159,8 +173,8 @@ def A(g): # # The MPS :math:`|\Psi\rangle` above is given by the rank-3 tensor :math:`A`. # We can think of this as an operator that acts on a bond site and simultaneously *creates* -# a physical site. In order to prepare the state in a circuit, which usually is not allowed -# to create qudits out of nowhere, we want to construct a unitary operation +# a physical site. However, a unitary quantum circuit is not allowed to +# create qudits out of nowhere. To fix this, we construct a unitary operation # from :math:`A` that already takes a physical site as input. # This means that we build a new rank-4 tensor :math:`U` with an additional # axis of dimension :math:`d`. Then we demand that :math:`U` acts like :math:`A` if this @@ -242,8 +256,11 @@ def A(g): print(np.linalg.norm(U_first_term, axis=0)) ###################################################################### -# We see that this fixed part has two norm-1 columns already, and we are -# able to complement this by :math:`C_\perp` using the same columns, but with shifted rows: +# We see that this fixed part has two norm-1 columns already, i.e., it maps the input states +# :math:`|00\rangle` and :math:`|10\rangle` to normalized, orthogonal quantum states. +# We can complement this by mapping :math:`|01\rangle` and :math:`|11\rangle` to two other +# normalized states that are mutually orthogonal again. This turns out to be easy for the +# small example here and leads to the following: E_01 = np.array([[0, 1], [0, 0]]) # |0><1| E_11 = np.array([[0, 0], [0, 1]]) # |1><1| @@ -257,66 +274,47 @@ def A(g): ###################################################################### # Great! This means that we already found the unitary :math:`U` for this MPS. -# Combining the terms above on paper, we see that -# -# .. math:: -# -# U &= A^0 \otimes (|0\rangle\!\langle 0| + |1\rangle\!\langle 1|) -# + A^1\otimes (|1\rangle\!\langle 0| + |0\rangle\!\langle 1|)\\ -# &= A^0 \otimes \mathbb{I} + A^1\otimes X -# -# This looks a lot like a :class:`~.pennylane.CNOT` gate. -# Removing it from :math:`U` we find +# We will implement it as the matrix that it is via :class:`~.pennylane.QubitUnitary`. +# If you're curious, you can try to decompose it into elementary gates (hint: you only +# need two!). # -# .. math:: -# -# U \operatorname{CNOT} -# &= (A^0 E_{00}) \otimes \mathbb{I} + (A^0E_{11}) \otimes X + (A^1 E_{00}) \otimes X + (A^1E_{11}) \otimes \mathbb{I}\\ -# &= \left(\begin{matrix} \eta & -\eta\sqrt{-g} \\ \eta\sqrt{-g} & \eta \end{matrix}\right)\otimes \mathbb{I}, -# -# The remaining operation is an -# :class:`~.pennylane.RY` rotation by the angle :math:`2\arccos(\eta)`, so -# that we can easily code up the unitary as a small circuit template: +# Now let's code up the entire sequential preparation circuit (we will be applying +# the unitary :math:`U` to the second instead of the first bond qubit). We implement the +# measurement \& postselection step as a separate function ``project_measure`` for +# easier reusability later on. -def U_template(eta, bond_idx, phys_idx): - qml.CNOT([bond_idx, phys_idx]) - qml.RY(2 * np.arccos(eta), bond_idx) +def label_fn(self, *_, **__): + """Report the physical wire this operation acts on in its label.""" + return f"$U^{{({self.wires[1]})}}$" -eta = 1 / np.sqrt(1 - g) -template_matrix = qml.matrix(U_template, wire_order=["bond", "phys"])(eta, "bond", "phys") -print(f"The template reproduces U: {np.allclose(template_matrix, U)}") - -###################################################################### -# Excellent, now that we have a circuit that realizes the unitary :math:`U`, let's code up the entire -# sequential preparation circuit. We implement the measurement \& postselection step as a separate -# function ``project_measure``. - - -def prepare_bell(wire_0, wire_1): - """Prepare two qubits in the Bell state (|00>+|11>)/sqrt(2).""" - qml.Hadamard(wire_0) - qml.CNOT([wire_0, wire_1]) +qml.QubitUnitary.label = label_fn def sequential_preparation(g, wires): """Prepare the example MPS \Psi(g) on N qubits where N is the length of the passed wires minus 2. The bond qubits are still entangled.""" eta = 1 / np.sqrt(1 - g) - bond_wires = [wires[0], wires[-1]] # Bond qubits [0, N+1] - phys_wires = wires[1:-1] # Physical qubits [1, 2, ... N] - prepare_bell(*bond_wires) + # Define bond qubits [0, N+1] and physical qubits [1, 2, ... N] + bond0, bond1 = wires[0], wires[-1] + phys_wires = wires[1:-1] + + # Prepare bond qubits in the Bell state (|00>+|11>)/sqrt(2) + qml.Hadamard(bond0) + qml.CNOT([bond0, bond1]) + + # Apply unitary U to second bond qubit and each physical qubit for phys_wire in phys_wires: - U_template(eta, bond_wires[1], phys_wire) + u = qml.QubitUnitary(U, wires=[bond1, phys_wire]) def project_measure(wire_0, wire_1): """Measure two qubits in the Bell basis and postselect on (|00>+|11>)/sqrt(2).""" - # Move bond qubits to Bell basis by undoing the preparation + # Move bond qubits from Bell basis to computational basis qml.CNOT([wire_0, wire_1]) qml.Hadamard(wire_0) - # Measure and postselect |00> + # Measure in computational basis and postselect |00> qml.measure(wire_0, postselect=0) qml.measure(wire_1, postselect=0) @@ -349,6 +347,8 @@ def sequential_circuit(N, g): # Fusion of MPS states with mid-circuit measurements # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # +# TODO IMAGE +# # The next ingredient for the constant-depth MPS preparation circuit is to fuse the product # state of two MPS states together into one (entangled) MPS. Recall that a single sequential # creation of this final MPS would pass a bond site between the first and second group of physical @@ -359,21 +359,28 @@ def sequential_circuit(N, g): # together simultaneously is at the heart of going from linear to constant circuit depth. # # Consider a state :math:`|\Phi\rangle=|\Phi_1\rangle\otimes|\Phi_2\rangle`, where -# :math:`|\Phi_{1,2}\rangle` are MPS prepared with the sequential technique from -# above on :math:`K` and :math:`L` physical qudits, respectively, but with the bond -# qudits left unmeasured. In particular, the postselection step has not been performed +# +# .. math:: +# +# |\Phi_{r}\rangle = \frac{1}{\sqrt{D}} \sum_{j,k} +# \sum_{\vec{m}} |\vec{m}\rangle \langle j|A^{m_1} A^{m_2}\cdots A^{m_{N_r}}|k\rangle |jk\rangle_D +# +# for :math:`r=1,2` are MPS on :math:`N_r` qudits, respectively. We assume that they have been +# prepared with the sequential technique from above, but that the bond +# qudits have not been measured yet. In particular, the postselection step has not been performed # yet. Using the form of :math:`|\psi_1\rangle` from the recipe above, we can write # this state as # # .. math:: # -# |\Phi\rangle = \frac{1}{D} \sum_{i,j,\ell,p} -# \sum_{\vec{m}} |\vec{m}\rangle |ij\ell p\rangle_D -# \langle i|A^{m_1} \cdots A^{m_K}|j\rangle_D -# \langle \ell|A^{m_{K+1}} \cdots A^{m_{K+L}}|p\rangle_D. +# |\Phi\rangle = \frac{1}{D} \sum_{j,k,\ell,p} +# \sum_{\vec{m}} |\vec{m}\rangle |jk\ell p\rangle_D +# \langle j|A^{m_1} \cdots A^{m_{N_1}}|k\rangle_D +# \langle \ell|A^{m_{{N_1}+1}} \cdots A^{m_{{N_1}+{N_2}}}|p\rangle_D. # # Now we want to measure two bond qudits in an entangled basis, one of -# :math:`|\Phi_1\rangle` and :math:`|\Phi_2\rangle` each. Assume this basis +# :math:`|\Phi_1\rangle` and :math:`|\Phi_2\rangle` each (e.g., the ones indexed +# by :math:`k` and :math:`\ell` above). Assume this basis # to be given in the form of (orthogonal) states # # .. math:: @@ -411,13 +418,18 @@ def sequential_circuit(N, g): # # This choice must seem arbitrary at this point, but will become clear # later on. There, we will see that in general the matrices :math:`B^k` and the MPS -# tensor :math:`A` have to "play well" together. For the choice above, we find the matrices +# tensor :math:`A` have to "play well" together. +# As an example, this choice leads to :math:`B^1_{j\ell} = 1` for +# :math:`(j,\ell)=(0, 1)` and :math:`(j, \ell)=(1,0)`, and :math:`B^1_{j\ell}=0` +# otherwise (note that the factor :math:`\frac{1}{\sqrt{2}}` is not part of :math:`B^k` +# in the general equation above). This means that :math:`B^1=X`, the Pauli matrix! +# Similarly, we find # # .. math:: # # B^0 = I \quad B^1 = X \quad B^2 = Z \quad B^3 = Y, # -# i.e., the standard Pauli matrices. +# i.e., all standard Pauli matrices. # The measurement procedure can then be coded up in the following short function, # which is similar to the ``project_measure`` function from the sequential preparation # routine. Note, however, that instead of postselecting, we record the measurement @@ -581,6 +593,8 @@ def push_and_correct(op_id, phys_wires): # # We summarize the algorithm in the following sketch, which again follows [#smith]_. # +# TODO: ADAPT WORDING IN IMAGE FROM PAPER TO DEMO +# # .. image:: ../_static/demonstration_assets/constant_depth_mps_prep/algorithm.png # :width: 75% # :align: center @@ -623,10 +637,6 @@ def correct_end_bond(bond_idx, op_id): qml.cond(op_id[0], qml.Z)(bond_idx) # Apply X if correction op is X or Y qml.cond(op_id[1], qml.X)(bond_idx) - # op_int = np.array([2, 1]) @ op_id - # qml.cond(op_int == 1, qml.X)(bond_idx) - # qml.cond(op_int == 2, qml.Z)(bond_idx) - # qml.cond(op_int == 3, qml.Y)(bond_idx) def constant_depth(N, g, q): @@ -702,25 +712,25 @@ def constant_depth_circuit(N, g, q): ((-0.45, -0.35), 1.7, 4.7, "#FF87EB"), # Bond entangling 1 ((-0.45, 4.65), 1.7, 4.7, "#FF87EB"), # Bond entangling 2 ((-0.45, 9.65), 1.7, 4.7, "#FF87EB"), # Bond entangling 3 - ((1.65, 0.55), 5.7, 3.8, "#FFE096"), # Sequential prep 1 - ((1.65, 5.55), 5.7, 3.8, "#FFE096"), # Sequential prep 2 - ((1.65, 10.55), 5.7, 3.8, "#FFE096"), # Sequential prep 3 - ((7.65, 3.6), 1.7, 1.65, "#D7A2F6"), # Fuse 1 and 2 - ((11.65, 8.65), 1.7, 1.65, "#D7A2F6"), # Fuse (1,2) and 3 - ((9.65, 3.6), 9.7, 4.75, "#70CEFF"), # Push and correct b/w 1 and 2 - ((13.65, 8.65), 9.7, 4.75, "#70CEFF"), # Push and correct b/w (1, 2) and 3 - ((23.6, 13.6), 1.75, 0.8, "#C756B2"), # Correct pushed bond operator - ((25.7, -0.35), 2.7, 14.75, "#B5F2ED"), # Projective measurement + ((1.55, 0.55), 2.85, 3.8, "#FFE096"), # Sequential prep 1 + ((1.55, 5.55), 2.85, 3.8, "#FFE096"), # Sequential prep 2 + ((1.55, 10.55), 2.85, 3.8, "#FFE096"), # Sequential prep 3 + ((4.7, 3.6), 1.65, 1.65, "#D7A2F6"), # Fuse 1 and 2 + ((8.7, 8.65), 1.65, 1.65, "#D7A2F6"), # Fuse (1,2) and 3 + ((6.65, 3.6), 9.7, 4.75, "#70CEFF"), # Push and correct b/w 1 and 2 + ((10.65, 8.65), 9.7, 4.75, "#70CEFF"), # Push and correct b/w (1, 2) and 3 + ((20.6, 13.6), 1.75, 0.8, "#C756B2"), # Correct pushed bond operator + ((22.7, -0.35), 2.7, 14.75, "#B5F2ED"), # Projective measurement ] labels = [ ("Step 1a", 0.5, 14.75), - ("Step 1b", 4.5, 14.75), - ("Step 2", 8.5, 5.5), - ("Step 2", 12.5, 10.5), - ("Step 3", 20.5, 7.5), - ("Step 4", 24.5, 13), - ("Step 5", 27, 14.75), + ("Step 1b", 3.0, 14.75), + ("Step 2", 5.5, 5.75), + ("Step 2", 9.5, 10.75), + ("Step 3", 17.5, 7.5), + ("Step 4", 21.5, 13), + ("Step 5", 24, 14.75), ] for xy, width, height, color in box_data: @@ -738,126 +748,26 @@ def constant_depth_circuit(N, g, q): p_seq = sequential_circuit(N, g) p_const = constant_depth_circuit(N, g, q) print(f"The two probabilities agree for {g=:.1f}: {np.allclose(p_seq, p_const)}") +plt.show() ###################################################################### -# Nice, we have a constant-depth circuit that reproduces the existing sequential +# Nice, we arrived at a *constant-depth* circuit that reproduces the existing sequential # preparation circuit, which has linear depth! -# Not that while the dynamic operations of step 3 seem to have a linear depth, +# Note that while the dynamic operations of step 3 appear visually to have a linear depth, # they can be applied in parallel because they are only controlled by classical # registers. # -# Correlation length -# ~~~~~~~~~~~~~~~~~~ -# -# When introducing the example MPS :math:`|\Psi(g)\rangle`, we mentioned that the -# parameter :math:`g` controls the correlation length of the state. As extreme examples, -# we can produce a zero-correlation length state for :math:`g=-1` and a long-range -# correlated state, namely the GHZ state, for :math:`g\to 0`. We can visualize -# these states via their probability distributions (note the logarithmic axis). -# - -N = 6 -g = -1 -uniform_probs = constant_depth_circuit(N, g, q) -g = -1e-8 -ghz_probs = constant_depth_circuit(N, g, q) -ints = list(range(2**N)) -fig, ax = plt.subplots() -label_plus = "zero correlation length ($|+\\rangle^N$)" -label_ghz = "long correlation length (GHZ)" -ax.bar(ints, ghz_probs, width=1, color="#ff87eb", label=label_ghz, edgecolor="k") -ax.bar(ints, uniform_probs, width=1, color="#70ceff", label=label_plus, edgecolor="0.5") -ax.set_yscale("log") -ax.set_ylim([1e-3, 1]) -_ = ax.legend() - -###################################################################### -# In a more nuanced setting, let us produce the MPS in question for a series of -# values for :math:`g`, and measure the correlation length ourselves. -# For this, we measure the two-body correlators -# -# .. math:: -# -# C(j) = \langle Z_1 Z_j\rangle - \langle Z_1\rangle \langle Z_j\rangle -# -# for an increasing distance :math:`j`. - - -@qml.qnode(dev, interface="numpy") -def pre_correlations(N, g, q): - phys_wires = constant_depth_ansatz(N, g, q) - single_obs = [qml.expval(qml.Z(i)) for i in phys_wires] - # Exclude the product Z(1) @ Z(1) = I - prod_obs = [qml.expval(qml.Z(1) @ qml.Z(i)) for i in phys_wires[1:]] - return single_obs + prod_obs - - -def correlations(N, g, q): - # Compute expectation values for correlators - expvals = pre_correlations(N, g, q) - single_expvals = expvals[:N] - prod_expvals = expvals[N:] - # Exclude the self-correlation C(1) - return [p - single_expvals[0] * s for p, s in zip(prod_expvals, single_expvals[1:])] - - -N = 12 -q = 6 -gs = [-0.2, -0.4, -0.6, -0.8] -fig, ax = plt.subplots() -colors = ["#70ceff", "#ffe096", "#c756b2", "#e1eba8"] -plot_kwargs = dict(ls="", marker="o", markersize=7, markeredgecolor="k") -all_correls = [] -for g, c in zip(gs, colors): - correls = correlations(N, g, q) - ax.plot(list(range(N - 1)), correls, color=c, label=f"$g={g}$", **plot_kwargs) - all_correls.append(correls) -ax.set_yscale("log") -ax.set_xlabel("$j$") -ax.set_ylabel("$C(j)$") -ax.set_ylim([5e-8, 1.5]) -ax.legend() -plt.show() - -###################################################################### -# Note that due to the periodic boundary conditions, we have :math:`C(N-j)=C(j)`. -# This is why the correlations increase again after :math:`j=\frac{N}{2}`. -# To obtain the correlation length, we fit an exponential to the measured correlators -# up until :math:`j=\frac{N}{2}` and invert the decay rate. -# We can also compare this result to the analytic result for the correlation -# length :math:`\xi` mentioned in the beginning: - - -def exp_fn(x, a, b): - return a * np.exp(x * b) - - -mid_point = N // 2 + 1 -distances = list(range(1, mid_point)) -print(f"Correlation lengths xi\n======================") -for g, correls in zip(gs, all_correls): - popt, pcov = curve_fit(exp_fn, distances, correls[1:mid_point]) - xi_numerical = -1 / popt[1] - xi_predicted = 1 / abs(np.log((1 + g) / (1 - g))) - print(f"{g=:.1f}") - print(f"Numerical: {xi_numerical:.5f}") - print(f"Predicted: {xi_predicted:.5f}") - print() - -###################################################################### -# As we can see, the produced MPS states indeed have the predicted correlation lengths, -# up to some numerical imprecisions. -# # Conclusion # ---------- # -# We successfully implemented a constand-depth dynamic quantum circuit that +# We successfully implemented a constant-depth dynamic quantum circuit that # prepares an MPS with a parametrized correlation length. In particular, # we saw that a dynamic quantum circuit can reach quantum states in constant depth # that require linear depth with purely unitary circuits. # The cost for this advantage are the auxiliary qubits we had to add as intermediate -# bond sites. Nonetheless, this constant-depth algorithm is an exciting advance in +# bond sites. +# This constant-depth algorithm is an exciting advance in # state preparation techniques, alleviating requirements of coherence times and # connectivity on hardware. # From fe0fb6192f6b8efd95ce277caf83d84dcf3de4e4 Mon Sep 17 00:00:00 2001 From: dwierichs Date: Thu, 5 Sep 2024 09:12:48 +0200 Subject: [PATCH 23/37] minor image update --- demonstrations/tutorial_chemical_reactions.py | 2 +- .../tutorial_constant_depth_mps_prep.py | 43 +++++++++++++++---- 2 files changed, 36 insertions(+), 9 deletions(-) diff --git a/demonstrations/tutorial_chemical_reactions.py b/demonstrations/tutorial_chemical_reactions.py index 1d69bd6196..62874c11e6 100644 --- a/demonstrations/tutorial_chemical_reactions.py +++ b/demonstrations/tutorial_chemical_reactions.py @@ -436,4 +436,4 @@ def circuit(parameters): # ----------------- # .. include:: ../_static/authors/varun_rishi.txt # -# .. include:: ../_static/authors/juan_miguel_arrazola.txt \ No newline at end of file +# .. include:: ../_static/authors/juan_miguel_arrazola.txt diff --git a/demonstrations/tutorial_constant_depth_mps_prep.py b/demonstrations/tutorial_constant_depth_mps_prep.py index ea772972a6..59d220fd15 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.py +++ b/demonstrations/tutorial_constant_depth_mps_prep.py @@ -1,3 +1,18 @@ +# TODOs +# +# - Adapt wording in algorithm picture to demo wording +# +# - Create sketch of algorithm preview that is reduced c.f. full algorithm picture +# +# - Create sketch for sequential preparation +# +# - Create sketch for fusion step +# +# - Adapt algorithm and algorithm preview pictures to the fusion step picture +# +# - Comment on bond measurements being the same for sequential and const-depth algorithm, +# and how that works (brief!) +# r"""Constant-depth preparation of matrix product states with dynamic circuits ============================================================================= @@ -44,8 +59,11 @@ Alongside this introduction we describe and code up each building block for a specific MPS. Then we combine the building blocks into the constant-depth algorithm by Smith et al. and run it for the example MPS. +As a preview, this is what the algorithm will look like schematically: -TODO: INCLUDE ALGORITHM OVERVIEW IMAGE HERE ALREADY +.. image:: ../_static/demonstration_assets/constant_depth_mps_prep/algorithm_preview.png + :width: 75% + :align: center Building blocks --------------- @@ -128,7 +146,7 @@ C_{+}(j) = \langle Z(0) Z(j) \rangle_{+} - \lange Z(0)\rangle_{+}\lange Z(j)\rangle_{+} = 0, so that correlations decay "instantaneously". The correlation length vanishes. -Long-range correlated states require linear-depth circuits if we are restricted to unitary +Long-range correlated states require linear-depth circuits if we are restricted to unitary operations. Therefore, the constant-depth circuit for :math:`|\Psi(0)\rangle` will demonstrate that dynamic quantum circuits, which are not purely unitary, are more powerful than unitary operations alone! @@ -164,8 +182,6 @@ def A(g): # Sequential preparation circuit # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # -# TODO IMAGE -# # An MPS like the one above can be prepared in linear depth using an established technique # by Schön et al. [#schoen]_. We introduce this technique here because the new # constant-depth construction uses it as a (parallelized) subroutine, and because we will @@ -243,6 +259,14 @@ def A(g): # physical qubit, and a final basis change to measure the Bell basis. Overall, this amounts to a linear # operation count and circuit depth. # +# These steps are visualized in the following sketch, with the bond qudits starting on the right. +# Note that the sketch deviates from standard circuit diagrams in that it passes one bond qudits from +# the right to the left, instead of keeping the position of the qudits fixed. +# +# .. image:: ../_static/demonstration_assets/constant_depth_mps_prep/sequential_prep.png +# :width: 75% +# :align: center +# # **Example** # # For our example MPS :math:`|\Psi(g)\rangle`, let us first find the unitary :math:`U`. @@ -347,8 +371,6 @@ def sequential_circuit(N, g): # Fusion of MPS states with mid-circuit measurements # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # -# TODO IMAGE -# # The next ingredient for the constant-depth MPS preparation circuit is to fuse the product # state of two MPS states together into one (entangled) MPS. Recall that a single sequential # creation of this final MPS would pass a bond site between the first and second group of physical @@ -358,6 +380,13 @@ def sequential_circuit(N, g): # As a matter of fact, chaining beads on smaller strings and the fact that we can knot them # together simultaneously is at the heart of going from linear to constant circuit depth. # +# Before we dive into the mathematical details, we visualize the idea behind the fusion step +# schematically: +# +# .. image:: ../_static/demonstration_assets/constant_depth_mps_prep/fusion.png +# :width: 75% +# :align: center +# # Consider a state :math:`|\Phi\rangle=|\Phi_1\rangle\otimes|\Phi_2\rangle`, where # # .. math:: @@ -593,8 +622,6 @@ def push_and_correct(op_id, phys_wires): # # We summarize the algorithm in the following sketch, which again follows [#smith]_. # -# TODO: ADAPT WORDING IN IMAGE FROM PAPER TO DEMO -# # .. image:: ../_static/demonstration_assets/constant_depth_mps_prep/algorithm.png # :width: 75% # :align: center From b9a1f985fabb59f454c53cba9e4a19038baf5fe7 Mon Sep 17 00:00:00 2001 From: dwierichs Date: Tue, 10 Sep 2024 14:53:50 +0200 Subject: [PATCH 24/37] polish --- ...rial_constant_depth_mps_prep.metadata.json | 6 ++--- .../tutorial_constant_depth_mps_prep.py | 27 ++++++------------- 2 files changed, 11 insertions(+), 22 deletions(-) diff --git a/demonstrations/tutorial_constant_depth_mps_prep.metadata.json b/demonstrations/tutorial_constant_depth_mps_prep.metadata.json index 4c405f4b97..bba90467c8 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.metadata.json +++ b/demonstrations/tutorial_constant_depth_mps_prep.metadata.json @@ -2,11 +2,11 @@ "title": "Constant-depth preparation of matrix product states with dynamic circuits", "authors": [ { - "id": "david_wierichs" + "username": "dwierichs" } ], - "dateOfPublication": "2024-09-10T00:00:00+00:00", - "dateOfLastModification": "2024-09-10T00:00:00+00:00", + "dateOfPublication": "2024-09-25T00:00:00+00:00", + "dateOfLastModification": "2024-09-25T00:00:00+00:00", "categories": [ "Quantum Computing", "Algorithms" diff --git a/demonstrations/tutorial_constant_depth_mps_prep.py b/demonstrations/tutorial_constant_depth_mps_prep.py index 59d220fd15..554f8baec6 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.py +++ b/demonstrations/tutorial_constant_depth_mps_prep.py @@ -1,18 +1,3 @@ -# TODOs -# -# - Adapt wording in algorithm picture to demo wording -# -# - Create sketch of algorithm preview that is reduced c.f. full algorithm picture -# -# - Create sketch for sequential preparation -# -# - Create sketch for fusion step -# -# - Adapt algorithm and algorithm preview pictures to the fusion step picture -# -# - Comment on bond measurements being the same for sequential and const-depth algorithm, -# and how that works (brief!) -# r"""Constant-depth preparation of matrix product states with dynamic circuits ============================================================================= @@ -259,11 +244,11 @@ def A(g): # physical qubit, and a final basis change to measure the Bell basis. Overall, this amounts to a linear # operation count and circuit depth. # -# These steps are visualized in the following sketch, with the bond qudits starting on the right. -# Note that the sketch deviates from standard circuit diagrams in that it passes one bond qudits from -# the right to the left, instead of keeping the position of the qudits fixed. +# These steps are visualized in the following sketch, with the bond qudits starting at the top. +# Note that the sketch deviates from standard circuit diagrams in that it passes one bond qudit from +# the top to the bottom, instead of keeping the position of the qudits fixed. # -# .. image:: ../_static/demonstration_assets/constant_depth_mps_prep/sequential_prep.png +# .. image:: ../_static/demonstration_assets/constant_depth_mps_prep/sequential.png # :width: 75% # :align: center # @@ -629,6 +614,10 @@ def push_and_correct(op_id, phys_wires): # It is important to remember that showing the existence of suitable operator pushing # relations (and finding them explicitly) is a crucial step in general, which goes # beyond the scope of the demo. +# +# The projective measurement step at the end is the same as for the sequential +# preparation algorithm, and therefore is *probabilistic*, with the same +# success probability. # # Block size :math:`q` # ~~~~~~~~~~~~~~~~~~~~ From 17680f28bfd72e7ff812027ffe7e2626bc326b90 Mon Sep 17 00:00:00 2001 From: dwierichs Date: Tue, 10 Sep 2024 14:54:05 +0200 Subject: [PATCH 25/37] [skip ci] From 02bcc0cfd8636daf4f3c4a0fdb8d28be31198f5c Mon Sep 17 00:00:00 2001 From: dwierichs Date: Thu, 19 Sep 2024 09:23:26 +0200 Subject: [PATCH 26/37] update images --- ...humbnail_large_constant_depth_mps_prep.png | Bin 96717 -> 49464 bytes .../thumbnail_constant_depth_mps_prep.png | Bin 62696 -> 49464 bytes .../constant_depth_mps_prep/algorithm.png | Bin 444775 -> 164926 bytes .../algorithm_preview.png | Bin 0 -> 213540 bytes .../constant_depth_mps_prep/fusion.png | Bin 0 -> 69087 bytes 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z4@}Sh|Ly<18iVg*V`Jsg28M=e{tWgIYxFnOfvh9XD(=C@S2`zOGAF-%E6<^x_M&w( zM~H|Iu8ZR^C3GFVva2`I7gji&qp0_=y$5OQ?Tlj6tT(Hk21$|W|!wBYA= zZ{6I&N&NBiQ8r%uLUPz!^uJ$u2)p~i|Gz&T7C+QCVQntZb%*#VnTv{&X%g2T{4daN BCyoFB diff --git a/demonstrations/tutorial_constant_depth_mps_prep.py b/demonstrations/tutorial_constant_depth_mps_prep.py index 554f8baec6..177f5e2189 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.py +++ b/demonstrations/tutorial_constant_depth_mps_prep.py @@ -44,12 +44,16 @@ Alongside this introduction we describe and code up each building block for a specific MPS. Then we combine the building blocks into the constant-depth algorithm by Smith et al. and run it for the example MPS. -As a preview, this is what the algorithm will look like schematically: +As a rough preview, this is what the algorithm will look like schematically: .. image:: ../_static/demonstration_assets/constant_depth_mps_prep/algorithm_preview.png :width: 75% :align: center +The algorithm consists of a parallelized preparation step of states +on fewer qubits, a fusion step connecting them to a defective state on all qubits, +and finally correcting the defects to obtain the target state. + Building blocks --------------- From 51b4ac9121ba0ccd4b46745d1c668f4d38bd417a Mon Sep 17 00:00:00 2001 From: dwierichs Date: Thu, 19 Sep 2024 09:24:34 +0200 Subject: [PATCH 27/37] undo unrelated --- demonstrations/tutorial_chemical_reactions.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_chemical_reactions.py b/demonstrations/tutorial_chemical_reactions.py index 62874c11e6..1d69bd6196 100644 --- a/demonstrations/tutorial_chemical_reactions.py +++ b/demonstrations/tutorial_chemical_reactions.py @@ -436,4 +436,4 @@ def circuit(parameters): # ----------------- # .. include:: ../_static/authors/varun_rishi.txt # -# .. include:: ../_static/authors/juan_miguel_arrazola.txt +# .. include:: ../_static/authors/juan_miguel_arrazola.txt \ No newline at end of file From d941edb975ab60726c1a99ecae299921ae517ffc Mon Sep 17 00:00:00 2001 From: dwierichs Date: Thu, 19 Sep 2024 10:10:43 +0200 Subject: [PATCH 28/37] additional references, image sizes --- .../tutorial_constant_depth_mps_prep.py | 31 ++++++++++++++----- 1 file changed, 24 insertions(+), 7 deletions(-) diff --git a/demonstrations/tutorial_constant_depth_mps_prep.py b/demonstrations/tutorial_constant_depth_mps_prep.py index 177f5e2189..09983a984d 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.py +++ b/demonstrations/tutorial_constant_depth_mps_prep.py @@ -125,14 +125,14 @@ .. math:: - C_{\text{GHZ}}(j) = \langle Z(0) Z(j) \rangle_{\text{GHZ}} - \lange Z(0)\rangle_{\text{GHZ}}\lange Z(j)\rangle_{\text{GHZ}} = 1. + C_{\text{GHZ}}(j) = \langle Z(0) Z(j) \rangle_{\text{GHZ}} - \langle Z(0)\rangle_{\text{GHZ}}\langle Z(j)\rangle_{\text{GHZ}} = 1. That is, the correlation does not decay with the distance :math:`j`, but the correlation length is infinite. For the product state, on the other hand, we get .. math:: - C_{+}(j) = \langle Z(0) Z(j) \rangle_{+} - \lange Z(0)\rangle_{+}\lange Z(j)\rangle_{+} = 0, + C_{+}(j) = \langle Z(0) Z(j) \rangle_{+} - \langle Z(0)\rangle_{+}\langle Z(j)\rangle_{+} = 0, so that correlations decay "instantaneously". The correlation length vanishes. Long-range correlated states require linear-depth circuits if we are restricted to unitary @@ -516,7 +516,7 @@ def prepare_and_unprepare(g): # Visually, we find: # # .. image:: ../_static/demonstration_assets/constant_depth_mps_prep/operator_pushing.png -# :width: 35% +# :width: 80% # :align: center # # For simplicity, we denote a push by :math:`B^k\mapsto (C^k, D^k)`. @@ -530,7 +530,7 @@ def prepare_and_unprepare(g): # measurement basis. In particular, the relations are # # .. image:: ../_static/demonstration_assets/constant_depth_mps_prep/operator_pushing_example.png -# :width: 75% +# :width: 90% # :align: center # # That is, in the notation above we have @@ -612,7 +612,7 @@ def push_and_correct(op_id, phys_wires): # We summarize the algorithm in the following sketch, which again follows [#smith]_. # # .. image:: ../_static/demonstration_assets/constant_depth_mps_prep/algorithm.png -# :width: 75% +# :width: 90% # :align: center # # It is important to remember that showing the existence of suitable operator pushing @@ -695,7 +695,9 @@ def constant_depth_ansatz(N, g, q): # Step 5: Perform projective measurement on outer-most bond sites. project_measure(*outer_bond_sites) # Collect wire ranges for physical wires, skipping bond wires - phys_wires = (range(block_wires * i + 1, block_wires * (i + 1) - 1) for i in range(num_blocks)) + phys_wires = ( + range(block_wires * i + 1, block_wires * (i + 1) - 1) for i in range(num_blocks) + ) # Turn ranges to lists and sum them together return sum(map(list, phys_wires), start=[]) @@ -774,7 +776,7 @@ def constant_depth_circuit(N, g, q): ###################################################################### # Nice, we arrived at a *constant-depth* circuit that reproduces the existing sequential # preparation circuit, which has linear depth! -# Note that while the dynamic operations of step 3 appear visually to have a linear depth, +# Note that while the dynamic operations of steps 2 and 3 appear visually to have a linear depth, # they can be applied in parallel because they are only controlled by classical # registers. # @@ -791,6 +793,10 @@ def constant_depth_circuit(N, g, q): # state preparation techniques, alleviating requirements of coherence times and # connectivity on hardware. # +# We want to note that there are other MPS preparation algorithms with depths that only +# scale logarithmically in the qubit count [#malz1]_. They, too, can be improved further by +# using mid-circuit measurements and dynamic circuit operations [#malz2]_ +# # For more information, consider the original article, the mentioned reviews, as well # as our demos on dynamic circuits and tensor network states. # @@ -820,6 +826,17 @@ def constant_depth_circuit(N, g, q): # `closed access `__, 2011. # `arXiv:1008.3477 `__, 2010. # +# .. [#malz1] +# +# Z. Wei, D. Malz, J. I. Cirac +# "Efficient adiabatic preparation of tensor network states", Physical Review Research, **5**, L022037, +# `open access `__, 2023. +# +# .. [#malz2] +# +# Z. Wei, D. Malz, J. I. Cirac +# "Preparation of Matrix Product States with Log-Depth Quantum Circuits", Physical Review Letters, **132**, 040404, +# `open access `__, 2024. # # About the author # ---------------- From a8b11cfd11bf0a1a409d92eaff13f56aaf0c0d4e Mon Sep 17 00:00:00 2001 From: dwierichs Date: Thu, 19 Sep 2024 12:11:11 +0200 Subject: [PATCH 29/37] polish --- demonstrations/tutorial_constant_depth_mps_prep.py | 8 +++++--- 1 file changed, 5 insertions(+), 3 deletions(-) diff --git a/demonstrations/tutorial_constant_depth_mps_prep.py b/demonstrations/tutorial_constant_depth_mps_prep.py index 09983a984d..4c54a1559f 100644 --- a/demonstrations/tutorial_constant_depth_mps_prep.py +++ b/demonstrations/tutorial_constant_depth_mps_prep.py @@ -248,9 +248,11 @@ def A(g): # physical qubit, and a final basis change to measure the Bell basis. Overall, this amounts to a linear # operation count and circuit depth. # -# These steps are visualized in the following sketch, with the bond qudits starting at the top. -# Note that the sketch deviates from standard circuit diagrams in that it passes one bond qudit from -# the top to the bottom, instead of keeping the position of the qudits fixed. +# Alongside the correspondence between :math:`U` acting on :math:`|0\rangle` and :math:`A`, +# these steps are visualized in the following sketch. +# Note that the sketch deviates from standard circuit diagrams: +# The bond qudits start at the top and throughout the circuit one of them is passed from +# the top to the bottom, instead of keeping the position of all qudits fixed. # # .. image:: ../_static/demonstration_assets/constant_depth_mps_prep/sequential.png # :width: 75% From 0df16720d0198537c9b1bb9264c3fe5818f017bf Mon Sep 17 00:00:00 2001 From: dwierichs Date: Thu, 19 Sep 2024 16:27:12 +0200 Subject: [PATCH 30/37] images update again 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