From 8c3ed01a265ddb02bdfb1ae57086bf3bb4a5a66b Mon Sep 17 00:00:00 2001 From: JosephRRB <36796950+JosephRRB@users.noreply.github.com> Date: Fri, 20 Sep 2024 06:02:56 -0700 Subject: [PATCH 1/2] GQE demo using Pennylane (#1119) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit ### Before submitting Please complete the following checklist when submitting a PR: - [ ] Ensure that your tutorial executes correctly, and conforms to the guidelines specified in the [README](../README.md). - [ ] Remember to do a grammar check of the content you include. - [ ] All tutorials conform to [PEP8 standards](https://www.python.org/dev/peps/pep-0008/). To auto format files, simply `pip install black`, and then run `black -l 100 path/to/file.py`. When all the above are checked, delete everything above the dashed line and fill in the pull request template. ------------------------------------------------------------------------------------------------------------ **Title:** Generative quantum eigensolver demo using Pennylane **Summary:** We use Pennylane to generate a static molecular dataset and calculate the corresponding energies to train a small GPT model as described by https://arxiv.org/abs/2401.09253. We show that as training progresses, the GPT model generates operator sequences whose predicted energies more accurately resembles the true energies calculated by Pennylane. In addition, the sampling process is shown to better generate the ground state for better performing models. - Story ticket: https://app.shortcut.com/xanaduai/story/64095/contribute-gqe-demo-as-a-pennylane-demo **Relevant references:** **Possible Drawbacks:** **Related GitHub Issues:** ---- If you are writing a demonstration, please answer these questions to facilitate the marketing process. * GOALS — Why are we working on this now? *Eg. Promote a new PL feature or show a PL implementation of a recent paper.* * AUDIENCE — Who is this for? *Eg. Chemistry researchers, PL educators, beginners in quantum computing.* * KEYWORDS — What words should be included in the marketing post? * Which of the following types of documentation is most similar to your file? (more details [here](https://www.notion.so/xanaduai/Different-kinds-of-documentation-69200645fe59442991c71f9e7d8a77f8)) - [ ] Tutorial - [ ] Demo - [ ] How-to --------- Co-authored-by: JosephRRB Co-authored-by: Josh Izaac Co-authored-by: Ivana Kurečić --- _static/authors/joseph_bunao.jpg | Bin 0 -> 409283 bytes _static/authors/joseph_bunao.txt | 4 + ...l_large_generative_quantum_eigensolver.png | Bin 0 -> 109349 bytes ...umbnail_generative_quantum_eigensolver.png | Bin 0 -> 102285 bytes ...umbnail_generative_quantum_eigensolver.png | Bin 0 -> 25014 bytes .../gqe_training/gqe_performance.png | Bin 0 -> 83489 bytes .../gqe_training/gqe_training_diagram.png | Bin 0 -> 122003 bytes .../gqe_training/gqe_training_loss.png | Bin 0 -> 65450 bytes .../gqe_training/paper_gqe_vqe.png | Bin 0 -> 110536 bytes demonstrations/gqe_training.metadata.json | 74 ++ demonstrations/gqe_training.py | 703 ++++++++++++++++++ 11 files changed, 781 insertions(+) create mode 100644 _static/authors/joseph_bunao.jpg create mode 100644 _static/authors/joseph_bunao.txt create mode 100644 _static/demo_thumbnails/large_demo_thumbnails/thumbnail_large_generative_quantum_eigensolver.png create mode 100644 _static/demo_thumbnails/opengraph_demo_thumbnails/OGthumbnail_generative_quantum_eigensolver.png create mode 100644 _static/demo_thumbnails/regular_demo_thumbnails/thumbnail_generative_quantum_eigensolver.png create mode 100644 _static/demonstration_assets/gqe_training/gqe_performance.png create mode 100644 _static/demonstration_assets/gqe_training/gqe_training_diagram.png create mode 100644 _static/demonstration_assets/gqe_training/gqe_training_loss.png create mode 100644 _static/demonstration_assets/gqe_training/paper_gqe_vqe.png create mode 100644 demonstrations/gqe_training.metadata.json create mode 100644 demonstrations/gqe_training.py diff --git a/_static/authors/joseph_bunao.jpg b/_static/authors/joseph_bunao.jpg new file mode 100644 index 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HcmV?d00001 diff --git a/_static/authors/joseph_bunao.txt b/_static/authors/joseph_bunao.txt new file mode 100644 index 0000000000..ef1b6ee68c --- /dev/null +++ b/_static/authors/joseph_bunao.txt @@ -0,0 +1,4 @@ +.. bio:: Joseph Bunao + :photo: ../_static/authors/joseph_bunao.jpg + + Joseph is a machine learning specialist at Xanadu. 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PennyLane data", + "authors": [ + { + "username": "Joseph" + }, + { + "username": "zy_n" + } + ], + "dateOfPublication": "2024-09-20T00:00:00+00:00", + "dateOfLastModification": "2024-09-20T00:00:00+00:00", + "categories": ["Quantum Machine Learning", "Quantum Chemistry", "Algorithms"], + "tags": [], + "previewImages": [ + { + "type": "thumbnail", + "uri": "_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_generative_quantum_eigensolver.png" + }, + { + "type": "large_thumbnail", + "uri": "/_static/demo_thumbnails/large_demo_thumbnails/thumbnail_large_generative_quantum_eigensolver.png" + } + ], + "seoDescription": "Learn how you can train a small GPT model using the generative quantum eigensolver (GQE) technique and PennyLane data.", + "doi": "", + "canonicalURL": "/qml/demos/gqe_training", + "references": [ + { + "id": "nakaji2024", + "type": "article", + "title": "The generative quantum eigensolver (GQE) and its application for ground state search", + "authors": "K. Nakaji, L. B. Kristensen, J. A. Campos-Gonzalez-Angulo, M. G. Vakili, H. Huang, M. Bagherimehrab, C. Gorgulla, F. Wong, A. McCaskey, J. S. Kim, T. Nguyen, P. Rao, A. Aspuru-Guzik", + "year": "2024", + "journal": "", + "url": "https://arxiv.org/abs/2401.09253" + }, + { + "id": "radford2019", + "type": "article", + "title": "Language Models are Unsupervised Multitask Learners", + "authors": "A. Radford, J. Wu, R. Child, D. Luan, D. Amodei, I. Sutskever", + "year": "2019", + "journal": "OpenAI blog", + "url": "https://openai.com/index/better-language-models/" + }, + { + "id": "vaswani2017", + "type": "article", + "title": "Attention is All you Need", + "authors": "A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, I. Polosukhin", + "year": "2017", + "journal": "Advances in Neural Information Processing Systems", + "url": "https://papers.nips.cc/paper_files/paper/2017/hash/3f5ee243547dee91fbd053c1c4a845aa-Abstract.html" + } + ], + "basedOnPapers": [ + "10.48550/arXiv.2401.09253" + ], + "referencedByPapers": [], + "relatedContent": [ + { + "type": "demonstration", + "id": "tutorial_vqe", + "weight": 1.0 + }, + { + "type": "demonstration", + "id": "tutorial_vqe_qng", + "weight": 1.0 + } + ], + "hardware": [] +} \ No newline at end of file diff --git a/demonstrations/gqe_training.py b/demonstrations/gqe_training.py new file mode 100644 index 0000000000..8fbc6dee3d --- /dev/null +++ b/demonstrations/gqe_training.py @@ -0,0 +1,703 @@ +r"""Generative quantum eigensolver (GQE) training using PennyLane data +=================================================== + +We will be demonstrating and evaluating a novel algorithm proposed by Nakaji et al. in their paper +`The generative quantum eigensolver (GQE) and its application for ground state search `__ +that employs a classical generative model of quantum circuits for the purpose of ground-state +energy estimation of any molecular Hamiltonian [#nakaji2024]_. +It has been proposed as a scalable alternative to the `variational quantum eigensolver (VQE) `__ approach, +where the quantum state is represented as a quantum circuit with tunable parameters which are then optimized during training in order to arrive at a +state minimizing the corresponding energy :math:`E`. Instead, in GQE, the structure of the quantum circuit is given by a trained generative model. + +We will primarily focus on offline training on a fixed training dataset -- thanks to the molecular data +available in `PennyLane Datasets `__. +By the end of the demo, we will show that the model gradually provides a better estimate for the energies and, in turn, +can sample energies close to the ground state energy calculated by PennyLane. + +This demo is organized as follows. Firstly, we compare the GQE and VQE algorithms, and afterwards, we dive deeper in +describing GPT-QE, i.e., the GQE algorithm which uses a GPT model, and its training. Next, we generate the training dataset for our GPT model using PennyLane, +and give details on our model architecture and training implementation. After that, we +evaluate the model throughout its training and discuss its performance in estimating the ground +state. And lastly, we discuss the results, potential ways optimizing the code, and its extension into the +"online training" phase. + +.. figure:: ../_static/demo_thumbnails/opengraph_demo_thumbnails/OGthumbnail_generative_quantum_eigensolver.png + :align: centeravailable + :width: 70% + :target: javascript:void(0) + +""" + +###################################################################### +# GQE vs. VQE +# ----------- +# Despite the relative success of VQE, there are some issues regarding its *trainability* for large problem +# instances [#nakaji2024]_. This shortcoming makes it less competitive against the performance of +# classical machine learning (ML) algorithms for large problems. To bypass this, the GQE algorithm was +# proposed. Specifically, GQE uses a classical generative model where quantum circuits are sampled as a +# sequence of unitaries from a given operator pool. This generative model is then trained so that it learns to +# predict quantum circuits that evolves an initial state to the states better approximating the ground state. +# +# The main difference between the two approaches is where the tunable parameters are embedded. +# That is, it is the classical GQE model that is being optimized as opposed to the variable +# quantum circuit of VQE. Potentially then, the :doc:`barren plateau ` landscape of +# VQE and the quantum gradient evaluation of large circuits will be sidestepped by GQE, thus becoming more amenable +# for larger problems. +# + + +############################################################################## +#.. figure:: ../_static/demonstration_assets/gqe_training/paper_gqe_vqe.png +# :align: center +# :width: 90% +# +# Figure 1: Diagrams of GQE and VQE from Fig. 1 in [#nakaji2024]_ + +###################################################################### +# GPT-QE background +# -------------------- +# +# In particular, the model architecture used by Nakaji et al. [#nakaji2024]_ for GQE was a generative pre-trained +# transformer (GPT) [#radford2019]_, [#vaswani2017]_. This choice is then reflected in the name GPT-QE. +# As language models, GPTs are successful in generating +# sequences of words that closely resemble human natural language. This performance is +# harnessed for quantum chemistry by constructing quantum states :math:`\rho` as a sequence of unitary operators +# which are, in turn, represented by quantum circuits. That is, we let :math:`\rho = U\rho_0 U^{\dagger}` +# for some fixed initial state :math:`\rho_0` and the aforementioned sequence is :math:`U = U_{j_N}U_{j_{N-1}}\cdots U_{j_1}`. +# The GPT model samples a sequence of integers :math:`j_1, j_2, ..., j_N` indexing a pool +# of operators :math:`U_j` generated using molecular data from `PennyLane Molecules `__. We interpret these +# integers as tokens and the pool as the vocabulary in the parlance for language models. +# The goal of training is then to minimize the corresponding energy +# :math:`E = \mbox{Tr}(\hat{H}\rho)`, where :math:`\hat{H}` is the Hamiltonian of the molecule in +# question. +# +# Each token :math:`j_i` is sampled from the distribution :math:`\exp(-\beta w_{j_i})`, where +# :math:`w_{j_i}` is the unnormalized log probability (or logit) returned by the GPT model for the token :math:`j_i` +# and :math:`\beta` is an inverse temperature representing a trade-off parameter between exploration and exploitation. We then +# observe that the probability of sampling a state through the method described above is +# proportional to :math:`\exp(-\beta w_{\mbox{sum}})`, where :math:`w_{\mbox{sum}} = \sum_{i=1}^N w_{j_i}` +# and the probability for the corresponding energy is :math:`\exp(-\beta E)`. We thus have a constraint +# for the total logit to be equal to the energy of the corresponding state: :math:`w_{\mbox{sum}} = E`, which +# can be imposed by training GPT-QE to minimize the loss function :math:`C = (w_{\mbox{sum}} - E)^2`. +# With this constraint satisfied, GPT-QE would then be sampling states of smaller energies with increasing +# likelihood. +# +# More concretely, we summarize the *pre-*training loop in Figure 2. This is called +# pre-training because it involves learning from a fixed dataset first before transitioning to the "real" training +# which utilizes the data generated by the model itself. In this demo, we will call the pre-training as offline +# training since the GPT model does not receive feedback from the sequences it samples, and online +# training if otherwise. + +############################################################################## +#.. figure:: ../_static/demonstration_assets/gqe_training/gqe_training_diagram.png +# :align: center +# :width: 90% +# +# Figure 2: Overview for offline training of GPT-QE + +###################################################################### +# Dataset construction via PennyLane +# ------------------------------------- +# +# Firstly, let us construct the static dataset we will use for offline training. We choose +# to generate our own dataset in order to illustrate the sequences and energies more concretely. +# Our dataset will be made from random sequences of tokens, which we recall corresponds to indices +# of a vocabulary of unitary operators. We then define an energy function in PennyLane to calculate +# the energy of a state corresponding to a token sequence. Applying the aforementioned function, +# we would then have a dataset of token sequences and energies for the GPT model offline training. + +###################################################################### +# Loading molecular information +# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +# +# For simplicity, let us consider the `hydrogen molecule `__ +# and load the corresponding dataset from PennyLane. +# Recall that we would need a vocabulary of operators :math:`U_j`, an initial state :math:`\rho_0`, and +# the Hamiltonian :math:`\hat{H}` for a hydrogen molecule. We also get the ground state energy for later comparison +# with the results. +# +# Specifically, the unitary operators :math:`U_j` are time evolution operators as prescribed +# in [#nakaji2024]_. The non-identity operators are generated in PennyLane using :class:`~.pennylane.SingleExcitation` +# and :class:`~.pennylane.DoubleExcitation` which then depend on the number of electrons and orbitals +# of the molecule. +# + +import numpy as np +import pennylane as qml + +def generate_molecule_data(molecules="H2"): + datasets = qml.data.load("qchem", molname=molecules) + + # Get the time set T + op_times = np.sort(np.array([-2**k for k in range(1, 5)] + [2**k for k in range(1, 5)]) / 160) + + # Build operator set P for each molecule + molecule_data = dict() + for dataset in datasets: + molecule = dataset.molecule + num_electrons, num_qubits = molecule.n_electrons, 2 * molecule.n_orbitals + singles, doubles = qml.qchem.excitations(num_electrons, num_qubits) + double_excs = [qml.DoubleExcitation(time, wires=double) for double in doubles for time in op_times] + single_excs = [qml.SingleExcitation(time, wires=single) for single in singles for time in op_times] + identity_ops = [qml.exp(qml.I(range(num_qubits)), 1j*time) for time in op_times] # For Identity + operator_pool = double_excs + single_excs + identity_ops + molecule_data[dataset.molname] = { + "op_pool": np.array(operator_pool), + "num_qubits": num_qubits, + "hf_state": dataset.hf_state, + "hamiltonian": dataset.hamiltonian, + "expected_ground_state_E": dataset.fci_energy + } + return molecule_data + +molecule_data = generate_molecule_data("H2") +h2_data = molecule_data["H2"] +op_pool = h2_data["op_pool"] +num_qubits = h2_data["num_qubits"] +init_state = h2_data["hf_state"] +hamiltonian = h2_data["hamiltonian"] +grd_E = h2_data["expected_ground_state_E"] +op_pool_size = len(op_pool) + +###################################################################### +# Defining the energy function +# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +# +# In PennyLane, we define the energy function :math:`E = \mbox{Tr}(\hat{H}U_{j_N}\cdots U_{j_1}\rho_0 U_{j_1}^{\dagger}\cdots U_{j_N}^{\dagger})` +# corresponding to Eq. 1 of [#nakaji2024]_. Here, ``energy_circuit`` takes in the operator sequence :math:`U_{j_1}, U_{j_2}, ..., U_{j_N}` +# and returns the energy of the corresponding quantum state. +# +# As a slight extension, we can also calculate the energies for each subsequence of +# operators to help with the training of the model. That is, for a sequence of three operators +# :math:`U_{j_1}, U_{j_2}, U_{j_3}` we compute the energies for :math:`U_{j_1}` and :math:`U_{j_1}, U_{j_2}` instead of just +# the full sequence of three operators, which was described in [#nakaji2024]_. This can be done simply in PennyLane, using +# :class:`~.pennylane.Snapshot` as shown below. +# + +dev = qml.device("default.qubit", wires=num_qubits) + +@qml.qnode(dev) +def energy_circuit(gqe_ops): + # Computes Eq. 1 from Nakaji et al. based on the selected unitary operators + qml.BasisState(init_state, wires=range(num_qubits)) # Initial state <-- Hartree Fock state + for op in gqe_ops: + qml.Snapshot(measurement=qml.expval(hamiltonian)) + qml.apply(op) # Applies each of the unitary operators + return qml.expval(hamiltonian) + +energy_circuit = qml.snapshots(energy_circuit) + +def get_subsequence_energies(op_seq): + # Collates the energies of each subsequence for a batch of sequences + energies = [] + for ops in op_seq: + es = energy_circuit(ops) + energies.append( + [es[k].item() for k in list(range(1, len(ops))) + ["execution_results"]] + ) + return np.array(energies) + +###################################################################### +# Token sequence generation with corresponding energies +# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +# +# With these ingredients, we can now construct a dataset containing sequences of tokens and their energies. +# Since we cannot feed the operators directly to the GPT model, we would need to tokenize +# them. The indices of ``op_pool`` seems to be a good candidate, but we instead choose the tokens to be +# the ``op_pool`` indices shifted by 1. This is so that we can define a special token ``0`` that tells +# the GPT model where the sequence starts. +# +# We generate a ``train_size`` number of random operator sequences of length ``seq_len`` for our +# purposes and calculate their energies (and their subsequences). +# + +# Generate sequence of indices of operators in vocab +train_size = 1024 +seq_len = 4 +train_op_pool_inds = np.random.randint(op_pool_size, size=(train_size, seq_len)) + +# Corresponding sequence of operators +train_op_seq = op_pool[train_op_pool_inds] + +# Corresponding tokens with special starting tokens +train_token_seq = np.concatenate([ + np.zeros(shape=(train_size, 1), dtype=int), # starting token is 0 + train_op_pool_inds + 1 # shift operator inds by one +], axis=1) + +# Calculate the energies for each subsequence in the training set +train_sub_seq_en = get_subsequence_energies(train_op_seq) + +###################################################################### +# GPT-QE offline training +# -------------------------- +# Having setup our training dataset, we can start implementing our offline training loop as +# illustrated in Figure 2. We outline our implementation below. + +###################################################################### +# GPT model implementation details +# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +# +# The GPT model we will use in this demo is mostly implemented in the `nanoGPT repo +# `__ (a reimplementation of the `OpenAI GPT-2 `__) as the +# `class `__ +# ``GPT`` with the model hyperparameters stored in the +# `dataclass `__ +# ``GPTConfig``. Namely, we will use 12 attention layers, 12 attention heads, and 768 embedding dimensions, +# which are equal to those described in [#nakaji2024]_. +# We can import from the nanoGPT repo directly by running the ``curl`` command (commented out below) in a Jupyter Notebook. +# Since nanoGPT is trained as a language model, its loss function and sampling method are +# defined differently. We then define the subclass ``GPTQE`` below to override some nanoGPT methods in order to make it more +# suitable for our case. +# + +# !curl -O https://raw.githubusercontent.com/karpathy/nanoGPT/master/model.py +from model import GPT, GPTConfig +import torch +from torch.nn import functional as F + +class GPTQE(GPT): + def forward(self, idx): + device = idx.device + b, t = idx.size() + pos = torch.arange(0, t, dtype=torch.long, device=device) # shape (t) + + # forward the GPT model itself + tok_emb = self.transformer.wte(idx) # token embeddings of shape (b, t, n_embd) + pos_emb = self.transformer.wpe(pos) # position embeddings of shape (t, n_embd) + x = self.transformer.drop(tok_emb + pos_emb) + for block in self.transformer.h: + x = block(x) + x = self.transformer.ln_f(x) + logits = self.lm_head(x) + return logits + + def calculate_loss(self, tokens, energies): + current_tokens, next_tokens = tokens[:, :-1], tokens[:, 1:] + # calculate the logits for the next possible tokens in the sequence + logits = self(current_tokens) + # get the logit for the actual next token in the sequence + next_token_mask = torch.nn.functional.one_hot( + next_tokens, num_classes=self.config.vocab_size + ) + next_token_logits = (logits * next_token_mask).sum(axis=2) + # calculate the cumulative logits for each subsequence + cumsum_logits = torch.cumsum(next_token_logits, dim=1) + # match cumulative logits to subsequence energies + loss = torch.mean(torch.square(cumsum_logits - energies)) + return loss + + @torch.no_grad() + def generate(self, n_sequences, max_new_tokens, temperature=1.0, device="cpu"): + idx = torch.zeros(size=(n_sequences, 1), dtype=int, device=device) + total_logits = torch.zeros(size=(n_sequences, 1), device=device) + for _ in range(max_new_tokens): + # if the sequence context is growing too long we must crop it at block_size + idx_cond = idx if idx.size(1) <= self.config.block_size else idx[:, -self.config.block_size:] + # forward the model to get the logits for the index in the sequence + logits = self(idx_cond) + # pluck the logits at the final step + logits = logits[:, -1, :] + # set the logit of the first token so that its probability will be zero + logits[:, 0] = float("inf") + # apply softmax to convert logits to (normalized) probabilities and scale by desired temperature + probs = F.softmax(-logits / temperature, dim=-1) + # sample from the distribution + idx_next = torch.multinomial(probs, num_samples=1) + # # Accumulate logits + total_logits += torch.gather(logits, index=idx_next, dim=1) + # append sampled index to the running sequence and continue + idx = torch.cat((idx, idx_next), dim=1) + return idx, total_logits + +###################################################################### +# However, it is important to note that the loss function ``calculate_loss``, that we defined is different from the one +# described in [#nakaji2024]_ which is :math:`(\exp(-w_{\mbox{sum}}) - \exp(-E))^2`. As described +# beforehand, we instead directly compute the mean squared error between :math:`w_{\mbox{sum}}` and :math:`E`. +# Since the exponential function is one-to-one, both loss functions would then impose the same minimum. +# Using the error between exponentials may even +# introduce numerical instabilities in the training since the loss would be taking differences of potentially large numbers. +# In addition to this change from [#nakaji2024]_, we also use the error between the cumulative sum of logits and the corresponding energy +# for each subsequence instead of just the error between total logits and the energy of an entire sequence. +# This addition will give more training data to the model and should help with logit matching the intermediate tokens. +# +# Since it was not explicitly shown in [#nakaji2024]_, another possible deviation we made is with the logit calculation during offline training. +# It seems that the logits were accumulated by looping through the sequential generation of tokens. +# In order to fill in the blanks, we implement the logit calculation in a manner that we think is efficient. +# Namely, we directly pass the fixed training sequences to the model and retrieve the relevant logits from it. +# This can be done because we are using a causal mask for the attention blocks so that the logits of the earlier tokens in the sequence are +# not affected by tokens in the later part of the sequence. Thus, having the same effect of the sequential token generation. +# +# We initialize our GPT model below and we see that it has around 85 million parameters. When saved, the model is ``324.25 MB`` in size. + +tokens = torch.from_numpy(train_token_seq).to("cuda") +energies = torch.from_numpy(train_sub_seq_en).to("cuda") + +gpt = GPTQE(GPTConfig( + vocab_size=op_pool_size + 1, + block_size=seq_len, + dropout=0.2, + bias=False +)).to("cuda") +opt = gpt.configure_optimizers( + weight_decay=0.01, learning_rate=5e-5, betas=(0.9, 0.999), device_type="cuda" +) + +###################################################################### +# .. rst-class:: sphx-glr-script-out +# +# .. code-block:: none +# +# number of parameters: 84.98M +# num decayed parameter tensors: 50, with 84,963,072 parameters +# num non-decayed parameter tensors: 25, with 19,200 parameters + +###################################################################### +# GPT offline training +# ~~~~~~~~~~~~~~~~~~~~ +# +# We now implement a training loop for our GPT model. This can be framed as a straightforward +# supervised learning problem. We sketch the steps for each training iteration/epoch below: +# +# 1. Shuffle the training set and split it into ``n_batches`` minibatches +# 2. For each minibatch, calculate the average loss, the gradients, and take an optimizer step +# 3. For each n-th iteration (n is 500 here), evaluate the GPT model: +# +# - Generate a batch of sequences and the predicted energies (total logits). Note that these are not necessarily same sequences as in the training set. +# +# - Calculate the true energies using PennyLane +# +# - Calculate the mean absolute error as a metric to track the learning progress and save the GPT model everytime the metric gets better +# + +# %%time +n_batches = 8 +train_inds = np.arange(train_size) + +losses = [] +pred_Es_t = [] +true_Es_t = [] +current_mae = 10000 +gpt.train() +for i in range(10000): + # Shuffle batches of the training set + np.random.shuffle(train_inds) + token_batches = torch.tensor_split(tokens[train_inds], n_batches) + energy_batches = torch.tensor_split(energies[train_inds], n_batches) + + # SGD on random minibatches + loss_record = 0 + for token_batch, energy_batch in zip(token_batches, energy_batches): + opt.zero_grad() + loss = gpt.calculate_loss(token_batch, energy_batch) + loss.backward() + opt.step() + loss_record += loss.item() / n_batches + losses.append(loss_record) + + if (i+1) % 500 == 0: + # For GPT evaluation + gpt.eval() + gen_token_seq, pred_Es = gpt.generate( + n_sequences=100, + max_new_tokens=seq_len, + temperature=0.001, # Use a low temperature to emphasize the difference in logits + device="cuda" + ) + pred_Es = pred_Es.cpu().numpy() + + gen_inds = (gen_token_seq[:, 1:] - 1).cpu().numpy() + gen_op_seq = op_pool[gen_inds] + true_Es = get_subsequence_energies(gen_op_seq)[:, -1].reshape(-1, 1) + + mae = np.mean(np.abs(pred_Es - true_Es)) + ave_E = np.mean(true_Es) + + pred_Es_t.append(pred_Es) + true_Es_t.append(true_Es) + + print(f"Iteration: {i+1}, Loss: {losses[-1]}, MAE: {mae}, Ave E: {ave_E}") + + if mae < current_mae: + current_mae = mae + torch.save(gpt, f"./seq_len={seq_len}/gqe.pt") + print("Saved model!") + + gpt.train() + +pred_Es_t = np.concatenate(pred_Es_t, axis=1) +true_Es_t = np.concatenate(true_Es_t, axis=1) + +###################################################################### +# .. rst-class:: sphx-glr-script-out +# +# .. code-block:: none +# +# Iteration: 500, Loss: 0.004496691238049528, MAE: 0.13945468622863236, Ave E: -1.1161227981406456 +# Saved model! +# Iteration: 1000, Loss: 0.001162520404255374, MAE: 0.11792013497926974, Ave E: -1.116178063434579 +# Saved model! +# Iteration: 1500, Loss: 0.0006311882560414964, MAE: 0.08421050347067748, Ave E: -1.1304435666682537 +# Saved model! +# Iteration: 2000, Loss: 0.0002220232025956396, MAE: 0.03313205549288038, Ave E: -1.13411711385679 +# Saved model! +# Iteration: 2500, Loss: 9.021296506465553e-05, MAE: 0.03720317687198404, Ave E: -1.1360217383940532 +# Iteration: 3000, Loss: 0.00011929328764308375, MAE: 0.010246824522607662, Ave E: -1.1355033629645301 +# Saved model! +# Iteration: 3500, Loss: 4.015137835017087e-05, MAE: 0.008332604993116905, Ave E: -1.1362737218253494 +# Saved model! +# Iteration: 4000, Loss: 0.00025425587370956726, MAE: 0.03346923599957368, Ave E: -1.13442109812976 +# Iteration: 4500, Loss: 4.590269966149363e-05, MAE: 0.0086580669691949, Ave E: -1.1344678899103924 +# Iteration: 5000, Loss: 2.7407370499136962e-05, MAE: 0.006680762382889203, Ave E: -1.136412143925528 +# Saved model! +# Iteration: 5500, Loss: 3.778071550021417e-05, MAE: 0.014272903220676704, Ave E: -1.1362969016861684 +# Iteration: 6000, Loss: 2.2792776141250974e-05, MAE: 0.007428675818214263, Ave E: -1.1367647064449693 +# Iteration: 6500, Loss: 1.9002385742602413e-05, MAE: 0.004431537870071902, Ave E: -1.135880723613281 +# Saved model! +# Iteration: 7000, Loss: 1.5268728079291623e-05, MAE: 0.002464256235883442, Ave E: -1.1356989137037925 +# Saved model! +# Iteration: 7500, Loss: 1.1030378864566936e-05, MAE: 0.007000517223791054, Ave E: -1.1360445255294285 +# Iteration: 8000, Loss: 7.638036884241474e-06, MAE: 0.0044611951680048586, Ave E: -1.1352658877947734 +# Iteration: 8500, Loss: 1.616690860258467e-05, MAE: 0.004094392133172753, Ave E: -1.1356437076129735 +# Iteration: 9000, Loss: 7.37882245331426e-06, MAE: 0.004240113290004896, Ave E: -1.1358971131175264 +# Iteration: 9500, Loss: 1.004411104422562e-05, MAE: 0.010631562300185794, Ave E: -1.1368761600775912 +# Iteration: 10000, Loss: 1.809862392776087e-05, MAE: 0.01987725166307399, Ave E: -1.1345492765523346 +# CPU times: user 2h 12min 24s, sys: 8.18 s, total: 2h 12min 32s +# Wall time: 2h 12min 32s + +###################################################################### +# With a preliminary look at the training logs above, we see that the offline training took 2 h and 12 min for 10,000 +# training iterations. The code execution time was measured by including the ``%%time`` magic command before the code block. +# We also note that the mean absolute error between the predicted and true energies for the +# generated sequences quickly became smaller during the earlier parts of the training but fluctuates more +# later on. As mentioned earlier, a model version is saved each time we get better performance. The best model version is then +# saved at the 7,000th iteration, where the mean absolute error is at its lowest. The other quantities we observe are the +# average true energies of the generated sequences. It is then promising to see that throughout training, this is +# close to the ground state energy ``grd_E=-1.1372633205048763 Ha``. +# + +###################################################################### +# GPT-QE results +# ----------------- +# Having finished the offline training, let's take a look at some of our results. + +###################################################################### +# Training loss curve +# ~~~~~~~~~~~~~~~~~~~ +# One of the first things we can look at is the training loss curve. Recall that for our case, the +# loss is the mean squared error between the cumulative sum of logits (predicted subsequence energies) and their +# corresponding true subsequence energies. So reducing this quantity allows the model to be better at giving +# correct energies and in turn, correctly sample sequences with lower energies. Since the raw loss values +# become very small, we instead plot the loss in log-scale below to magnify the loss trend. +# + +import holoviews as hv +import hvplot.pandas +import pandas as pd + +hvplot.extension('matplotlib') + +losses = pd.read_csv("./seq_len=4/trial7/losses.csv")["0"] +loss_fig = losses.hvplot( + title="Training loss progress", ylabel="loss", xlabel="Training epochs", logy=True +).opts(fig_size=600, fontscale=2, aspect=1.2) +loss_fig + +############################################################################## +#.. figure:: ../_static/demonstration_assets/gqe_training/gqe_training_loss.png +# :align: center +# :width: 100% +# +# Figure 3: The subsequence loss for each training iteration + +############################################################################## +# We see in Figure 3 that the loss +# continues to decrease until around the 4,000th iteration. There, the model was erroraneous but is quick +# to recover as training continues. This may signal that the GPT model started focusing on learning something +# erroraneous too quickly. So, more regularization noise (like ``dropout``) may be needed to help avoid this. + +###################################################################### +# Evaluation progress +# ~~~~~~~~~~~~~~~~~~~~~~~~~~~ +# +# We now track the performance of the GPT model throughout its training in Figure 4 below. As mentioned +# before, after every 500th iteration, we let the model generate a batch of sequences. Alongside, we +# also return the total logits (predicted energies) used in the sequence generation. In Figure 4, +# the average predicted energies correspond to the red markers and the distribution of predicted +# energies is represented by the red area. Once we have the generated sequences, we can also let PennyLane calculate the true sequence +# energies. Similarly then, the blue markers are the average true energies and the blue +# area represents the true energy distribution. +# + +df_true = pd.read_csv("./seq_len=4/trial7/true_Es_t.csv").iloc[:, 1:] +df_pred = pd.read_csv("./seq_len=4/trial7/pred_Es_t.csv").iloc[:, 1:] + +df_true.columns = df_true.columns.astype(int) +df_pred.columns = df_pred.columns.astype(int) + +df_trues_stats = pd.concat([df_true.mean(axis=0), df_true.min(axis=0), df_true.max(axis=0)], axis=1).reset_index() +df_trues_stats.columns = ["Training Iterations", "Ave True E", "Min True E", "Max True E"] + +df_preds_stats = pd.concat([df_pred.mean(axis=0), df_pred.min(axis=0), df_pred.max(axis=0)], axis=1).reset_index() +df_preds_stats.columns = ["Training Iterations", "Ave Pred E", "Min Pred E", "Max Pred E"] + +fig = ( + df_trues_stats.hvplot.scatter(x="Training Iterations", y="Ave True E", label="Mean True Energies") * + df_trues_stats.hvplot.line(x="Training Iterations", y="Ave True E", alpha=0.5, linewidth=1) * + df_trues_stats.hvplot.area(x="Training Iterations", y="Min True E", y2="Max True E", alpha=0.1) +) * ( + df_preds_stats.hvplot.scatter(x="Training Iterations", y="Ave Pred E", label="Mean Predicted Energies") * + df_preds_stats.hvplot.line(x="Training Iterations", y="Ave Pred E", alpha=0.5, linewidth=1) * + df_preds_stats.hvplot.area(x="Training Iterations", y="Min Pred E", y2="Max Pred E", alpha=0.1) +) +fig = fig * hv.Curve([[0, grd_E], [10000, grd_E]], label="Ground State Energy").opts(color="k", alpha=0.4, linestyle="dashed") +fig = fig.opts(ylabel="Sequence Energies", title="GQE Evaluations", fig_size=600, fontscale=2) +fig + +############################################################################## +#.. figure:: ../_static/demonstration_assets/gqe_training/gqe_performance.png +# :align: center +# :width: 100% +# +# Figure 4: True and predicted energies for sequences generated by the GPT model for each 500th training iteration + +############################################################################## +# We now see that the energies predicted by the model improve in accuracy, aligning more closely with the true +# energies during training. The increase in accuracy then allows the model to correctly sample states +# with lower energies. This is supported in Figure 4 where the sampled true energies +# get closer to the ground state energy (the dashed line). +# +# Note that at around the 4,000th iteration, the predicted energies are very far from the true energies. +# This makes sense considering our observation in Figure 3. Also note that at the 7,000th +# iteration, the averages of the predicted and true energies are the closest and even their respective +# spreads seem to have good overlap. This is when the best performing model version was saved, as seen in the training logs. +# For later iterations however, the predicted energies no longer improved. +# This may indicate that the GPT model has started overfitting on the training dataset in the later iterations. +# That is, the model became great at predicting the correct energies for the training set (as observed by the decreasing +# loss in Figure 3) but not great at generalizing on those outside the training set (like the sequences that the model +# generated on its own). One solution to avoid overfitting could then be online training. This is so that +# the GPT model is not restricted on a fixed dataset on which to overfit. +# + +###################################################################### +# Sequence generation comparison +# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +# +# Here, we compare some statistics of the true energies corresponding to sequences generated +# by a "random" model, the latest version of the model after all the training iterations, and the best model version +# saved based on the mean absolute error between the true and predicted energies during training +# (for our case, this is the model saved at the 7,000th iteration). +# Note that we consider the training set to be generated by a random model since the token +# sequences are just sampled uniformly. +# + +# Latest model +gen_token_seq_, _ = gpt.generate( + n_sequences=1024, + max_new_tokens=seq_len, + temperature=0.001, + device="cuda" +) +gen_inds_ = (gen_token_seq_[:, 1:] - 1).cpu().numpy() +gen_op_seq_ = op_pool[gen_inds_] +true_Es_ = get_subsequence_energies(gen_op_seq_)[:, -1].reshape(-1, 1) + +# Best model +loaded = torch.load("./seq_len=4/trial7/gqe.pt") +loaded_token_seq_, _ = loaded.generate( + n_sequences=1024, + max_new_tokens=seq_len, + temperature=0.001, + device="cuda" +) +loaded_inds_ = (loaded_token_seq_[:, 1:] - 1).cpu().numpy() +loaded_op_seq_ = op_pool[loaded_inds_] +loaded_true_Es_ = get_subsequence_energies(loaded_op_seq_)[:, -1].reshape(-1, 1) + +# Summary table +df_compare_Es = pd.DataFrame({ + "Source": ["Random", "Latest Model", "Best Model"], + "Aves": [train_sub_seq_en[:, -1].mean(), true_Es_.mean(), loaded_true_Es_.mean()], + "Mins": [train_sub_seq_en[:, -1].min(), true_Es_.min(), loaded_true_Es_.min()], + "Maxs": [train_sub_seq_en[:, -1].max(), true_Es_.max(), loaded_true_Es_.max()], + "Mins_error": [ + abs(train_sub_seq_en[:, -1].min() - grd_E), + abs(true_Es_.min() - grd_E), + abs(loaded_true_Es_.min() - grd_E), + ], +}) +df_compare_Es + +###################################################################### +# .. rst-class:: sphx-glr-script-out +# +# .. code-block:: none +# +# Source Aves Mins Maxs Mins_error +# 0 Random -1.114531 -1.136982 -1.027878 2.811117e-04 +# 1 Latest Model -1.132200 -1.137038 -1.125118 2.253048e-04 +# 2 Best Model -1.135560 -1.137263 -1.125118 7.712067e-07 + +############################################################################## +# We observe that the minimum energy corresponding to the random training set is already close +# to the ground state energy ``grd_E=-1.1372633205048763 Ha`` with an error of around ``2.81e-04 Ha``. But we notice that the maximum energy +# is relatively far so the random sequences give a wider spread of energies. +# +# The energies of the generated sequences from the latest and the best GPT model versions however have a narrower spread and so, +# the average and the minimum energies are very close to ``grd_E``, closer than those in the random +# training set. Namely, around a ``2.25e-04 Ha`` error for the minimum energy generated by the latest model version +# and a ``7.71e-07 Ha`` error for the best model version. It is then very interesting that +# a well-trained GPT model can generate sequences that are better than those in the training set, even though every +# sequence the model has seen just came from that set. That is, the model was able to generalize. +# +# Between the two GPT model versions, we see that the latest version is worse than the best version. The minimum energy +# error for the latest version has the same order of magnitude as the corresponding one for the random training set. +# Contrast this with the minimum energy error for the best version, which is 3 orders of magnitude smaller. +# This behavior is supported by our observation in Figure 4 where the performance of the model after +# the 7,000th iteration worsened. That is, the predicted energies started to deviate further from the true energies +# which in turn caused the states being sampled from these predicted energies to be different from the intended +# lower energy states. +# + +###################################################################### +# Conclusion +# ------------- +# +# In this demo, we see that GPT-QE is a viable alternative in estimating the ground-state energy +# of a hydrogen molecule. The best underlying GPT model version can generate a state whose energy is +# only around ``7.71e-07 Ha`` away from ground state energy, which is well below the chemical accuracy. Additionally, since the GPT model being optimized +# is completely detached from the quantum simulations, gradients across a potentially large quantum circuit +# don't need to be computed, for example. Thus, the machinery of classical ML can be harnessed without +# worrying too much about the quantum algorithm side of the problem. +# +# The reader can also experiment with other molecules from PennyLane and tweak several hyperparameters +# of the GPT model (like ``dropout``, and ``n_layer``) and include standard ML callbacks to its training +# (like an early stopping mechanism and a learning rate schedule). The code itself is also open to further +# optimization. For instance, using `PennyLane Lightning `__ +# to evaluate the energies faster. An online training loop can also be implemented similarly to our offline +# version, the reader would just need to sample sequences from the current GPT model instead of a fixed +# dataset for each training iteration. To facilitate exploration and exploitation, one would also define +# a schedule for the inverse temperature. That is, initially letting the GPT model sample more randomly through +# a high temperature, then gradually decreasing so that the GPT model focuses more on the higher probability +# states (low energies). +# + +###################################################################### +# Reference +# --------- +# +# .. [#nakaji2024] +# +# Kouhei Nakaji *et al.*, "The generative quantum eigensolver (GQE) and its application for ground state search". +# `arXiv:2401.09253 (2024) `__ +# +# .. [#radford2019] +# +# Alec Radford *et al.*, "Language Models are Unsupervised Multitask Learners". +# OpenAI blog, 1(8):9 (2019) +# +# .. [#vaswani2017] +# +# Ashish Vaswani *et al.*, "Attention is All you Need". +# Advances in Neural Information Processing Systems, 30 (2017) +# From a590798a67e424cf63398e81aecf9f4cf328c3cb Mon Sep 17 00:00:00 2001 From: Korbinian Kottmann <43949391+Qottmann@users.noreply.github.com> Date: Fri, 20 Sep 2024 15:40:52 +0200 Subject: [PATCH 2/2] [Demo] What is a matrix product state? (#1201) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit MPS simulation demo [sc-66753] --------- Co-authored-by: Pietropaolo Frisoni Co-authored-by: Ivana Kurečić --- .../thumbnail_large_mps_simulation.png | Bin 0 -> 27746 bytes .../OGthumbnail_mps_simulation.png | Bin 0 -> 26098 bytes .../thumbnail_mps_simulation.png | Bin 0 -> 14617 bytes .../demonstration_assets/mps/apply_gate.png | Bin 0 -> 102081 bytes _static/demonstration_assets/mps/area_law.png | Bin 0 -> 72344 bytes .../demonstration_assets/mps/cnot_split.png | Bin 0 -> 20546 bytes .../demonstration_assets/mps/final_expval.png | Bin 0 -> 59344 bytes .../mps/left_orthonormal.png | Bin 0 -> 48234 bytes .../mps/mixed_canonical_observable.png | Bin 0 -> 119866 bytes .../mps/non_local_cnot.png | Bin 0 -> 75921 bytes .../demonstration_assets/mps/norm_trivial.png | Bin 0 -> 84601 bytes .../demonstration_assets/mps/psi_to_mps_0.png | Bin 0 -> 63262 bytes 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"thumbnail", + "uri": "/_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_mps_simulation.png" + }, + { + "type": "large_thumbnail", + "uri": "/_static/demo_thumbnails/large_demo_thumbnails/thumbnail_large_mps_simulation.png" + } + ], + "seoDescription": "What is a matrix product state (MPS)? Here is all you need to know about MPS from a quantum computation and quantum simulation perspective.", + "doi": "", + "canonicalURL": "/qml/demos/tutorial_mps", + "references": [ + { + "id": "Baiardi", + "type": "preprint", + "title": "The Density Matrix Renormalization Group in Chemistry and Molecular Physics: Recent Developments and New Challenges", + "authors": "Alberto Baiardi, Markus Reiher", + "year": "2019", + "publisher": "", + "journal": "", + "doi": "10.48550/arXiv.1910.00137", + "url": "https://arxiv.org/abs/1910.00137" + }, + { + "id": "Patra", + "type": "preprint", + "title": "Efficient tensor network simulation of IBM's largest quantum processors", + "authors": "Siddhartha Patra, Saeed S. Jahromi, Sukhbinder Singh, Roman Orus", + "year": "2023", + "publisher": "", + "journal": "", + "doi": "10.48550/arXiv.2309.15642", + "url": "https://arxiv.org/abs/2309.15642" + }, + { + "id": "Jiang", + "type": "preprint", + "title": "Accurate determination of tensor network state of quantum lattice models in two dimensions", + "authors": "H. C. Jiang, Z. Y. Weng, T. Xiang", + "year": "2008", + "publisher": "", + "journal": "", + "doi": "10.48550/arXiv.0806.3719", + "url": "https://arxiv.org/abs/0806.3719" + }, + { + "id": "Vidal", + "type": "preprint", + "title": "Efficient classical simulation of slightly entangled quantum computations", + "authors": "Guifre Vidal", + "year": "2003", + "publisher": "", + "journal": "", + "doi": "10.48550/arXiv.quant-ph/0301063", + "url": "https://arxiv.org/abs/quant-ph/0301063" + }, + { + "id": "Schollwoeck", + "type": "preprint", + "title": "The density-matrix renormalization group in the age of Matrix Product States", + "authors": "Ulrich Schollwoeck", + "year": "2010", + "publisher": "", + "journal": "", + "doi": "10.48550/arXiv.1008.3477", + "url": "https://arxiv.org/abs/1008.3477" + } + ], + "basedOnPapers": ["10.48550/arXiv.1008.3477"], + "referencedByPapers": [], + "relatedContent": [ + { + "type": "demonstration", + "id": "tutorial_How_to_simulate_quantum_circuits_with_tensor_networks", + "weight": 1.0 + }, + { + "type": "demonstration", + "id": "tutorial_diffable-mitigation", + "weight": 1.0 + } + ] +} diff --git a/demonstrations/tutorial_mps.py b/demonstrations/tutorial_mps.py new file mode 100644 index 0000000000..7e0d891e3e --- /dev/null +++ b/demonstrations/tutorial_mps.py @@ -0,0 +1,698 @@ +r"""Introducing matrix product states for quantum practitioners +=============================================================== + +Matrix product states remain the workhorse for a broad range of modern classical quantum simulation techniques, +still to this day. Their unique features (like offering a canonical form) make them an incredibly neat tool +in terms of simplicity and algorithmic complexity. +In this demo, we are going to cover all the essentials you need to know in order to handle matrix product states, +and show how to use them to simulate quantum circuits. + +.. figure:: ../_static/demonstration_assets/how_to_simulate_quantum_circuits_with_tensor_networks/TN_MPS.gif + :align: center + :width: 90% + +Introduction +------------ + +Matrix product states (MPS) are an efficient representation of quantum states in one spatial dimension. +However, due to their unique features like offering a canonical form, they are employed in a variety of tasks beyond just 1D systems. + +The amount of entanglement the MPS can represent is user-controlled via a hyper-parameter, the so-called `bond dimension` :math:`\chi`. +If we allow :math:`\chi` to be of :math:`\mathcal{O}(2^{\frac{n}{2}})` for a system of :math:`n` qubits, we can write `any` state as an `exact` MPS. +To avoid exponentially large resources, however, one typically sets a finite bond dimension :math:`\chi` at the cost of introducing an approximation error. + +.. figure:: ../_static/demo_thumbnails/opengraph_demo_thumbnails/OGthumbnail_mps_simulation.png + :align: center + :width: 70% + +For some specific classes of states, this is provably sufficient to have faithful representations (see :ref:`Area Law`). +But because MPS come with a lot of powerful computational features that we are going to discuss later (in particular :ref:`canonical forms `), +they are still used in much more complex systems where these requirements do not hold anymore, and still yield good results. +For example, state-of-the-art `quantum chemistry `__ simulations were performed using MPS [#Baiardi]_ +and similar methods have been used to simulate experiments on the largest available quantum computers at the time [#Patra]_. + +It is known that there are more suitable tensor network states like projected entangled pair states (PEPS) for +more complex situations (see the :ref:`Area Law` section). +However, they all suffer from the need for significantly more complicated algorithms +and higher costs, as a lot of the things that make MPS so attractive, +like the availability of a canonical form, are not true anymore. +To put it plainly, it is often simply much easier to use readily available +MPS code and throw a large amount of resources into the bond dimension than to develop +more advanced tensor network methods. + +An exception to that are so-called `simple update` +methods [#Jiang]_, which use pseudo-canonical forms that allow a similarly simple +algorithmic complexity at the cost of not being optimal in its resources. +Reference [#Patra]_ is a good example of that. + +More advanced tensor network methods are continually being developed and optimized, though the biggest hindrance to the widespread use of already known advanced tensor network methods is the lack of reliable open-source implementations. +While the current state of affairs in academia is giving little incentive to change that, +MPS continue to be the workhorse for a wide variety of quantum simulation techniques. + +We are going to introduce the essentials of MPS in the first part of this demo. +Afterwards, in the second part, we will look at the specific application of simulating quantum circuits using MPS. + +Matrix product state essentials +------------------------------- + +Compression using singular value decomposition (SVD) +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +To understand MPS and the bond dimension, we first need to understand how one can generally use singular value decomposition (SVD) to do compression. +Any matrix :math:`M = \mathbb{C}^{M\times N}` can be singular-value-decomposed as + +.. math:: M = U \Lambda V^\dagger, + +where :math:`\Lambda` is the diagonal matrix of the :math:`r=\min(M, N)` real and non-negative singular values, +:math:`U \in \mathbb{C}^{M\times r}` is left-unitary, :math:`U^\dagger U = \mathbb{I}_r`, and +:math:`V^\dagger \in \mathbb{C}^{r\times N}` is right-unitary, :math:`V^\dagger (V^\dagger)^\dagger = \mathbb{I}_r`. +We say the columns and rows of :math:`U` and :math:`V^\dagger` are the left- and right-orthonormal +singular vectors, respectively. In the case of square and normal matrices, the singular values and singular vectors +are just the eigenvalues and eigenvectors. + +Small singular values and their corresponding singular vectors carry little information of the matrix. When the singular values have +a tail of small values, we can compress the matrix by throwing away both these numbers and the corresponding singular vectors. +The power of this approach is best seen by a small example. Let us load the image we have shown in the header of this demo and compress it. + +""" + +import numpy as np +import matplotlib.pyplot as plt + +# import image +img = plt.imread("../_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_mps_simulation.png") +# alternative: import image directly from url +# import urllib2 +# img_url = urllib2.urlopen("https://pennylane.ai/_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_mps_simulation.png") +# img = plt.imread(img_url) + +# only look at one color channel for demonstration purposes +img = img[:, :, 0] + +# Perform SVD +U, Lambda, Vd = np.linalg.svd(img) + +# Keep only the 50 largest singular values and vectors +chi = 50 +U_compressed = U[:, :chi] +Lambda_compressed = Lambda[:chi] +Vd_compressed = Vd[:chi] + +# Reconstruct the compressed image +compressed_img = U_compressed @ np.diag(Lambda_compressed) @ Vd_compressed + +fig, axs = plt.subplots(ncols=2) +ax = axs[0] +ax.imshow(img, vmin=0, vmax=1) +ax.set_title("Uncompressed image") + +ax = axs[1] +ax.imshow(compressed_img, vmin=0, vmax=1) +ax.set_title("Compressed image") + +plt.show() + +size_original = np.prod(img.shape) +size_compressed = np.prod(U_compressed.shape) + np.prod(Lambda_compressed.shape) + np.prod(Vd_compressed.shape) + +print(f"original image size: {size_original}, compressed image size: {size_compressed}, factor {size_original/size_compressed:.3f} saving") +############################################################################## +# +# +# The original image is :math:`334 \times 542` pixels, that we compress as :math:`334 \times 50` pixels in +# :math:`U`, :math:`50` pixels in :math:`\Lambda` and :math:`50 \times 542` pixels in :math:`V^\dagger`. +# This is possible because the information density in the image is low, as seen by the distribution of singular values. +# Let us visualize this by re-computing the singular values and plotting their distribution (note that they are automatically ordered in descending order). + + +_, Lambda, _ = np.linalg.svd(img) # recompute the full spectrum +plt.plot(Lambda) +plt.xlabel("index $i$") +plt.ylabel("$\\Lambda_i$") +plt.show() + +############################################################################## +# +# We are later going to do the same trick with state vectors. +# Note that the compressed information is encoded in :math:`U,` :math:`S` and :math:`V^\dagger`. +# If we want to retrieve the actual image :math:`M` (or state vector), we still need to reconstruct the full :math:`334 \times 542` pixels. +# Luckily, as we will later see in the case of MPS, we can retrieve all relevant information efficiently from the compressed components without ever +# having to reconstruct the full state vector. +# +# Turn any state into an MPS +# ~~~~~~~~~~~~~~~~~~~~~~~~~~ +# +# Formally, any quantum state :math:`|\psi\rangle \in \mathbb{C}^{2^n}` on :math:`n` qubits can be written as a matrix product state (MPS). +# The goal of this section will be to write an arbitrary state :math:`|\psi\rangle = \sum_{\sigma_1, .., \sigma_n} \psi_{\sigma_1, .., \sigma_n} |\sigma_1 .. \sigma_n\rangle` in the form +# +# .. math:: |\psi \rangle = \sum_{\sigma_1, .., \sigma_n} U^{\sigma_1} .. U^{\sigma_n} |\sigma_1 .. \sigma_n\rangle, +# +# where we decomposed the rank :math:`n` tensor :math:`\psi_{\sigma_1, .., \sigma_n}` into a product of matrices :math:`U^{\sigma_j}` +# for each value of :math:`\sigma_j` (equal to :math:`0` or :math:`1` for qubits). This is why it is called a **matrix product** state. +# While it historically makes sense to treat these indeed as matrices, given a concrete value of :math:`\sigma_j`, I find it more convenient +# to forget about the notion of matrices and just treat them as the collection of rank-3 tensors :math:`\{ U_{\mu_{i-1} \sigma_i \mu_i} \}` +# that have two so-called virtual indices, :math:`\mu_{i-1}, \mu_i`, and one so-called physical index, :math:`\sigma_j`. +# Whenever we write :math:`U^{\sigma_j}` with a superscript, we mean a matrix given a concrete value of :math:`\sigma_j`. +# In particular, we define :math:`\left(U^{\sigma_j}\right)_{\mu_{i-1} \mu_i} = U_{\mu_{i-1} \sigma_i \mu_i}` for notational convenience and to make it easier to follow along with the code. +# +# Graphically, this corresponds to splitting up the big rank-:math:`n` tensor into :math:`n` smaller tensors, +# similarly to our approach in the example of compressing an image using SVD. +# +# .. figure:: ../_static/demonstration_assets/mps/psi_to_mps_0.png +# :align: center +# :width: 70% +# +# .. note:: +# We are going to use combinations of tensor indices and treat them as one big index. +# In particular, the two indices :math:`\sigma_1 = \{0, 1\}` and :math:`\sigma_2 = \{0, 1\}` will have the combined index +# :math:`(\sigma_1 \sigma_2) = \{00, 01, 10, 11\}`. The actual order is a choice and does not matter for the analytic descriptions, +# but in practice, we just choose to do it in the same way as ``numpy`` arrays are reshaped for convenience. +# That way we don't have to worry about it and we also save on transpositions. +# +# The horizontal connections between the :math:`U` tensors are the matrix multiplications in the equation above. +# They are contractions over the virtual indices. The dangling vertical lines are the +# `physical` indices of the original state, :math:`\sigma_i`. +# +# Let us look at a concrete state vector with :math:`n=3` sites, so :math:`\psi_{\sigma_1 \sigma_2 \sigma_3}`, and decompose it as an MPS. + +n = 3 # three sites = three legs +psi = np.random.rand(2**3) +psi = psi / np.linalg.norm(psi) # random, normalized state vector +psi = np.reshape(psi, (2, 2, 2)) # rewrite psi as rank-n tensor + +############################################################################## +# +# +# We rewrite the tensor as a matrix with indices of the first site :math:`\sigma_1` and the combined indices of all remaining sites, :math:`(\sigma_2 \sigma_3)`. +# Now that we have a matrix, we can perform SVD to split off the first site. Mathematically, this is +# +# .. math:: \psi_{\sigma_1 \sigma_2 \sigma_3} \stackrel{\text{reshape}}{=} \psi_{\sigma_1, (\sigma_2 \sigma_3)} \stackrel{\text{SVD}}{=} \sum_{\mu_1} U_{\sigma_1 \mu_1} \Lambda_{\mu_1} V^\dagger_{\mu_1 (\sigma_2 \sigma_3)}. +# + +# reshape vector to matrix +psi = np.reshape(psi, (2, 2**(n-1))) +# SVD to split off first site +U, Lambda, Vd = np.linalg.svd(psi, full_matrices=False) + +############################################################################## +# +# We multiply the singular values onto :math:`V^\dagger` and call this the remainder state, :math:`\psi'_{\mu_1, (\sigma_2 \sigma_3)}`, +# so overall we have +# +# .. math:: \psi_{\sigma_1 \sigma_2 \sigma_3} = \sum_{\mu_1} U_{\sigma_1 \mu_1} \psi'_{\mu_1, (\sigma_2 \sigma_3)}. +# +# Graphically, this corresponds to the following. +# +# .. figure:: ../_static/demonstration_assets/mps/psi_to_mps_1.png +# :align: center +# :width: 70% +# +# Note that because :math:`\Lambda` is diagonal, it has the same virtual index on either side. +# +# We keep the :math:`U` tensors. We want to maintain the convention that they are of the shape ``(virtual_left, physical, virtual_right)``. +# Because there is no virtual index on the left for the first site, we introduce a dummy index of size ``1``. +# This is just to make the bookkeeping of the final MPS a bit simpler, as all tensors have the same shape structure. + +Us = [] +U = np.reshape(U, (1, 2, 2)) # mu1, s2, mu2 +Us.append(U) + +############################################################################## +# +# This procedure is repeated through all sites. The first step was special in that :math:`U_{\sigma_1 \mu_1}` is a vector for each value of :math:`\sigma_1`. +# When splitting up :math:`\psi'_{\mu_1, (\sigma_2 \sigma_3)}` we combine the virtual bond with the current site, and have all remaining sites be the other leg of the matrix we create for SVD. +# In particular, we do the following. +# +# .. math:: \psi'_{\mu_1, (\sigma_2 \sigma_3)} \stackrel{\text{reshape}}{=} \psi'_{(\mu_1 \sigma_2), (\sigma_3)} \stackrel{\text{SVD}}{=} \sum_{\mu_2} U_{\mu_1 \sigma_2 \mu_2} \Lambda_{\mu_2} V_{\mu_2 \sigma_3} +# + +psi_remainder = np.diag(Lambda) @ Vd # mu1 (s2 s3) +psi_remainder = np.reshape(psi_remainder, (2*2, 2)) # (mu1 s2), s3 +U, Lambda, Vd = np.linalg.svd(psi_remainder, full_matrices=False) + +U = np.reshape(U, (2, 2, 2)) # mu1, s2, mu2 +Us.append(U) + +U.shape, Lambda.shape, Vd.shape + +############################################################################## +# We again multiply the singular values onto the new :math:`V^\dagger` and take that as the remainder state :math:`\psi''`. +# The state overall now reads +# +# .. math:: \psi_{\sigma_1 \sigma_2 \sigma_3} = \sum_{\mu_1 \mu_2} U_{\sigma_1\mu_1} U_{\mu_1 \sigma_2 \mu_2} \psi''_{\mu_2 \sigma_3}. +# +# Graphically, this corresponds to +# +# .. figure:: ../_static/demonstration_assets/mps/psi_to_mps_2.png +# :align: center +# :width: 70% +# +# When the state is normalized, we are done. Otherwise, we can do the procedure one more time, again with a virtual dummy dimension on the right-most site. + +psi_remainder = np.diag(Lambda) @ Vd # mu1 (s2 s3) +psi_remainder = np.reshape(psi_remainder, (2*2, 1)) # (mu1 s2), s3 +U, Lambda, Vd = np.linalg.svd(psi_remainder, full_matrices=False) + +U = np.reshape(U, (2, 2, 1)) # mu1, s2, mu2 +Us.append(U) + +U.shape, Lambda.shape, Vd.shape + +############################################################################## +# Because our state vector was already normalized, the singular value in this last SVD is just ``1``, else it would yield the norm of ``psi`` +# (a good exercise to confirm by skipping the normalization step in the definition of ``psi`` above). +# +# The collected tensors :math:`U_{\mu_{i-1} \sigma_i \mu_i}` now make up the Matrix Product State and describe the original state :math:`|\psi\rangle` +# by appropriately contracting the virtual indices :math:`\mu_i`. We can briefly confirm this by reverse engineering the original state. +# +# +# .. figure:: ../_static/demonstration_assets/mps/psi_to_mps_3.png +# :align: center +# :width: 70% +# +# +# Due to the convention of +# the indices as ``(virtual_left, physical, virtual_right)``, the contraction is simple and we can use +# `np.tensordot `_ with ``axes=1``, indicating matrix-product-like contraction of the left-most and right-most index. +# This is another way of thinking of the obtained state as a **matrix product** state. + +print(f"Shapes of Us: {[_.shape for _ in Us]}") + +psi_reconstruct = Us[0] + +for i in range(1, len(Us)): + # contract the rightmost with the left most index + psi_reconstruct = np.tensordot(psi_reconstruct, Us[i], axes=1) + +print(f"Shape of reconstructed psi: {psi_reconstruct.shape}") +# remove dummy dimensions +psi_reconstruct = np.reshape(psi_reconstruct, (2, 2, 2)) +# original shape of original psi +psi = np.reshape(psi, (2, 2, 2)) + +np.allclose(psi, psi_reconstruct) + + +############################################################################## +# +# Up to this point, the description of the original state in terms of the MPS, made up by the three matrices :math:`U_{\mu_{i-1} \sigma_i \mu_i}`, is exact. +# With this construction, the sizes of the virtual bonds grow exponentially from :math:`2` to :math:`2^{n/2}` until the middle of the chain (to be confirmed here below). +# +# Just like in the example with images before, we can compress the state by only keeping +# the :math:`\chi` largest singular values with their respective singular vectors. +# This hyper-parameter :math:`\chi` is the bond dimension we mentioned earlier. It allows us +# to control the amount of entanglement the state can represent between everything that +# is left and right of the bond (more on that later). +# +# A full subroutine from :math:`|\psi\rangle` to its compressed MPS description is given by the following function ``dense_to_mps``. +# It is convenient to also keep the singular values for each bond to easily change the orthonormality of the tensors, but more on that in the next section on canonical forms. + +def split(M, bond_dim): + """Split a matrix M via SVD and keep only the ``bond_dim`` largest entries.""" + U, S, Vd = np.linalg.svd(M, full_matrices=False) + bonds = len(S) + Vd = Vd.reshape(bonds, 2, -1) + U = U.reshape((-1, 2, bonds)) + + # keep only chi bonds + chi = np.min([bonds, bond_dim]) + U, S, Vd = U[:, :, :chi], S[:chi], Vd[:chi] + return U, S, Vd + +def dense_to_mps(psi, bond_dim): + """Turn a state vector ``psi`` into an MPS with bond dimension ``bond_dim``.""" + Ms = [] + Ss = [] + + psi = np.reshape(psi, (2, -1)) # split psi[2, 2, 2, 2..] = psi[2, (2x2x2...)] + U, S, Vd = split(psi, bond_dim) # psi[2, (2x2x..)] = U[2, mu] S[mu] Vd[mu, (2x2x2x..)] + + Ms.append(U) + Ss.append(Ss) + bondL = Vd.shape[0] + psi = Vd + + for _ in range(n-2): + psi = np.reshape(psi, (2*bondL, -1)) # reshape psi[2 * bondL, (2x2x2...)] + U, S, Vd = split(psi, bond_dim) # psi[2, (2x2x..)] = U[2, mu] S[mu] Vd[mu, (2x2x2x..)] + Ms.append(U) + Ss.append(Ss) + + psi = np.tensordot(np.diag(S), Vd, 1) + bondL = Vd.shape[0] + + # dummy step on last site + psi = np.reshape(psi, (-1, 1)) + U, _, _ = np.linalg.svd(psi, full_matrices=False) + + U = np.reshape(U, (-1, 2, 1)) + Ms.append(U) + + return Ms, Ss + +############################################################################## +# Let us look at a larger state. First, let us observe how the dimension scales exponentially when we don't truncate the bonds. + +n = 12 +bond_dim = 10000 + +psi = np.random.rand(*[2]*n) +psi = psi/np.linalg.norm(psi) +Ms, Ss = dense_to_mps(psi, bond_dim) + +[M.shape for M in Ms] + +############################################################################## +# When setting a finite bond dimension :math:`\chi \leq 2^{n/2}`, +# we see the virtual bonds grow at the boundaries until reaching +# the maximal bond dimension and staying constant thereafter until +# reaching the other side. + +Ms, Ss = dense_to_mps(psi, 5) + +[M.shape for M in Ms] + +############################################################################## +# +# This was all to conceptually understand the relationship between dense vectors and a compressed matrix product state. +# We want to use MPS for many sites, where it is often not possible to write down the exponentially large state vector in the first place. +# In that case, we would simply start from an MPS description in terms of :math:`n` :math:`\chi \times 2 \times \chi` tensors. +# Luckily, we can obtain all relevant information without ever reconstructing the full state vector. +# +# .. _Canonical Forms: +# +# Canonical forms +# ~~~~~~~~~~~~~~~ +# +# In the above construction, we unknowingly already baked in a very useful feature of our MPS because all the :math:`U` matrices from the SVD +# are left-orthonormal (highlighted by the pink color in the illustrations of the left-orthonormal :math:`U` tensors). In particular, they satisfy +# +# .. math:: \sum_{\sigma_i} \left(U^{\sigma_i} \right)^\dagger U^{\sigma_i} = \mathbb{I}, +# +# which is a compact matrix notation treating :math:`U^{\sigma_i}` as a matrix for a concrete value of :math:`\sigma_i`. +# Making the coefficient in the matrix multiplication explicit we have +# +# .. math:: \sum_{\sigma_i \mu_{i-1}} U^{*}_{\mu_{i-1} \sigma_i \mu'_i} U_{\mu_{i-1} \sigma_i \mu_i} = \mathbb{I}_{\mu'_i \mu_i}. +# +# Note that we only use the complex conjugation :math:`U^{*}` instead of Hermitian conjugate :math:`U^\dagger` +# because we can just choose the indices to contract over accordingly. +# +# Let us briefly confirm that: + +for i in range(len(Ms)): + id_ = np.tensordot(Ms[i].conj(), Ms[i], axes=([0, 1], [0, 1])) + is_id = np.allclose(id_, np.eye(len(id_))) + print(f"U[{i}] is left-orthonormal: {is_id}") + +############################################################################## +# This is a very powerful identity as it tells us that contracting a site of the MPS from the left is just the identity. +# +# .. figure:: ../_static/demonstration_assets/mps/left_orthonormal.png +# :align: center +# :width: 30% +# +# This means that computing the norm, which is just contracting the MPS with itself, becomes trivial. +# +# .. figure:: ../_static/demonstration_assets/mps/norm_trivial.png +# :align: center +# :width: 70% +# +# The fact that we went through the MPS from left to right was a choice. +# We could have equivalently gone through the MPS from right to left and obtained a right-canonical state by keeping the right-orthonormal :math:`V^\dagger` of the decompositions. +# +# When computing expectation values, it is convenient to have the MPS in a mixed canonical form. +# Take some single-site observable :math:`O_i` for which we want to compute the expectation value. +# The best way to do this is to have the MPS such that all sites left of site :math:`i` are left-canonical and all sites right of it are right-canonical. +# That way, the contraction :math:`\langle \psi | O | \psi \rangle` for local expectation values reduces to contractions on just a single site, +# because all other contractions are just the identity. +# We call that single tensor :math:`\Theta`, it is all we need to compute the expectation value and we shall see how we can easily obtain it while not having to care about all other sites. +# +# .. figure:: ../_static/demonstration_assets/mps/mixed_canonical_observable.png +# :align: center +# :width: 70% +# +# We can obtain such a mixed canonical form by starting from our left-canonical MPS and going through the sites from right to left, thereby right-canonizing all sites until the observable. +# However, if we keep track of the singular values at all times, we can switch any site tensor from left- to right-orthonormal by just multiplying with the singular values. +# This is the so-called Vidal form introduced in [#Vidal]_ and it works like this: Even though we are not going to use them in our representation, it makes sense to introduce the "bare" local :math:`\Gamma`-tensors :math:`\{\Gamma^{\sigma_i}\}` in terms of +# +# .. math:: \Gamma^{\sigma_i} = \left(\Lambda^{[i-1]}\right)^{-1} U^{\sigma_i} = \left(V^\dagger\right)^{\sigma_i} \left(\Lambda^{[i]}\right)^{-1} +# +# and write the MPS in terms of those bare :math:`\Gamma`-tensors with the singular values connecting them. We can then just sub-select and recombine parts to get either right- or left-canonical tensors in the following manner. +# +# .. figure:: ../_static/demonstration_assets/mps/vidal.png +# :align: center +# :width: 70% +# +# In particular, to compute the expectation value described just above, we need the central tensor that we named :math:`\Theta`. +# We are not going to actually store the :math:`\Gamma`-tensors but continue to use the left-orthonormal :math:`U`-tensors. +# So all we need to do is construct :math:`\Theta_{\mu_{i-1} \sigma_i \mu_i} = \sum_{\tilde{\mu}_i} U_{\mu_{i-1} \sigma_i \tilde{\mu}_i} \Lambda^{[i]}_{\tilde{\mu}_i \mu_i}`. This has two advantages: 1) +# we only need to perform one contraction with the singular values from the right and 2) we avoid having to compute any +# inverses of the singular values, which numerically can become messy for very small singular values. +# +# Finally, the local observable expectation value simply becomes +# +# .. math:: \langle \psi | O | \psi \rangle = \text{tr}\left[ \sum_{\sigma_i \tilde{\sigma}_i} \Theta^{\sigma_i} O^{\sigma_i \tilde{\sigma}_i} \Theta^{*\tilde{\sigma}_i} \right], +# +# or, graphically, the following. +# +# .. figure:: ../_static/demonstration_assets/mps/final_expval.png +# :align: center +# :width: 50% +# +# The canonical form essentially allows us to treat local operations locally and remove all redundancy on other sites. This will come in handy later when we look at simulating quantum circuits with MPS. +# It also enables the very powerful density matrix renormalization group (DMRG) algorithm. Here, one constructs the ground state of a Hamiltonian by iteratively sweeping through the MPS back and forth, solving +# the eigenvalue problem locally at each site (with all other sites "frozen"). This works extremely well in practice and is hence still one of the workhorses of classical quantum simulation to this day. For a very good review +# on DMRG with MPS, see [#Schollwoeck]_. +# +# .. _Area Law: +# +# Entanglement and area laws +# ~~~~~~~~~~~~~~~~~~~~~~~~~~ +# +# Entanglement is best quantified via bipartitions of states. In particular, separating a full system :math:`|\psi\rangle` and a sub-system :math:`\rho_\text{sub}`, the von Neumann entanglement entropy is given by +# +# .. math:: S(\rho_\text{sub}) = -\text{tr}\left[\rho_\text{sub} \log\left(\rho_\text{sub}\right) \right]. +# +# In an MPS, the singular values of the bonds naturally encode the entanglement of bipartitions between all sites left vs all sites right of the bond. +# In particular, the von Neumann entanglement entropy at bond :math:`i` is given by +# +# .. math:: S(\rho_{1:i}) = S(\rho_{i+1:n}) = - \sum_{\mu_i=1}^{\chi} \Lambda^2_{\mu_i} \log\left( \Lambda_{\mu_i}^2 \right). +# +# Given a bond dimension :math:`\chi`, the maximal entanglement entropy we can obtain is for the all-equal distribution of singular values, :math:`\Lambda_i^2 \equiv 1/\chi`. +# The entanglement entropy is thus bounded by +# +# .. math:: S(\rho_{1:i}) \leq \log(\chi) = \text{const}. +# +# This is the area law of entanglement for one-dimensional systems. A "volume" in one spatial dimension is a line, and its surface area, :math:`\partial V`, two points, so constant in the system size (see note below). +# +# .. note:: +# +# Ground states of local and gapped Hamiltonians are known to satisfy the area law of entanglement. +# This law states that the entanglement entropy of a sub-system grows with its surface area :math:`\partial V` instead of its volume :math:`V`. +# For one-dimensional systems, the volume is just a line and its surface area just a constant. The entanglement +# between any such sub-system in an MPS with a finite bond dimension :math:`\chi` is naturally bounded by :math:`\log(\chi)=\text{const.}`, +# so MPS satisfy the area law of entanglement for one-dimensional systems. +# +# Projected entangled pair states (PEPS) are the natural generalization of MPS to regular 2D or 3D grids as well as more general graph connectivities and are known to fulfill the +# respective area laws of entanglement, making them the correct ansätze for local and gapped Hamiltonians in those cases. For example, for a 2D +# PEPS with a square subsystem of volume :math:`L \times L` and bond dimension :math:`\chi`, the entanglement entropy between the square and the rest of the system is bounded by +# :math:`\log(\chi^L) = L \log(\chi)`, which is proportional to the circumference :math:`\propto L`, and not its area. +# +# .. figure:: ../_static/demonstration_assets/mps/area_law.png +# :align: center +# :width: 50% + + + +############################################################################## +# +# +# Quantum simulation with MPS +# --------------------------- +# +# We can use MPS to classically simulate quantum algorithms. This is a very useful tool for as long as +# real quantum devices are noisy and costly to use. +# An MPS simulator works very similarly to a state vector simulator like :class:`~DefaultQubit`, the only difference is that the underlying state +# is encoded in an MPS. +# +# Applying local gates +# ~~~~~~~~~~~~~~~~~~~~ +# +# Due to its canonical form, applying local gates onto an MPS is very straightforward and little work. +# Let us walk through the example of applying a :math:`\text{CNOT}` gate on neighboring sites on the underlying MPS state of the simulator. +# +# Graphically, this is happening in three steps as illustrated here. +# +# .. figure:: ../_static/demonstration_assets/mps/apply_gate.png +# :align: center +# :width: 65% +# +# In the first step, we simply contract the appropriate physical indices of the MPS with those of the :math:`\text{CNOT}` matrix (which we reshape to :math:`2\times 2\times 2\times 2`). +# Note that we also take the bond singular values into account. The result is a big blob with two virtual indices and two physical indices. +# We then just split this blob in the same way we split up the dense state vector and keep track of its singular values to restore the canonical form. +# +# Applying non-local gates +# ~~~~~~~~~~~~~~~~~~~~~~~~ +# +# The situation gets a bit more complicated when we apply a :math:`\text{CNOT}` (or any multi-site) gate on non-neighboring sites. We have two possibilities to handle this. +# We can do what a quantum computer would do, which is swap out sites until the appropriate indices are neighboring, perform the operation, and then un-swap the sites. +# Alternatively, we can construct a so-called matrix product operator (MPO) that acts on all sites in between with an identity. This is done in the following way. +# +# First, we split the matrix of the gate into tensors that act locally. This is done again using SVD. In general, the MPO bond dimension for a two-qubit gate is maximally 4, +# but in some cases like :math:`\text{CNOT}` it is :math:`2`, so we can do a lossless compression by only keeping the two non-zero singular values and tossing the zeros. After we have done that, we can multiply +# the singular values onto either site as we do not have a use for them in the MPO other than for doing the compression. +# +# .. figure:: ../_static/demonstration_assets/mps/cnot_split.png +# :align: center +# :width: 50% +# +# Now if we want to apply the CNOT gate on non-neighboring sites, we fill the intermediate sites with identities :math:`\mathbb{I} = \delta_{\sigma_i \sigma'_i} \delta_{\mu_{i-1} \mu_i}`, and contract that larger +# MPO with the MPS. +# +# .. figure:: ../_static/demonstration_assets/mps/non_local_cnot.png +# :align: center +# :width: 80% +# +# Here we just need to be careful to contract in the right order, otherwise we might end up with unnecessarily large tensors in intermediate steps. For example, if we first contract all physical indices we get a big blob +# that is exponentially large in the number of intermediate sites. While it is in general NP-hard to find the optimal contraction path, +# for MPS the optimal path is known. The way to do it is by alternating between the physical index and the corresponding two virtual indices, going either from left to right or, equivalently, from right to left (see Figure 21 in [#Schollwoeck]_). +# +# +# Running simulations and setting the bond dimension +# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +# +# While we focussed on the specific case of a :math:`\text{CNOT}` gate, this concept is readily generalized to arbitrary two- or multi-qubit gates. +# With that, we are now ready to run some quantum circuit simulations. We don't have to code up all contractions by hand. Instead, we can use +# PennyLane's :class:`~pennylane.devices.default_tensor.DefaultTensor` device that takes care of all of this under the hood. All we need to do is set the bond dimension and tell the device whether +# it should use the swap-unswap or MPO method for applying non-local gates. This is done via the keyword argument ``contract``, where we can choose between ``"swap+split"`` (what I called swap-unswap), +# ``"nonlocal"`` (what I called the MPO method), and ``"auto-mps"``, which uses swap-unswap for 2-qubit gates and the MPO method for 3 and more qubits. +# +# Aside from that, we can basically use the device like any other state vector simulator device. +# Let us run a VQE example from using the `PennyLane Datasets `__ data for the :math:`H_6` molecule. What you will see here is mostly boilerplate code in PennyLane for using ``default.tensor``, and you can +# see our :doc:`demo on how to use default.tensor ` for more details. + +import pennylane as qml + +[dataset] = qml.data.load("qchem", molname="H6", bondlength=1.3, basis="STO-3G") + +H = dataset.hamiltonian # molecular Hamiltonian in qubit basis +n_wires = len(H.wires) # number of qubits + +def circuit(): + qml.BasisState(dataset.hf_state, wires=H.wires) # Hartree–Fock initial state + for op in dataset.vqe_gates: # Applying all pre-optimized VQE gates + qml.apply(op) + return qml.expval(H) # expectation value of molecular Hamiltonian + +# set up device with hyper-parameters and kwargs +mps = qml.device("default.tensor", wires=n_wires, method="mps", max_bond_dim=30, contract="auto-mps") + +# Create the QNode to execute the circuit on the device, and call it (w/o arguments) +res = qml.QNode(circuit, mps)() + +# Compare MPS simulation result with pre-optimized state-vector result +res, dataset.vqe_energy + +############################################################################## +# We've set the bond dimension to ``30`` kind of arbitrarily, and saw that we are pretty close to the exact result +# (note that exact here refers to the simulation method, not the ground state energy of the VQE result itself). +# But when dealing with systems of sizes where we don't have the means to compare to an exact result, how do know +# that our simulation results make sense, and are not scrambled by the errors introduced by a small bond dimension? +# +# The answer is **finite-size scaling** (sometimes also called bond dimension scaling or just extrapolation). This is a standard method +# in tensor network simulations and originates from condensed matter physics and quantum phase transitions. +# The idea is to run the same simulation with an increasing bond dimension and check that it saturates and converges to an extrapolated value. +# In spirit, this is similar to :doc:`zero noise extrapolation `. +# +# We choose a range of bond dimensions and plot the results for the simulation against them, keeping in mind that +# the maximum bond dimension of a system of :math:`n` qubits is :math:`2^{\frac{n}{2}}`. + +bond_dims = 2**np.arange(2, (n_wires//2)+1) # maximum required bond dimension is 2**(n_wires//2) = 64 +ress = [] + +for bond_dim in bond_dims: + mps = qml.device("default.tensor", wires=n_wires, method="mps", max_bond_dim=bond_dim, contract="auto-mps") + res = qml.QNode(circuit, mps)() + ress.append(res) + + +plt.plot(bond_dims, ress, "x:", label="mps sim") +plt.hlines(dataset.vqe_energy, bond_dims[0], bond_dims[-1], label="exact result") +plt.xscale("log") +plt.xlabel("bond dim $\\chi$") +plt.xticks(bond_dims, bond_dims) +plt.legend() +plt.show() + + +############################################################################## +# We see that already for :math:`\chi = 32` we have pretty accurate results. +# We might even get away with :math:`\chi = 16` for some qualitative simulations at the cost of some approximation error. +# +# Setting the bond dimension is an important part of performing MPS simulations. Finite-size scaling is a +# quantitative tool to orient ourselves and choose a suitable bond dimension for our simulations. +# We can extrapolate the exact result for every simulation run, +# but this is of course more expensive as it requires multiple executions at different bond dimensions, and is not guaranteed to converge. +# Sometimes we are also just happy to get a cheap qualitative result for large systems that may or may not be 100% accurate. +# +# .. note:: +# ``default.tensor`` is based on `quimb `_. Both +# ``quimb`` and ``default.tensor`` are under active development, so in case you encounter some rough edges, please +# submit a `GitHub issue `_ in the PennyLane repository. + + +############################################################################## +# +# Conclusion +# ---------- +# +# In this demo we introduced the basics of matrix product states (MPS) and saw how the existence of a canonical form simplifies a lot of the contractions. +# This fact can also be used for simulations of quantum circuits with local and non-local gates. +# We showed how to run quantum circuits using PennyLane's :class:`~pennylane.devices.default_tensor.DefaultTensor` device and how to systematically find an appropriate bond dimension. +# +# While MPS are mathematically known to be well-suited to describe a particular class of states (those that fulfill the area law of entanglement in 1D), we can +# also simulate more complex systems by throwing some extra resources into the bond dimension. + + + +############################################################################## +# +# References +# ---------- +# +# .. [#Baiardi] +# +# Alberto Baiardi, Markus Reiher +# "The Density Matrix Renormalization Group in Chemistry and Molecular Physics: Recent Developments and New Challenges" +# `arXiv:1910.00137 `__, 2019. +# +# .. [#Patra] +# +# Siddhartha Patra, Saeed S. Jahromi, Sukhbinder Singh, Roman Orus +# "Efficient tensor network simulation of IBM's largest quantum processors" +# `arXiv:2309.15642 `__, 2023. +# +# .. [#Jiang] +# +# H. C. Jiang, Z. Y. Weng, T. Xiang +# "Accurate determination of tensor network state of quantum lattice models in two dimensions" +# `arXiv:0806.3719 `__, 2008. +# +# .. [#Vidal] +# +# Guifre Vidal +# "Efficient classical simulation of slightly entangled quantum computations" +# `arXiv:quant-ph/0301063 `__, 2003. +# +# .. [#Schollwoeck] +# +# Ulrich Schollwoeck +# "The density-matrix renormalization group in the age of Matrix Product States" +# `arXiv:1008.3477 `__, 2010. +# + + +############################################################################## +# About the author +# ---------------- +#

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