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Entangling Quantum Generative Adversarial Networks

Murphy Yuezhen Niu, Alexander Zlokapa, Michael Broughton, Sergio Boixo, Masoud Mohseni, Vadim Smelyanskyi, Hartmut Neven

arXiv:2105.00080v2quant-ph

TL;DR

Earlier quantum GANs can fail to converge, motivating EQ-GAN, which entangles true and generated data in adversarial training. The paper reports convergence, robustness to discriminator gate errors, and an approximate-QRAM application to quantum-neural-network training.

  • Problem

    Earlier QuGANs can oscillate through mode collapse and may have a non-unique Nash equilibrium.

    Source: INTRODUCTION · p. 1

  • Method

    EQ-GAN uses a discriminator that entangles true quantum data with the generator output during adversarial training.

    Source: INTRODUCTION · p. 1

  • Results

    EQ-GAN converges to the global optimal Nash equilibrium, is more robust to discriminator gate errors than supervised learning, and its QRAM superposition achieves better final QNN accuracy at equal gradient-evaluation budgets.

    Source: INTRODUCTION · p. 1, APPLICATION TO QRAM · p. 5

  • Takeaways & Limitations

    An adversarially learned approximate QRAM can represent classical data as a superposition for quantum-neural-network applications.

    Source: INTRODUCTION · p. 1

    Mixed-state learning requires a factor-of-two overhead in qubit count under Choi’s theorem.

    Source: Hybrid EQ-GAN for Mixed-State · p. 10

Abstract

from arXiv · show

Generative adversarial networks (GANs) are one of the most widely adopted semisupervised and unsupervised machine learning methods for high-definition image, video, and audio generation. In this work, we propose a new type of architecture for quantum generative adversarial networks (entangling quantum GAN, EQ-GAN) that overcomes some limitations of previously proposed quantum GANs. Leveraging the entangling power of quantum circuits, EQ-GAN guarantees the convergence to a Nash equilibrium under minimax optimization of the discriminator and generator circuits by performing entangling operations between both the generator output and true quantum data. We show that EQ-GAN has additional robustness against coherent errors and demonstrate the effectiveness of EQ-GAN experimentally in a Google Sycamore superconducting quantum processor. By adversarially learning efficient representations of quantum states, we prepare an approximate quantum random access memory (QRAM) and demonstrate its use in applications including the training of quantum neural networks.

INTRODUCTION

The section motivates fully quantum GANs for learning quantum states and introduces EQ-GAN, which entangles true and generated data to address convergence problems in earlier QuGANs. It also previews robustness to gate errors and an approximate-QRAM application for quantum neural networks.

INTRODUCTION

  • Quantum-computer-generated distributions are thought to be classically hard to sample under plausible conjectures, motivating quantum circuits that learn to reproduce quantum states.The section frames the task as learning a generator circuit from quantum data.Source: INTRODUCTION · p. 1

PRIOR ART

  • Earlier QuGANs can oscillate between states through mode collapse, and may have a non-unique Nash equilibrium.Near-matching states can make the optimal discriminator nearly orthogonal to the true state, prompting generator updates that overshoot and sustain oscillation.Source: INTRODUCTION · p. 1, PRIOR ART · p. 2

INTRODUCTION

  • EQ-GAN converges to the global optimal Nash equilibrium, including on problem instances where QuGAN failed.The discriminator entangles true and generated data.Source: INTRODUCTION · p. 1
  • EQ-GAN is more robust to discriminator gate errors than supervised learning with an exact swap test.Gate errors in the supervised discriminator can cause convergence to the incorrect optimum.Source: INTRODUCTION · p. 1
  • An EQ-GAN can create an approximate QRAM from a superposition of classical data, with an application that improves quantum-neural-network classification performance.The passage presents this as an application of the architecture in quantum machine learning for classical data.Source: INTRODUCTION · p. 1

CONVERGENCE OF EQ-GAN

EQ-GAN trains a discriminator on the joint true and generated states so its measurement can act as a fidelity test. In the swap-test ansatz, discriminator optimization selects the perfect-swap angle, and the generator’s optimum is reached when its state equals the true state.

CONVERGENCE OF EQ-GAN

  • The discriminator receives both the true state σ and generated state ρ(θg), and can measure their fidelity on the joint system.The EQ-GAN objective adversarially optimizes both state generation and the fidelity measurement.Source: CONVERGENCE OF EQ-GAN · p. 2
  • The implementation uses a destructive ancilla-free swap test because a traditional swap test spans 2n qubits and incurs prohibitive depth overhead on locally connected devices.The ancilla shown in Fig. 1 is included for clarity.Source: CONVERGENCE OF EQ-GAN · p. 2, Figure 1 · p. 3
  • For the swap-test ansatz, the distinguishing maximum is uniquely achieved at θ = π/2, the perfect-swap angle.The discriminator is optimized toward this angle by decreasing the probability of measuring |0⟩.Source: CONVERGENCE OF EQ-GAN · p. 3
  • At ρ(θg) = σ, the generator cannot improve, and this is the global optimum of EQ-GAN.The figure caption identifies this as the global optimum.Source: CONVERGENCE OF EQ-GAN · p. 3, Figure 1 · p. 3
  • EQ-GAN converges on problem instances unreachable by a fully trained and properly parameterized QuGAN.The text also notes that poor circuit parameterization can make gradient optimization difficult through a non-convex loss landscape, an issue shared with QuGANs and classical GANs.Source: CONVERGENCE OF EQ-GAN · p. 3

LEARNING TO SUPPRESS ERRORS

EQ-GAN uses an adversarially adjustable swap-test discriminator to mitigate gate errors while learning a quantum state. On a Sycamore device, it achieved higher state overlap and significantly lower state-fidelity error than a perfect swap test.

LEARNING TO SUPPRESS ERRORS

  • Gate errors can shift the unique optimum of a frozen perfect swap test away from the true data state, whereas EQ-GAN can learn an error-correcting discriminator parameterization.The approach is agnostic to the precise parameterization of a perfect swap test.Source: LEARNING TO SUPPRESS ERRORS · pp. 3–4
  • Free single-qubit Z rotations after the CZ let the discriminator learn to compensate for gate errors while the generator varies X and Z rotations toward the target state.The device example learns the single-qubit state 1/√2(|0⟩+|1⟩).Source: LEARNING TO SUPPRESS ERRORS · p. 4, Figure 2 · p. 4
  • Noisy gradient measurements and minimax optimization can perturb training away from the optimum even after convergence to the Nash equilibrium.Both methods may fail to remain at the optimal point.Source: LEARNING TO SUPPRESS ERRORS · p. 4
  • On a Sycamore device, EQ-GAN achieved significantly lower state-fidelity error than a perfect swap test and converged to higher state overlap.The comparison reports average error over multiple runs; the figure marks convergence at the iteration where discriminator loss is minimized.Source: Table I · p. 4, Figure 3 · p. 4

APPLICATION TO QRAM

The paper trains a shallow EQ-GAN circuit to approximate a superposition representing classical data, forming a variational QRAM. In a quantum neural network test, QRAM-based training achieved better accuracy than individual-sample training at the same circuit-evaluation budget.

APPLICATION TO QRAM

  • Preparing an arbitrary superposition of n classical data states takes O(n) operations at best, potentially eliminating quantum speedup, so the paper uses EQ-GAN to learn an approximate superposition.The state amplitudes are proportional to the square roots of the empirical data probabilities.Source: APPLICATION TO QRAM · p. 4
  • A shallow peak ansatz trained by EQ-GAN approximates a two-peak empirical distribution because exact encoding requires a very deep circuit and multiple-control rotations.The illustrated dataset has 120 samples, with balanced training and test sets of 60 each.Source: APPLICATION TO QRAM · p. 5, Figure 5 · p. 5
  • At equal circuit-evaluation counts, the QNN trained on variational QRAM achieved better test accuracy than training on individual samples, despite using an approximate distribution.Accuracy was evaluated on the raw dataset, although QRAM-based training did not directly access it.Source: APPLICATION TO QRAM · p. 5, Table II · p. 5

CONCLUSION

The conclusion presents EQ-GAN as a way to address non-convexity and mode collapse while suppressing device errors in laboratory quantum computing. It highlights variational QRAM as an experimental application for improving classical-data classification with a quantum neural network.

CONCLUSION

  • The authors propose EQ-GAN to address non-convexity and mode collapse, using a Hilbert-Schmidt-norm cost function rather than trace distance.They note that alternatives such as Wasserstein metrics may also improve convergence.Source: CONCLUSION · p. 5
  • A shallow discriminator is reported to suppress device errors and ensure robust convergence in laboratory quantum-computer runs.The conclusion frames this as relevant to near-term quantum-computing applications.Source: CONCLUSION · p. 5
  • The paper reports an experimental EQ-GAN application in which QRAM improves quantum-neural-network classification accuracy on classical data.The authors identify efficient, high-fidelity QRAM preparation as a direction for quantum generative models.Source: CONCLUSION · p. 5

Mode Collapse Example of QuGAN

The QuGAN example shows how minimax updates can cycle instead of reaching equilibrium. The paper generalizes this failure to cycles across multiple states.

Mode Collapse Example of QuGAN

  • Under full minimax optimization, QuGAN alternates between states ρ and ρ′ rather than reaching the Nash equilibrium.The discriminator and generator updates return the system to the two states in succession.Source: Mode Collapse Example of QuGAN · p. 7
  • A k-fold composition Tσ^(k)(ρ)=ρ for k>1 is sufficient for oscillation among k states.Source: Mode Collapse Example of QuGAN · p. 7
  • For an n-qubit system, a π/3 separation between target and initial states on the generalized Bloch sphere can produce this cycle condition.Source: Mode Collapse Example of QuGAN · p. 7

Training EQ-GAN

EQ-GAN converges on instances where QuGAN fails, and is steadier under the compared training configurations. A two-phase procedure pretrains the discriminator as a swap test before adversarial training.

Training EQ-GAN

  • EQ-GAN converges on problem instances where QuGAN fails, while QuGAN remains more unstable under both full and partial training.The compared full-training setup uses 50 generator epochs followed by 50 discriminator epochs per iteration; partial training uses one epoch per iteration.Source: Training EQ-GAN · p. 7, Figure 7 · p. 7
  • For the state σ example, full optimization makes QuGAN oscillate between states with constant overlap 3/4.Source: Training EQ-GAN · p. 7
  • The proposed two-phase procedure first trains the generator against a frozen swap-test discriminator, then allows the discriminator to vary adversarially.In the presence of gate errors, the second phase may yield a unitary closer to a true swap test.Source: Training EQ-GAN · p. 8
  • With biased rotation noise, the perfect swap test forces convergence to an incorrect state, unlike the EQ-GAN comparison.Source: Figure 8 · p. 8

EQ-GAN Discriminator Architecture

The discriminator uses an ancilla-free swap test for state discrimination, adapted to the device’s native gates and connectivity constraints.

EQ-GAN Discriminator Architecture

  • For Sycamore, each CNOT is decomposed into H-CZ-H operations, and trainable RZ rotations after CZ gates mitigate modeled phase errors.The free angles are optimized by gradient descent during adversarial training.Source: EQ-GAN Discriminator Architecture · p. 8
  • The discriminator provably converges to the best state discriminator, motivating early stopping when its loss indicates that this discriminator has been achieved.Source: EQ-GAN Discriminator Architecture · p. 8
  • The ancilla-free swap test rewrites controlled swaps as CNOT and Toffoli gates and uses classical post-processing with an ancillary classical bit.It is used to perform state discrimination.Source: Figure 9 · p. 8

QNN Architecture

The QNN uses a four-qubit data state on Sycamore and compares training by sampling individual examples with training on class superpositions prepared by a pretrained EQ-GAN.

QNN Architecture

  • The QNN uses a rank-4 entangling gate decomposed for the native CZ gate, and its four-qubit data-state layout reflects Sycamore’s planar connectivity.The figure shows the QNN architecture and its device layout, including a readout qubit interacting with the data qubits.Source: QNN Architecture · p. 8, Figure 10 · p. 9, Figure 11 · p. 9
  • Training compares one sampled example per iteration with a superposition of an entire class per iteration, using a shallow EQ-GAN approximation rather than an exact dataset superposition.The paper contrasts this approximation with Gaussian and exact-superposition preparations that require controlled rotations and, for planar layouts, additional swaps.Source: QNN Architecture · p. 9
  • The comparison equalizes device access at 60 iterations for 60 examples, with sampling trained for one epoch and superposition evaluated 30 times per class.Source: QNN Architecture · p. 9
  • Learning rates are tuned by Bayesian optimization, and final QNN performance is evaluated over 50 trials with computed standard deviations.Each parameter query averages 10 trials; the final rates are 10^-3.93 for sampling and 10^-1.83 for superposition.Source: QNN Architecture · p. 9

Adversarial Learning Without Errors

In a noiseless example, a perfect swap test cannot train from a zero-angle generator initialization, but EQ-GAN learns the data state by allowing the discriminator to change.

Adversarial Learning Without Errors

Hybrid EQ-GAN for Mixed-State

For mixed-state learning, the paper describes a hybrid classical-quantum representation that samples state components, while noting a qubit overhead for the Choi-based approach.

Hybrid EQ-GAN for Mixed-State

  • Mixed-state learning with the described Choi-based approach requires a factor-of-two overhead in qubit count.The quantum GAN does not require a random latent input, unlike classical GANs.Source: Hybrid EQ-GAN for Mixed-State · p. 10
  • A hybrid network represents a mixed state using a classical distribution over component states and a quantum circuit that prepares each associated state.Sampling from the classical distribution and preparing the associated state outputs a probabilistic mixture, avoiding the possible double-exponential overhead of learning the full quantum channel.Source: Hybrid EQ-GAN for Mixed-State · p. 10
  • The mixed-state EQ-GAN jointly processes true data σ, generated data ρ, and an ancilla, using ancilla measurement for its cost function.The generated input ρ is prepared by a unitary circuit, and the discriminator applies a joint unitary transformation.Source: Figure 15 · p. 10

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