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Noise robustness and experimental demonstration of a quantum generative adversarial network for continuous distributions

Abhinav Anand, Jonathan Romero, Matthias Degroote, Alán Aspuru-Guzik

arXiv:2006.01976v2quant-phcs.LG

TL;DR

The paper examines whether hybrid quantum generative adversarial networks can learn continuous distributions despite noise in quantum devices. It simulates noisy HQGANs and trains one on Rigetti’s Aspen-4-2Q-A, reporting training under moderate noise, with purely dephasing noise an exception in simulations.

  • Problem

    Evidence is limited on how noise in imperfect quantum devices affects variational quantum circuits for generative modelling.

    Source: §I · p. 1

  • Method

    The authors simulate a hybrid quantum generative adversarial network with noise added to its quantum generator, then train the HQGAN on Rigetti’s Aspen-4-2Q-A.

    Source: §I · p. 1, B. Simulation setup · p. 3, §IV · p. 8

  • Results

    Noisy simulations approximately converged to the target distribution except under purely dephasing noise, while numerical and physical-device demonstrations showed training under moderate noise.

    Source: §2 · p. 5, §IV · p. 8

  • Takeaways & Limitations

    The authors report empirical evidence that HQGAN training can use fewer input samples per epoch and fewer circuit evaluations for expectation values.

    Source: §3 · p. 6, §IV · p. 8

Abstract

from arXiv · show

The potential advantage of machine learning in quantum computers is a topic of intense discussion in the literature. Theoretical, numerical and experimental explorations will most likely be required to understand its power. There has been different algorithms proposed to exploit the probabilistic nature of variational quantum circuits for generative modelling. In this paper, we employ a hybrid architecture for quantum generative adversarial networks (QGANs) and study their robustness in the presence of noise. We devise a simple way of adding different types of noise to the quantum generator circuit, and numerically simulate the noisy hybrid quantum generative adversarial networks (HQGANs) to learn continuous probability distributions, and show that the performance of HQGANs remain unaffected. We also investigate the effect of different parameters on the training time to reduce the computational scaling of the algorithm and simplify its deployment on a quantum computer. We then perform the training on Rigetti's Aspen-4-2Q-A quantum processing unit, and present the results from the training. Our results pave the way for experimental exploration of different quantum machine learning algorithms on noisy intermediate scale quantum devices.

I. INTRODUCTION

Quantum correlations motivate interest in quantum sampling and machine learning, but imperfect device control limits implementation to proof-of-principle experiments. This work investigates noise robustness in HQGANs for classical probability distributions before implementing the model on a quantum computer.

I. INTRODUCTION

  • Imperfect control in available quantum devices limits implementation of proposed algorithms to proof-of-principle experiments, motivating hybrid quantum-classical approaches.Hybrid approaches combine quantum resources with classical computers.Source: §I · p. 1
  • Parameterized quantum circuits optimized by cost functions are used in hybrid quantum-classical algorithms for noisy intermediate-scale devices, including QGANs.Source: §I · p. 1
  • Prior QGAN studies included proof-of-principle simulations and experiments, but noise robustness and potential advantages over classical counterparts still required investigation.The cited demonstrations focused on quantum state estimation or learning distributions loaded into quantum states.Source: §I · p. 1
  • The paper investigates noise effects on an HQGAN for classical probability distributions through simulations with realistic parameters, before implementing the model on a quantum computer.Source: §I · pp. 1–2

II. THEORY OF HYBRID QUANTUM GENERATIVE ADVERSARIAL NETWORKS

A GAN trains a generator and discriminator adversarially to model a target distribution. This work uses a hybrid variant with a quantum generator and classical discriminator to learn classical data distributions.

II. THEORY OF HYBRID QUANTUM GENERATIVE ADVERSARIAL NETWORKS

  • A prototypical GAN pairs a generator that maps prior samples toward the real distribution with a discriminator that distinguishes generated from real samples.Source: §II · p. 2
  • GAN training alternates optimization of the discriminator and generator loss functions, adjusting their parameters after each optimization round.Source: §II · p. 2
  • This work uses a hybrid quantum GAN in which a quantum circuit serves as generator and a neural network as discriminator to learn a classical target distribution.QGAN configurations can also use quantum circuits for both networks or place the quantum circuit in the discriminator.Source: §II · p. 2

A. HQGAN architecture

The HQGAN retains the generator–discriminator structure of a regular GAN while using a two-qubit quantum circuit as its generator. Its encoding and variational elements produce a measured output that serves as generated data for the classical discriminator.

A. HQGAN architecture

  • The generator is a two-qubit circuit with an encoding element and a variational element whose parameterized rotation angles are optimized during training.The encoding uses two layers of single-qubit rotations and tensorial mapping for nonlinearities; the variational element includes rotations and CNOT gates.Source: A. HQGAN architecture · p. 2
  • The generated data is the first qubit’s expectation value measured in the σz basis.Source: A. HQGAN architecture · p. 2
  • The discriminator is a classical four-layer feed-forward network with two hidden layers of 50 units each.The hidden layers use ELU activations and the output layer uses a sigmoid activation.Source: A. HQGAN architecture · p. 2

B. Simulation setup

The setup trains a hybrid quantum GAN with a variational quantum generator under simulated hardware noise. It varies noise models and computational parameters to assess robustness and training cost.

B. Simulation setup

  • Training used PyQuil circuits, PyTorch autograd gradients, Adam optimization, and one-sided label smoothing, with a uniform prior over [−1, 1].Expectation values used 1,000 circuit evaluations unless stated otherwise; training metrics used 100 prior samples unless stated otherwise.Source: B. Simulation setup · pp. 2–3
  • The target distribution was generated with θg = [0.35, 2.10, 5.06] from 1,000 prior samples, and simulations began from nearby generator parameters.The initial parameters were θg = [0.31, 1.89, 4.56].Source: B. Simulation setup · p. 3, Figure 2 · p. 3
  • Noisy identity gates were inserted after generator gates, with the number set by the gate-time ratio n = tgate/tI−gate.The noisy gates were implemented using Forest pragmas and Kraus maps.Source: B. Simulation setup · p. 3, Figure 3 · p. 3
  • The simulations modeled amplitude damping, dephasing, their combination, and combined damping, dephasing, and readout noise.The combined decoherence channel multiplies the damping and dephasing Kraus operators; readout noise uses conditional assignment probabilities.Source: B. Simulation setup · p. 3
  • Single-source simulations used p = 0.09, while combined-noise simulations used lower probabilities closer to experimental parameters.At p = 0.09, combining multiple noise sources produced very poor performance; this was the highest single-source level with reasonably functioning HQGANs.Source: B. Simulation setup · p. 4, Table I · p. 4
  • The study examined how training-sample count and circuit evaluations per expectation value affect optimization and total circuit workload.Reducing either parameter was considered as a way to lower computational overhead and quantum-device wall-time.Source: B. Simulation setup · p. 4

C. Simulation results

The simulation results are assessed over a 4,500-epoch training cutoff using distribution and optimization metrics, with violin plots used to compare generated distributions.

C. Simulation results

  • Training runs were capped at 4,500 epochs to enable fair comparison across HQGAN simulations.The cutoff exceeded the average number of iterations required in the simulations.Source: C. Simulation results · p. 4
  • Evaluation tracked KL divergence, generator and discriminator losses, gradient norms, and distribution means and standard deviations.Source: C. Simulation results · p. 4
  • Violin plots show the post-training distributions, with median, interquartile range, remaining non-outlier range, and kernel-density area encoded.The plotted samples use optimized generator parameters and 1,000 generated points.Source: C. Simulation results · p. 4

1. Noiseless simulations

In the noiseless simulation, training brings the generator distribution toward the target. The authors also report agreement with earlier simulations and an additional run from zero initial parameters.

1. Noiseless simulations

  • After 4,500 epochs, the noiseless generator approximated the target distribution, with similar median, interquartile range, and overall distribution.Figure 4 compares the target with the simulated distribution before and after training.Source: §1 · p. 4, Figure 4 · p. 5
  • The noiseless result agrees with the authors’ earlier numerical simulations [13].Source: §1 · p. 5
  • The authors also ran an ideal simulation initialized with all generator parameters set to zero; its results are reported in the appendix.Source: §1 · p. 5

2. Noisy simulations

Noisy HQGAN simulations generally approximated the target distribution, but strong pure dephasing was a notable exception. The tested realistic noise combinations showed little deviation, and moderate noise was associated with fewer epochs to convergence.

2. Noisy simulations

  • Across the noisy simulations, all cases except pure dephasing approximately converged to the target, though noisy distributions showed visible tails.Source: §2 · p. 5
  • At p = 0.09, amplitude-damping training recovered the target distribution’s shape and moments despite noise.Before training, the channel drove the population average toward expectation value zero; after training, a visible part of the distribution centered around 1.0.Source: §2 · pp. 5–6
  • At p = 0.09, pure dephasing left the final distribution distorted, with target-like modes near 0.5 and 0.0 but a spurious mode near -0.6.The distribution became symmetric and centered at zero; the authors suggest the channel’s lack of transitions and resulting incoherent superposition could explain its spread.Source: §2 · p. 6
  • With realistic combined-noise probabilities, generated distributions showed no significant deviation from the ideal simulation, and moderate noise was associated with fewer epochs to convergence.The combined-noise probabilities were significantly smaller than the p = 0.09 used in the earlier simulations.Source: §2 · p. 6

3. Input samples and Circuit evaluations

Reducing the number of training samples per epoch or circuit evaluations for expectation values had little effect on the generated distributions in the tested simulations. The authors identify both as tunable ways to reduce runtime with minimal effect on accuracy.

3. Input samples and Circuit evaluations

  • With 25, 50, 75, or 100 input samples per epoch, distributions converged to a good approximation of the target after 2000 epochs.Source: §3 · p. 6
  • The number of circuit evaluations used to estimate expectation values had no evident effect on training across the tested simulations.The simulations used 100, 250, 500, or 1000 circuit evaluations per expectation-value calculation, with 25 input samples per epoch.Source: §3 · p. 6
  • Smaller input-sample and circuit-evaluation counts were identified as ways to reduce experiment runtime with minimal effect on accuracy.The authors treat the number of input samples as a tunable algorithm hyperparameter.Source: §3 · p. 7

III. IMPLEMENTATION ON QUANTUM HARDWARE

The authors trained an HQGAN on Rigetti’s Aspen-4-2Q-A processor, where the generated distribution achieved good overlap with the target. Training metrics converged, and results from the experiment were similar to simulation results.

III. IMPLEMENTATION ON QUANTUM HARDWARE

  • The QPU experiment used 25 samples per epoch and 250 circuit runs per expectation-value evaluation.These hyperparameter choices drew on the noisy-simulation findings to optimize QPU runtime.Source: §III · p. 7
  • After 2993 QPU training epochs, the final generated distribution achieved good overlap with the target.The training ran in approximately 30 two-hour slots; Figure 6 shows distributions at different training epochs.Source: §III · p. 7, Figure 6 · p. 7
  • During QPU training, KL divergence decreased to nearly 0, the cost functions converged to approximately ln(0.5) ≈ 0.7, and distribution means, standard deviations, and gradient norms converged.Figure 7 tracks these metrics over training epochs.Source: §III · pp. 7–8, Figure 7 · p. 8
  • The experiment’s final distribution and training statistics were similar to those from simulation when computed using 1000 samples from the generator with QPU-optimized parameters.The authors relate the result to optimal parameter resilience (OPR), in which variational parameters remain unaffected by different types of incoherent noise.Source: §III · pp. 7–8
  • The authors conclude that the procedure succeeded under moderate noise, adding practical evidence for resilience of parameterized-circuit optimization on NISQ devices.Source: §III · p. 8

IV. CONCLUSION

The paper demonstrates HQGAN training on a quantum computer and reports that it can be carried out under moderate noise. The authors frame this as proof of principle, with practical applications such as image, speech, and text generation left as directions for further work.

IV. CONCLUSION

  • Simulations and physical-device demonstrations showed that HQGAN training can be carried out under moderate noise, with Aspen-4-2Q-A results similar to classical simulation.The authors also report empirical evidence that fewer input samples per epoch and fewer circuit evaluations can be used for training.Source: §IV · p. 8
  • The authors describe these demonstrations as a first step toward using quantum resources in classical machine-learning pipelines, with practical image, speech, and text generation identified as future directions.Source: §IV · p. 8

APPENDIX

With zero-initialized generator parameters, the ideal simulation converges to the target distribution, though it takes more training rounds. The appendix also plots training metrics across several noise-model simulations.

I. Ideal simulation

  • With θg initialized to [0.0, 0.0, 0.0], the ideal simulation converges to the target distribution after more training rounds, as shown in the violin plot and metrics.The plotted distribution is shown before and after 6500 epochs.Source: §I · p. 10, Figure 8 · p. 10, Figure 9 · p. 11

II. Other simulation Metrics

  • Metrics are plotted for ideal, amplitude-damping, phase-damping, combined amplitude-and-phase-damping, and combined damping with readout-noise simulations.The plots show KL divergence, discriminator and generator objectives, gradient norms, and distribution means and standard deviations during training.Source: Figure 10 · p. 11, Figure 11 · p. 11, Figure 12 · p. 11
  • For each noise-model metrics figure, statistics use 100 samples from the generator with optimized parameters after training.Source: Figure 10 · p. 11, Figure 11 · p. 11, Figure 12 · p. 11
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