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There are some classes of + +# quantum citcuits that we know can be efficiently simulated, and this work attempts to fulfil part of + +# this niche. + +# + +# IQPopt is a package designed for fast optimization of parameterized Instantaneous Quantum Polynomial + +# (IQP) circuits using JAX, resulting in a linear scaling on the number of qubits. These circuits are + +# a commonly studied class of quantum circuits comprised of commuting gates, for which we can + +# efficiently evaluate expectation values of tensor Pauli operators, but have become well known due to + +# a number of results related to the hardness of sampling their output distributions [1-3]. As a + +# consequence, IQP circuits are prime candidates for potential quantum advantages for tasks related to + +# sampling, and have been the focus of a number of theoretical [4-6] and experimental [7,8] works + +# aiming towards this goal. + +# + +# In this demo we present the IQPopt package, which contains some tools that allow us to approximate + +# expectation values of PauliZ operators for these circuits and, therefore, to evaluate them in + +# generative optimization tasks. + +# + +###################################################################### +# .. figure:: ../_static/demonstration_assets/iqpopt/iqpopt_main.png + +# :alt: IQP circuit optimization + +# + +# IQP circuit optimization + +# + +###################################################################### +# Creating a circuit + +# ------------------ + +# + +###################################################################### +# These are circuits comprised of gates :math:`\text{exp}(i\theta_j X_j)`, where the generator + +# :math:`X_j` is a tensor product of Pauli X operators acting on some subset of qubits and + +# :math:`\theta_j` is a trainable parameter. Input states and measurements are diagonal in the + +# computational (Z) basis. + +# + +# To define such a circuit (with input state :math:`\vert 0 \rangle`) we need to specify the number of + +# qubits and the parameterized gates + +# + +import iqpopt as iqp +from iqpopt.utils import local_gates + +n_qubits = 3 +gates = local_gates(n_qubits, 2) # all gates with max weight 2 + +circuit = iqp.IqpSimulator(n_qubits, gates) + +###################################################################### +# A set of gates is specified by a list + +# + +# .. raw:: html + +# + +#

    + +# + +# gates = [gen1, gen2, gen3] + +# + +# .. raw:: html + +# + +#
    + +# + +# that contains the generators of the gates. Since all gates have Pauli X type generators, a generator + +# is represented by a list that specifies which qubits are acted on by an X operator. For example, + +# + +# .. raw:: html + +# + +#
    + +# + +# gen1 = [0,1] + +# + +# .. raw:: html + +# + +#
    + +# + +# corresponds to a gate with generator :math:`X\otimes X` acting on the first two qubits of the + +# circuit. For a three qubit system the corresponding parameterized gate is therefore + +# :math:`\exp(i\theta_1 X\otimes X\otimes \mathbb{I})`. + +# + +# The previous example prepares a three qubit circuit with all single and two qubit gates via the + +# function local_gates, i.e. + +# + +# .. raw:: html + +# + +#
    + +# + +# gates = [[0], [1], [2], [0,1], [0,2], [1,2]] + +# + +# .. raw:: html + +# + +#
    + +# + +###################################################################### +# Estimating expectation values + +# ----------------------------- + +# + +###################################################################### +# The class method ``IqpSimulator.op_expval()`` is used to provide estimates of expectation values. + +# The function requires a parameter array ``params``, a PauliZ operator specified by its binary + +# representation ``op``, the number of samples ``n_samples`` that controls the precision of the + +# approximation (the more the better), and a JAX pseudo random number generator key to seed the + +# randomness of the sampling. + +# + +# Below is a simple example implementing this for the three qubit circuit defined above, which returns + +# the expectation value estimate as well as its standard error. + +# + +import jax +import jax.numpy as jnp +import numpy as np + +op = jnp.array([0,1,0]) # Pauli Z on the second qubit +params = jnp.array(np.random.rand(len(circuit.gates))) +n_samples = 1000 +key = jax.random.PRNGKey(42) + +expval, std = circuit.op_expval(params, op, n_samples, key) + +print(expval, std) + +###################################################################### +# The function also allows for fast batch evaluation of expectation values. If we specify a batch of + +# operators ``ops`` by an array we can batch evaluate the expectation values and errors in parallel + +# with the same syntax. + +# + +ops = jnp.array([[1,0,0],[0,1,0],[0,0,1]]) # batch of single qubit Pauli Zs + +expvals, stds = circuit.op_expval(params, ops, n_samples, key) + +print(expvals, stds) + +###################################################################### +# Testing the expectation values with Pennylane + +# --------------------------------------------- + +# + +###################################################################### +# When using new software, it is always good practice to check your results if possible. In our case + +# we have Pennylane, a well tested library that does this calculations for a small number of qubits. + +# On the `package's github `__ you can find already made functions + +# that calculate the same expectation values as ``.op_expval()``, taking the necessary info from an + +# ``IqpSimulator`` object. + +# + +# Let's first download these functions so we don't have to build them again: + +# + +import requests + +req = requests.get("https://raw.githubusercontent.com/XanaduAI/iqpopt/refs/heads/main/tests/pennylane_functions.py?token=GHSAT0AAAAAACYVZBT5V4YNS5U2YD6COCQYZ2YM32Q") + +with open("pennylane_functions.py", "wb") as f: + f.write(req.content) + +###################################################################### +# Using the same ``params`` and ``op`` as before (but using ``numpy`` arrays instead of + +# ``jax.numpy``), let's see the pennylane's result. + +# + +from pennylane_functions import penn_op_expval + +penn_expval = penn_op_expval(circuit, np.array(params), np.array(op)) + +print(penn_expval) + +###################################################################### +# Since the calculation on ``iqpopt``\ 's side is stochastic, the results are not exactly the same, + +# but as we can see, they are within std error. You can try increasing ``n_samples`` in order to + +# obtain closer approximations. + +# + +###################################################################### +# Sampling and probabilities + +# -------------------------- + +# + +###################################################################### +# We can also view a parameterized IQP circuit as a generative model that generates samples of binary + +# vectors according to the distribution :raw-latex:`\begin{align} + +# q_{\boldsymbol{\theta}}(\boldsymbol{x}) \equiv q(\boldsymbol{x}\vert\boldsymbol{\theta})=\vert\bra{\boldsymbol{x}}U(\boldsymbol{\theta})\ket{0}\vert^2. + +# \end{align}` For low amounts of qubits, we can use Penylane's arsenal and know the output + +# probabilities of the circuit as well as sample from it. Note that there is not an efficient + +# algorithm to do these, so, for large numbers of qubits, these methods will either take too long or + +# not work at all. + +# + +# These functions are already implemented in the ``IqpSimulator`` object, so we don't have to download + +# them. + +# + +sample = circuit.sample(params, shots=1) +print(sample) + +probabilities = circuit.probs(params) +print(probabilities) + +###################################################################### +# The ``.probs()`` method works as it does in pennylane, the returned array of probabilities is in + +# lexicographic order. + +# + +###################################################################### +# Optimizing a circuit + +# -------------------- + +# + +###################################################################### +# Circuits can be optimized via a separate ``Trainer`` class. To instantiate a trainer object we first + +# define a loss function (also called an objective function), an optimizer and an initial stepsize for + +# the gradient descent. Continuing our example from before, below we define a simple loss function + +# that is a sum of expectation values returned by ``.op_expval()`` . + +# + +def loss_fn(params, circuit, ops, n_samples, key): + expvals = circuit.op_expval(params, ops, n_samples, key)[0] + return jnp.sum(expvals) + +optimizer = "Adam" +stepsize = 0.001 + +trainer = iqp.Trainer(optimizer, loss_fn, stepsize) + +###################################################################### +# Any differentiable loss function expressible in JAX can be defined, but must have a first argument + +# ``params`` that corresponds to the optimization parameters of the circuit. To minimize the loss + +# function, we call the method ``.train()`` , which requires the number of iterations ``n_iters`` and + +# the initial arguments of the loss function ``loss_kwargs`` given as a dictionary object. This + +# dictionary must contain a key ``params`` whose corresponding value specifies the initial parameters. + +# + +import matplotlib.pyplot as plt + +n_iters = 1000 +params_init = np.random.normal(0, 1, len(circuit.gates)) +n_samples = 100 + +loss_kwargs = { + "params": params_init, + "circuit": circuit, + "ops": ops, + "n_samples": n_samples, + "key": key, +} + +trainer.train(n_iters, loss_kwargs) + +trained_params = trainer.final_params +plt.plot(trainer.losses) # plot the loss curve +plt.show() + +###################################################################### +# Generative machine learning tools + +# --------------------------------- + +# + +# The package contains a module ``gen_qml`` with functionality to train and evaluate generative models + +# expressed as ``IqpSimulator`` circuits. + +# + +###################################################################### +# Training via the maximum mean discrepancy loss + +# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +# + +# The Maximum Mean Discrepancy (MMD) [9] is an integral probability metric that measures the + +# similarity between two probability distributions, and can serve as a loss function to train + +# generative models. It has one parameter called ``sigma``, the bandwidth of the used gaussian kernel. + +# + +# This metric is usually calculated with samples from both probability distributions and we have an + +# implementation ready for this case in the ``gen_qml`` module. + +# + +import iqpopt.gen_qml as gen +from iqpopt.gen_qml.utils import median_heuristic + +n_bits = 100 +# toy datasets of low weight bitstrings +X1 = np.random.binomial(1, 0.05, size=(100, n_bits)) +X2 = np.random.binomial(1, 0.07, size=(100, n_bits)) + +sigma = median_heuristic(X1) # bandwidth for MMD + +mmd = gen.mmd_loss_samples(X1, X2, sigma) +print(mmd) + +###################################################################### +# This metric can be determined only with expectation values of PauliZ operators, so, when instead of + +# samples, we have an ``IqpSimulator`` object, we can also calculate it comparing the circuit's + +# distribution with the ``ground_truth``. + +# + +# The implementation in the ``gen_qml`` module only needs an additional parameter ``n_ops`` that + +# controls the accuracy of this value. For each of these ``n_ops``, an expectation value will be + +# calculated with ``n_samples``. The higher the number of operators the better. + +# + +# Let's see it in an example with 20 qubits: + +# + +n_qubits = 20 + +# toy dataset of low weight bitstrings (from ground truth p) +ground_truth = np.random.binomial(1, 0.2, size=(100, n_qubits)) + +gates = local_gates(n_qubits, 2) +circuit = iqp.IqpSimulator(n_qubits, gates) + +params = np.random.normal(0, 1/np.sqrt(n_qubits), len(gates)) +sigma = median_heuristic(ground_truth) # bandwidth for MMD + +mmd = gen.mmd_loss_iqp(params, + circuit, + ground_truth, + sigma, + n_ops=1000, + n_samples=1000, + key=jax.random.PRNGKey(42)) +print(mmd) + +###################################################################### +# Now, similar to what we did a few sections back in *Optimizing a circuit*, this function can be used + +# with a ``Trainer`` object to train a quantum generative model given as a parameterized IQP circuit. + +# + +# Using the last defined 20 qubits circuit: + +# + +X_train = np.random.binomial(1, 0.2, size=(1000, n_qubits)) +params_init = np.random.normal(0, 1/np.sqrt(n_qubits), len(gates)) +loss = gen.mmd_loss_iqp # MMD loss + +loss_kwargs = { + "params": params_init, + "iqp_circuit": circuit, + "ground_truth": X_train, # samples from ground truth distribution + "sigma": sigma, + "n_ops": 1000, + "n_samples": 1000, + "key": jax.random.PRNGKey(42), +} + +trainer = iqp.Trainer("Adam", loss, stepsize=0.01) +trainer.train(n_iters=250, loss_kwargs=loss_kwargs) + +trained_params = trainer.final_params +plt.plot(trainer.losses) +plt.show() + +###################################################################### +# Kernel Generalized Empirical Likelihood + +# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +# + +# In order to probe how well a generative model captures a distribution of high-dimensional data, we + +# can use the Kernel Generalized Empirical Likelihood [10]. This metric is a measure of the closeness + +# of two distributions while also serving as a tool to evaluate mode dropping. + +# + +# As before, if samples are available from the generative model, we have a straightforward + +# implementation of the KGEL test: + +# + +n_qubits = 10 + +# toy dataset of low weight bitstrings +X = np.random.binomial(1, 0.2, size=(1000, n_qubits)) + +n_witness = 10 +T = X[-n_witness:] # witness points (sampled from the ground truth distribution) +X = X[:-n_witness] # ground truth for the kgel test + +# model samples +Y = np.random.binomial(1, 0.3, size=(1000, n_qubits)) + +sigma = median_heuristic(X) #bandwidth for KGEL +kgel, p_kgel = gen.kgel_opt_samples(T, X, Y, sigma) +print(kgel) + +###################################################################### +# To evaluate the KGEL for a model given by a ``IqpSimulator`` object, we need to use another + +# implementation based on the calculation of expectation values of PauliZ operators. Let's see how + +# well the last trained circuit matches the ground truth with the KGEL test: + +# + +n_witness = 10 +T = X_train[-n_witness:] # witness points (sampled from the ground truth distribution) +X = X_train[:-n_witness] # ground truth for the kgel test + +sigma = median_heuristic(X) +kgel, p_kgel = gen.kgel_opt_iqp(circuit, + trained_params, + T, + X, + sigma, + n_ops=1000, + n_samples=1000, + key=jax.random.PRNGKey(42)) +print(kgel) + +###################################################################### +# References: + +# + +# [1] Michael J Bremner, Richard Jozsa, and Dan J Shepherd. "Classical simulation of commuting + +# quantum computations implies collapse of the polynomial hierarchy". In: Proceedings of the Royal + +# Society A: Mathematical, Physical and Engineering Sciences 467.2126 (2011), pp. 459-472 (page 1). + +# + +# [2] Michael J Bremner, Ashley Montanaro, and Dan J Shepherd. "Average-case complexity versus + +# approximate simulation of commuting quantum computations". In: Physical review letters 117.8 (2016), + +# p. 080501 (page 1). + +# + +# [3] Simon C Marshall, Scott Aaronson, and Vedran Dunjko. "Improved separation between quantum and + +# classical computers for sampling and functional tasks". In: arXiv preprint arXiv:2410.20935 (2024) + +# (page 1). + +# + +# [4] Michael J Bremner, Ashley Montanaro, and Dan J Shepherd. "Achieving quantum supremacy with + +# sparse and noisy commuting quantum computations". In: Quantum 1 (2017), p. 8 (page 1). + +# + +# [5] Tomoyuki Morimae and Suguru Tamaki. "Additive-error fine-grained quantum supremacy". In: Quantum + +# 4 (2020), p. 329 (page 1). + +# + +# [6] Louis Paletta, Anthony Leverrier, Alain Sarlette, Mazyar Mirrahimi, and Christophe Vuillot. + +# "Robust sparse IQP sampling in constant depth". In: Quantum 8 (2024), p. 1337 (page 1). + +# + +# [7] Dolev Bluvstein, Simon J Evered, Alexandra A Geim, Sophie H Li, Hengyun Zhou, Tom Manovitz, + +# Sepehr Ebadi, Madelyn Cain, Marcin Kalinowski, Dominik Hangleiter, et al. "Logical quantum processor + +# based on reconfigurable atom arrays". In: Nature 626.7997 (2024), pp. 58-65 (pages 1, 2). + +# + +# [8] Dominik Hangleiter, Marcin Kalinowski, Dolev Bluvstein, Madelyn Cain, Nishad Maskara, Xun Gao, + +# Aleksander Kubica, Mikhail D Lukin, and Michael J Gullans. "Fault-tolerant compiling of classically + +# hard IQP circuits on hypercubes". In: arXiv preprint arXiv:2404.19005 (2024) (page 1). + +# + +# [9] Arthur Gretton, Karsten M. Borgwardt, Malte J. Rasch, Bernhard Schölkopf, and Alexander Smola. + +# "A Kernel Two-Sample Test". In: Journal of Machine Learning Research 13.25 (2012), pp. 723-773. url: + +# http://jmlr.org/papers/v13/gretton12a.html (page 14). + +# + +# [10] Suman Ravuri, Mélanie Rey, Shakir Mohamed, and Marc Deisenroth. Understanding Deep Generative + +# Models with Generalized Empirical Likelihoods. 2023. arXiv: 2306.09780 [cs.LG]. url: + +# https://arxiv.org/abs/2306.09780 (pages 3, 16). + +# + +###################################################################### +# About the author +# ---------------- +# # .. include:: ../_static/authors/erik_recio.txt From 88b7164611a860ded6e53a1b1cbb26470341313e Mon Sep 17 00:00:00 2001 From: Erik Recio Date: Thu, 12 Dec 2024 16:51:42 +0100 Subject: [PATCH 2/7] 2nd version of the iqpopt tutorial --- .../thumbnail_iqpopt.png | Bin 0 -> 253431 bytes .../iqpopt/sample_time.png | Bin 0 -> 82518 bytes demonstrations/tutorial_iqpopt.metadata.json | 32 ++ demonstrations/tutorial_iqpopt.py | 489 ++++++++++++------ 4 files changed, 363 insertions(+), 158 deletions(-) create mode 100644 _static/demo_thumbnails/regular_demo_thumbnails/thumbnail_iqpopt.png create mode 100644 _static/demonstration_assets/iqpopt/sample_time.png create mode 100644 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"erik.recio" + } + ], + "dateOfPublication": "2024-12-12T00:00:00+00:00", + "dateOfLastModification": "2024-12-12T00:00:00+00:00", + "categories": [ + "Optimization" + ], + "tags": [], + "previewImages": [ + { + "type": "thumbnail", + "uri": "/_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_iqpopt.png" + } + ], + "seoDescription": "Showing how to use the package IQPopt and optimize IQP circuits really fast.", + "doi": "", + "references": [], + "basedOnPapers": [], + "referencedByPapers": [], + "relatedContent": [ + { + "type": "tutorial", + "id": "tutorial_iqpopt_package", + "weight": 1.0 + } + ] +} diff --git a/demonstrations/tutorial_iqpopt.py b/demonstrations/tutorial_iqpopt.py index 4d7db84349..3d8a9ca532 100644 --- a/demonstrations/tutorial_iqpopt.py +++ b/demonstrations/tutorial_iqpopt.py @@ -53,9 +53,9 @@ # ###################################################################### -# Creating a circuit +# Creating an IQP circuit with pennylane -# ------------------ +# -------------------------------------- # @@ -70,22 +70,9 @@ # -# To define such a circuit (with input state :math:`\vert 0 \rangle`) we need to specify the number of +# To define such a circuit, we need a way to represent the parameterized gates. A set of gates will be -# qubits and the parameterized gates - -# - -import iqpopt as iqp -from iqpopt.utils import local_gates - -n_qubits = 3 -gates = local_gates(n_qubits, 2) # all gates with max weight 2 - -circuit = iqp.IqpSimulator(n_qubits, gates) - -###################################################################### -# A set of gates is specified by a list +# specified by a list # @@ -109,9 +96,11 @@ # -# that contains the generators of the gates. Since all gates have Pauli X type generators, a generator +# that contains the generators of the gates (each generator with an independent trainable parameter). -# is represented by a list that specifies which qubits are acted on by an X operator. For example, +# Since all gates have Pauli X type generators, a generator will be represented by a list that + +# specifies which qubits are acted on by an X operator. For example, # @@ -143,9 +132,7 @@ # -# The previous example prepares a three qubit circuit with all single and two qubit gates via the - -# function local_gates, i.e. +# Here is how to prepare a three qubit circuit with all the possible single and two qubit gates # @@ -169,113 +156,343 @@ # -###################################################################### -# Estimating expectation values +# In order to be more general and have more than one generator with the same trainable parameter, we -# ----------------------------- +# are going to group the generators with a tuple. Therefore, if the first and second generators share + +# the same parameter, then # -###################################################################### -# The class method ``IqpSimulator.op_expval()`` is used to provide estimates of expectation values. +# .. raw:: html -# The function requires a parameter array ``params``, a PauliZ operator specified by its binary +# -# representation ``op``, the number of samples ``n_samples`` that controls the precision of the +#
    -# approximation (the more the better), and a JAX pseudo random number generator key to seed the +# -# randomness of the sampling. +# gates = [([0], [1]), ([2],), ([0,1],), ([0,2],), ([1,2],)] # -# Below is a simple example implementing this for the three qubit circuit defined above, which returns +# .. raw:: html -# the expectation value estimate as well as its standard error. +# + +#
    # -import jax -import jax.numpy as jnp +# As we can see, the rest of the generators are alone in their own tuple. This would be the final form + +# of our gates parameter, which will define our circuit from now on. + +# + +# Let's now put it in practice and build a pennylane circuit out of this ``gates`` parameter. The + +# circuit will be build thanks to the following identity: + +# + +# .. math:: \text{exp}(i\theta_j X_j) = H^{\otimes n} \text{exp}(i\theta_j Z_j) H^{\otimes n} + +# + +# where :math:`H` is the Hadamard matrix and :math:`Z_j` is the same tensor product of Paulis, but now + +# instead of Pauli X there are Pauli Z with the corresponding identities. + +# + +# Our pennyalne circuit (with input state :math:`\vert 0 \rangle`) would therefore be the following. + +# + +import pennylane as qml import numpy as np -op = jnp.array([0,1,0]) # Pauli Z on the second qubit -params = jnp.array(np.random.rand(len(circuit.gates))) -n_samples = 1000 -key = jax.random.PRNGKey(42) +def penn_iqp_gates(params: np.ndarray, gates: list, n_qubits: int): + """IQP circuit in pennylane form. -expval, std = circuit.op_expval(params, op, n_samples, key) + Args: + params (np.ndarray): The parameters of the IQP gates. + gates (list): The gates representation of the IQP circuit. + n_qubits (int): The total number of qubits in the circuit. + """ -print(expval, std) + for i in range(n_qubits): + qml.Hadamard(i) + + for par, gate in zip(params, gates): + for gen in gate: + qml.MultiRZ(2*par, wires=gen) + + for i in range(n_qubits): + qml.Hadamard(i) ###################################################################### -# The function also allows for fast batch evaluation of expectation values. If we specify a batch of +# Now, we want to measure our circuit. As we said, all measurements are diagonal in the computational -# operators ``ops`` by an array we can batch evaluate the expectation values and errors in parallel +# (Z) basis, and we will therefore represent them with a bitstring. 1s, wherever there are Pauli Z, -# with the same syntax. +# and 0s, wherever there are identities. In this way, an operator such as: # -ops = jnp.array([[1,0,0],[0,1,0],[0,0,1]]) # batch of single qubit Pauli Zs +# .. math:: O = Z \otimes I \otimes Z -expvals, stds = circuit.op_expval(params, ops, n_samples, key) +# -print(expvals, stds) +# will be converted to + +# + +# .. raw:: html + +# + +#
    + +# + +# op = [1,0,1] + +# + +# .. raw:: html + +# + +#
    + +# + +# The final circuit that allow us to measure these type of expectation values is the following. + +# + +def penn_obs(op: np.ndarray) -> qml.operation.Observable: + """Returns a pennylane observable from a bitstring representation. + + Args: + op (np.ndarray): Bitstring representation of the Z operator. + + Returns: + qml.Observable: Pennylane observable. + """ + for i, z in enumerate(op): + if i==0: + if z: + obs = qml.Z(i) + else: + obs = qml.I(i) + else: + if z: + obs @= qml.Z(i) + return obs + +def penn_iqp_circuit(params: np.ndarray, gates: list, op: np.ndarray, n_qubits: int) -> qml.measurements.ExpectationMP: + """Defines the circuit that calculates the expectation value of the operator with the IQP circuit with pennylane tools. + + Args: + params (np.ndarray): The parameters of the IQP gates. + gates (list): The gates representation of the IQP circuit. + op (np.ndarray): Bitstring representation of the Z operator. + n_qubits (int): The total number of qubits in the circuit. + + Returns: + qml.measurements.ExpectationMP: Pennylane circuit with an expectation value. + """ + penn_iqp_gates(params, gates, n_qubits) + obs = penn_obs(op) + return qml.expval(obs) + +def penn_iqp_op_expval(params: np.ndarray, gates: list, op: np.ndarray, n_qubits: int) -> float: + """Calculates the expectation value of the operator with the IQP circuit with pennylane tools. + + Args: + params (np.ndarray): The parameters of the IQP gates. + gates (list): The gates representation of the IQP circuit. + op (np.ndarray): Bitstring representation of the Z operator. + n_qubits (int): The total number of qubits in the circuit. + + Returns: + float: Expectation value. + """ + dev = qml.device("lightning.qubit", wires=n_qubits) + penn_iqp_circuit_exe = qml.QNode(penn_iqp_circuit, dev) + return penn_iqp_circuit_exe(params, gates, op, n_qubits) ###################################################################### -# Testing the expectation values with Pennylane +# With this, we can now calculate all expectation of Pauli Z tensors of any IQP circuit we can think -# --------------------------------------------- +# of. Let's see an example. # +n_qubits = 3 +gates = [([0], [1]), ([2],), ([0,1],), ([0,2],), ([1,2],)] # same gates as in the example above +op = np.array([1,0,1]) # operator ZIZ +params = np.random.rand(len(gates)) # random parameters for all the gates (remember we have 5 gates with 6 generators in total) + +penn_op_expval = penn_iqp_op_expval(params, gates, op, n_qubits) +print(penn_op_expval) + ###################################################################### -# When using new software, it is always good practice to check your results if possible. In our case +# Estimating expectation values with IQPopt + +# ----------------------------------------- + +# + +# With this new package we can perform the same calculations our pennylane circuit above is able to + +# do. Although, this time, only with approximations instead of exact values. The gain is that we can -# we have Pennylane, a well tested library that does this calculations for a small number of qubits. +# work with a very large amount of qubits. -# On the `package's github `__ you can find already made functions +# -# that calculate the same expectation values as ``.op_expval()``, taking the necessary info from an +# After some calculations, one can arrive at the following expression when calculating the expectation -# ``IqpSimulator`` object. +# value of any tensor product of Pauli Z operators # -# Let's first download these functions so we don't have to build them again: +# .. math:: \langle Z_{\boldsymbol{a}} \rangle = \mathbb{E}_{\boldsymbol{z}\sim U}\Big[ \cos\Big(\sum_j \theta_{j}(-1)^{\boldsymbol{g}_{j}\cdot \boldsymbol{z}}(1-(-1)^{\boldsymbol{g}_j\cdot \boldsymbol{a}}\Big) \Big], # -import requests +# where :math:`\boldsymbol{a}` is the bitstring form of the operator we want to calculate the + +# expectation value of, :math:`\boldsymbol{z}` are bitstring samples from the multi-qubit system taken + +# from the uniform distribution, :math:`\theta_{j}` are the trainable parameters and + +# :math:`\boldsymbol{g}_{j}` are the different generators also in bitstring form. The dots are dot + +# products between the bitstrings taken as vectors of 1s and 0s. + +# -req = requests.get("https://raw.githubusercontent.com/XanaduAI/iqpopt/refs/heads/main/tests/pennylane_functions.py?token=GHSAT0AAAAAACYVZBT5V4YNS5U2YD6COCQYZ2YM32Q") +# We could classically calculate this expression exactly, but it would require an exponentially large + +# amount of :math:`\boldsymbol{z}` samples with the increasing number of qubits. To aliviate this, we + +# can replace the expectation with an empirical mean and compute an unbiased estimate of + +# :math:`\langle Z_{\boldsymbol{a}} \rangle` efficiently. That is, if we sample a batch of :math:`s` + +# bit strings :math:`\{\boldsymbol{z}_i\}` from the uniform distribution and compute the sample mean + +# -with open("pennylane_functions.py", "wb") as f: - f.write(req.content) +# .. math:: \hat{\langle Z_{\boldsymbol{a}}\rangle} = \frac{1}{s}\sum_{i}\cos\Big(\sum_j \theta_j(-1)^{\boldsymbol{g}_j\cdot \boldsymbol{z}_i}(1-(-1)^{\boldsymbol{g}_j\cdot \boldsymbol{a}})\Big), + +# + +# we obtain an unbiased estimate :math:`\hat{\langle Z_{\boldsymbol{a}}\rangle}` of + +# :math:`\langle Z_{\boldsymbol{a}}\rangle`; i.e. we have that + +# :math:`\mathbb{E}[\hat{\langle Z_{\boldsymbol{a}}\rangle}] = \langle Z_{\boldsymbol{a}}\rangle`. The + +# error of this approximation is well known since, by the central limit theorem, the standard + +# deviation of the sample mean of a bounded random variable decreases as + +# :math:`\mathcal{O}(1/\sqrt{s})`. + +# + +# Let's see now how to use the IQPopt package to calculate expectation values, using the same + +# arguments we used in the previous example. First, we create the circuit object with ``IqpSimulator`` + +# only with the number of qubits of the circuit and the ``gates`` parameter already explained: + +# + +import iqpopt as iqp + +small_circuit = iqp.IqpSimulator(n_qubits, gates) ###################################################################### -# Using the same ``params`` and ``op`` as before (but using ``numpy`` arrays instead of +# Now, we use the class method ``IqpSimulator.op_expval()``, which is the one that provides these + +# estimates. This function requires a parameter array ``params``, a PauliZ operator specified by its + +# binary representation ``op``, a new parameter ``n_samples`` (the number of samples :math:`s`) that + +# controls the precision of the approximation (the more the better), and a JAX pseudo random number -# ``jax.numpy``), let's see the pennylane's result. +# generator key to seed the randomness of the sampling. It returns the expectation value estimate as + +# well as its standard error. # -from pennylane_functions import penn_op_expval +# Using the same ``params`` and ``op`` as before: + +# -penn_expval = penn_op_expval(circuit, np.array(params), np.array(op)) +import jax -print(penn_expval) +n_samples = 1000 +key = jax.random.PRNGKey(42) + +expval, std = small_circuit.op_expval(params, op, n_samples, key) + +print(expval, std) ###################################################################### -# Since the calculation on ``iqpopt``\ 's side is stochastic, the results are not exactly the same, +# Since the calculation on ``iqpopt``\ 's side is stochastic, the result is not exactly the same than -# but as we can see, they are within std error. You can try increasing ``n_samples`` in order to +# the one obtained with pennylane. But, as we can see, they are within std error. You can try -# obtain closer approximations. +# increasing ``n_samples`` in order to obtain closer approximations. # +# This function also allows for fast batch evaluation of expectation values. If we specify a batch of + +# operators ``ops`` by an array, we can batch evaluate the expectation values and errors in parallel + +# with the same syntax. + +# + +ops = np.array([[1,0,0],[0,1,0],[0,0,1]]) # batch of single qubit Pauli Zs + +expvals, stds = small_circuit.op_expval(params, ops, n_samples, key) + +print(expvals, stds) + +###################################################################### +# With pennylane it would be very time consuming to pass the 30 qubit mark, but with IQPopt, we can + +# easily go way further than that. + +# + +from iqpopt.utils import local_gates + +n_qubits = 1000 +gates = local_gates(n_qubits, 1) # 1000 single qubit generators with independent trainable parameters + +large_circuit = iqp.IqpSimulator(n_qubits, gates) + +params = np.random.rand(len(gates)) +op = np.random.randint(0, 2, n_qubits) +n_samples = 1000 +key = jax.random.PRNGKey(42) + +expval, std = large_circuit.op_expval(params, op, n_samples, key) + +print(expval, std) + ###################################################################### # Sampling and probabilities @@ -286,43 +503,66 @@ ###################################################################### # We can also view a parameterized IQP circuit as a generative model that generates samples of binary -# vectors according to the distribution :raw-latex:`\begin{align} +# vectors according to the distribution -# q_{\boldsymbol{\theta}}(\boldsymbol{x}) \equiv q(\boldsymbol{x}\vert\boldsymbol{\theta})=\vert\bra{\boldsymbol{x}}U(\boldsymbol{\theta})\ket{0}\vert^2. +# -# \end{align}` For low amounts of qubits, we can use Penylane's arsenal and know the output +# .. math:: q_{\boldsymbol{\theta}}(\boldsymbol{x}) \equiv q(\boldsymbol{x}\vert\boldsymbol{\theta})=\vert\bra{\boldsymbol{x}}U(\boldsymbol{\theta})\ket{0}\vert^2. -# probabilities of the circuit as well as sample from it. Note that there is not an efficient +# -# algorithm to do these, so, for large numbers of qubits, these methods will either take too long or +# For a low amount of qubits, we can use Pennylane's arsenal and know the output probabilities of the -# not work at all. +# circuit as well as sample from it. Note that there is not an efficient algorithm to do these, so, + +# for large numbers of qubits, these methods will either take too long or not work at all. # -# These functions are already implemented in the ``IqpSimulator`` object, so we don't have to download +# These functions are already implemented in the ``IqpSimulator`` object. The ``.probs()`` method -# them. +# works as it does in pennylane, the returned array of probabilities is in lexicographic order. # -sample = circuit.sample(params, shots=1) +sample = small_circuit.sample(params, shots=1) print(sample) -probabilities = circuit.probs(params) +probabilities = small_circuit.probs(params) print(probabilities) +sample = large_circuit.sample(params, shots=1) +print(sample) + +probabilities = large_circuit.probs(params) +print(probabilities) + +###################################################################### +# As we can see, we can't sample or know the probabilities of the circuit for the large one. The only + +# efficient approximation algorithm we have is for the calculation of expectation values. In the + +# following figure, you can see how the time scales with the different methods when we increase the + +# number of qubits. The calculation of expectation values is the only one that remains linear instead + +# of increasing exponentially. + +# + ###################################################################### -# The ``.probs()`` method works as it does in pennylane, the returned array of probabilities is in +# .. figure:: ../_static/demonstration_assets/iqpopt/sample_time.png -# lexicographic order. +# :alt: Time vs n_qubits # +# Time vs n_qubits + ###################################################################### -# Optimizing a circuit +# Optimizing an IQPopt circuit -# -------------------- +# ---------------------------- # @@ -331,12 +571,14 @@ # define a loss function (also called an objective function), an optimizer and an initial stepsize for -# the gradient descent. Continuing our example from before, below we define a simple loss function +# the gradient descent. Continuing our ``small_circuit`` example from before, below we define a simple -# that is a sum of expectation values returned by ``.op_expval()`` . +# loss function that is a sum of expectation values returned by ``.op_expval()`` . # +import jax.numpy as jnp + def loss_fn(params, circuit, ops, n_samples, key): expvals = circuit.op_expval(params, ops, n_samples, key)[0] return jnp.sum(expvals) @@ -362,12 +604,12 @@ def loss_fn(params, circuit, ops, n_samples, key): import matplotlib.pyplot as plt n_iters = 1000 -params_init = np.random.normal(0, 1, len(circuit.gates)) +params_init = np.random.normal(0, 1, len(small_circuit.gates)) n_samples = 100 loss_kwargs = { "params": params_init, - "circuit": circuit, + "circuit": small_circuit, "ops": ops, "n_samples": n_samples, "key": key, @@ -499,81 +741,20 @@ def loss_fn(params, circuit, ops, n_samples, key): plt.plot(trainer.losses) plt.show() -###################################################################### -# Kernel Generalized Empirical Likelihood - -# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -# - -# In order to probe how well a generative model captures a distribution of high-dimensional data, we - -# can use the Kernel Generalized Empirical Likelihood [10]. This metric is a measure of the closeness - -# of two distributions while also serving as a tool to evaluate mode dropping. - -# - -# As before, if samples are available from the generative model, we have a straightforward - -# implementation of the KGEL test: - -# - -n_qubits = 10 - -# toy dataset of low weight bitstrings -X = np.random.binomial(1, 0.2, size=(1000, n_qubits)) - -n_witness = 10 -T = X[-n_witness:] # witness points (sampled from the ground truth distribution) -X = X[:-n_witness] # ground truth for the kgel test - -# model samples -Y = np.random.binomial(1, 0.3, size=(1000, n_qubits)) - -sigma = median_heuristic(X) #bandwidth for KGEL -kgel, p_kgel = gen.kgel_opt_samples(T, X, Y, sigma) -print(kgel) - -###################################################################### -# To evaluate the KGEL for a model given by a ``IqpSimulator`` object, we need to use another - -# implementation based on the calculation of expectation values of PauliZ operators. Let's see how - -# well the last trained circuit matches the ground truth with the KGEL test: - -# - -n_witness = 10 -T = X_train[-n_witness:] # witness points (sampled from the ground truth distribution) -X = X_train[:-n_witness] # ground truth for the kgel test - -sigma = median_heuristic(X) -kgel, p_kgel = gen.kgel_opt_iqp(circuit, - trained_params, - T, - X, - sigma, - n_ops=1000, - n_samples=1000, - key=jax.random.PRNGKey(42)) -print(kgel) - ###################################################################### # References: # -# [1] Michael J Bremner, Richard Jozsa, and Dan J Shepherd. "Classical simulation of commuting +# [1] Michael J Bremner, Richard Jozsa, and Dan J Shepherd. "Classical simulation of com- muting -# quantum computations implies collapse of the polynomial hierarchy". In: Proceedings of the Royal +# quantum computations implies collapse of the polynomial hierarchy". In: Proceed- ings of the Royal # Society A: Mathematical, Physical and Engineering Sciences 467.2126 (2011), pp. 459-472 (page 1). # -# [2] Michael J Bremner, Ashley Montanaro, and Dan J Shepherd. "Average-case complexity versus +# [2] Michael J Bremner, Ashley Montanaro, and Dan J Shepherd. "Average-case complex- ity versus # approximate simulation of commuting quantum computations". In: Physical review letters 117.8 (2016), @@ -623,7 +804,7 @@ def loss_fn(params, circuit, ops, n_samples, key): # -# [9] Arthur Gretton, Karsten M. Borgwardt, Malte J. Rasch, Bernhard Schölkopf, and Alexander Smola. +# [9] Arthur Gretton, Karsten M. Borgwardt, Malte J. Rasch, Bernhard Schölkopf, and Alexan- der Smola. # "A Kernel Two-Sample Test". In: Journal of Machine Learning Research 13.25 (2012), pp. 723-773. url: @@ -631,14 +812,6 @@ def loss_fn(params, circuit, ops, n_samples, key): # -# [10] Suman Ravuri, Mélanie Rey, Shakir Mohamed, and Marc Deisenroth. Understanding Deep Generative - -# Models with Generalized Empirical Likelihoods. 2023. arXiv: 2306.09780 [cs.LG]. url: - -# https://arxiv.org/abs/2306.09780 (pages 3, 16). - -# - ###################################################################### # About the author # ---------------- From eb603ee8bd25abe9803e7b084e536c36fc117741 Mon Sep 17 00:00:00 2001 From: Erik Recio Date: Thu, 12 Dec 2024 17:03:19 +0100 Subject: [PATCH 3/7] metadata update --- demonstrations/tutorial_iqpopt.metadata.json | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_iqpopt.metadata.json b/demonstrations/tutorial_iqpopt.metadata.json index 2afab7a270..9787db2439 100644 --- a/demonstrations/tutorial_iqpopt.metadata.json +++ b/demonstrations/tutorial_iqpopt.metadata.json @@ -24,7 +24,7 @@ "referencedByPapers": [], "relatedContent": [ { - "type": "tutorial", + "type": "demonstration", "id": "tutorial_iqpopt_package", "weight": 1.0 } From 249c6a6b4de1844d7d8494532e76f8b9bc3c2f8f Mon Sep 17 00:00:00 2001 From: Erik Date: Thu, 16 Jan 2025 19:06:14 +0100 Subject: [PATCH 4/7] update --- Makefile | 1 + _static/authors/erik_recio.jpg | Bin 417476 -> 1024748 bytes demonstrations/tutorial_iqpopt.metadata.json | 2 +- demonstrations/tutorial_iqpopt.py | 451 +++++++++++-------- pyproject.toml | 1 + 5 files changed, 264 insertions(+), 191 deletions(-) diff --git a/Makefile b/Makefile index ff5c334968..5ae4651475 100644 --- a/Makefile +++ b/Makefile @@ -79,5 +79,6 @@ environment: $$PYTHON_VENV_PATH/bin/python -m pip install --extra-index-url https://test.pypi.org/simple/ PennyLane-Catalyst --pre --upgrade;\ $$PYTHON_VENV_PATH/bin/python -m pip install --extra-index-url https://test.pypi.org/simple/ PennyLane-Lightning --pre --upgrade;\ $$PYTHON_VENV_PATH/bin/python -m pip install --upgrade git+https://github.com/PennyLaneAI/pennylane.git#egg=pennylane;\ + $$PYTHON_VENV_PATH/bin/python -m pip install 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