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Variational Quantum Generative Modeling by Sampling Expectation Values of Tunable Observables

Kevin Shen, Andrii Kurkin, Adrián Pérez-Salinas, Elvira Shishenina, Vedran Dunjko, Hao Wang

TL;DR

This work advances quantum generative modeling by introducing Observable-Tunable Expectation Value Samplers (OT-EVS), which expand expressivity beyond fixed observables through a tunable linear combination of observables $A_m = \sum_l \alpha_{m,l} O_l$. By employing a shadow-frugal parameterization and classical shadows, OT-EVS achieves favorable sample complexity, enabling efficient estimation of expectation values for $k$-local Pauli observables. An adapted adversarial training scheme prioritizes updates to the observables, reducing quantum-resource usage while maintaining performance, and three training regimes (Joint, Asynchronous, Decoupled) are analyzed. Theoretical expressivity results, along with extensive numerical experiments on synthetic data and MNIST/Fashion-MNIST, demonstrate that tunable observables improve performance over fixed-observable EVS and that classical-shadow-based sampling can substantially reduce the measurement burden. These findings support the practical potential of continuous quantum generative models that operate with limited quantum hardware resources.

Abstract

Expectation Value Samplers (EVSs) are quantum generative models that can learn high-dimensional continuous distributions by measuring the expectation values of parameterized quantum circuits. However, these models can demand impractical quantum resources for good performance. We investigate how observable choices affect EVS performance and propose an Observable-Tunable Expectation Value Sampler (OT-EVS), which achieves greater expressivity than standard EVS. By restricting the selectable observables, it is possible to use the classical shadows measurement scheme to reduce the sample complexity of our algorithm. In addition, we propose an adversarial training method adapted to the needs of OT-EVS. This training prioritizes classical updates of observables, minimizing the more costly updates of quantum circuit parameters. Numerical experiments, using an original simulation technique for correlated shot noise, confirm our model's expressivity and sample efficiency advantages compared to previous designs. We envision our proposal to encourage the exploration of continuous generative models running with few quantum resources.

Variational Quantum Generative Modeling by Sampling Expectation Values of Tunable Observables

TL;DR

This work advances quantum generative modeling by introducing Observable-Tunable Expectation Value Samplers (OT-EVS), which expand expressivity beyond fixed observables through a tunable linear combination of observables . By employing a shadow-frugal parameterization and classical shadows, OT-EVS achieves favorable sample complexity, enabling efficient estimation of expectation values for -local Pauli observables. An adapted adversarial training scheme prioritizes updates to the observables, reducing quantum-resource usage while maintaining performance, and three training regimes (Joint, Asynchronous, Decoupled) are analyzed. Theoretical expressivity results, along with extensive numerical experiments on synthetic data and MNIST/Fashion-MNIST, demonstrate that tunable observables improve performance over fixed-observable EVS and that classical-shadow-based sampling can substantially reduce the measurement burden. These findings support the practical potential of continuous quantum generative models that operate with limited quantum hardware resources.

Abstract

Expectation Value Samplers (EVSs) are quantum generative models that can learn high-dimensional continuous distributions by measuring the expectation values of parameterized quantum circuits. However, these models can demand impractical quantum resources for good performance. We investigate how observable choices affect EVS performance and propose an Observable-Tunable Expectation Value Sampler (OT-EVS), which achieves greater expressivity than standard EVS. By restricting the selectable observables, it is possible to use the classical shadows measurement scheme to reduce the sample complexity of our algorithm. In addition, we propose an adversarial training method adapted to the needs of OT-EVS. This training prioritizes classical updates of observables, minimizing the more costly updates of quantum circuit parameters. Numerical experiments, using an original simulation technique for correlated shot noise, confirm our model's expressivity and sample efficiency advantages compared to previous designs. We envision our proposal to encourage the exploration of continuous generative models running with few quantum resources.

Paper Structure

This paper contains 11 sections, 6 theorems, 34 equations, 8 figures, 1 table, 3 algorithms.

Table of Contents

  1. abstract
  2. Introduction
  3. Result
  4. Discussion
  5. Methods
  6. Declarations
  7. Data Availability
  8. Code Availability
  9. Competing interests
  10. Author Contributions
  11. Acknowledgements

Key Result

Proposition 1

For any OF-EVS $H_{\bm{\theta}} \coloneq (\mathbb{P}_z, U_{\bm{\bm{\theta}}}, (Q_m)^M_{m=1})$ and any OT-EVS $G_{\bm{\theta}, \bm{\alpha}} \coloneq (\mathbb{P}_z, U_{\bm{\bm{\theta}}}, (O_l)^L_{l=1}, \bm{\alpha})$,

Figures (8)

  • Figure 1: Training performance of OT-EVS with a two-layer $8$-qubit sequential circuit using the (a) Joint (b) Asynchronous (c) Decoupled method on the synthetic dataset. The interquartile mean and bootstraped $95\%$ confidence intervals over $20$ trials for the estimated KL divergence after $50k$ training iterations are shown.
  • Figure 2: Training performance of OT-EVS and OF-EVS with varying numbers of circuit ansatz layers on synthetic datasets. (a) Sequential ansatz. (b) Brickwork ansatz. The interquartile mean and bootstraped $95\%$ confidence intervals over $20$ trials for the estimated KL divergence after $50k$ training iterations are shown.
  • Figure 3: (a) Training performance of OT-EVS with $1$-local general observables (using the Decoupled method) and OF-EVS with $1$-local Pauli-Z observables with different numbers of measurements on the MNIST dataset. The mean and standard deviation over $5$ trials are shown. Samples generated by an $1$-local general observable OT-EVS trained with $2^2L$ (b) and $2^{10}L$ (c) conventional measurements. Samples generated by an $1$-local general observable OT-EVS trained with $2^2L$ (d) and $2^{10}L$ (e) classical shadows.
  • Figure 4: Training performance of OT-EVS with $1$-local Pauli-Z observables, $1$-local general observables or $2$-local general observables (using the Decoupled method) and a three-layer $512$-neuron MLP on the MNIST dataset (a) and Fashion-MNIST dataset (b). Samples generated by a $12$-qubit $1$-local Pauli-Z observable OT-EVS (c), a $12$-input MLP (d), a $12$-qubit $1$-local general observable OT-EVS (e), and a $12$-qubit $2$-local general observable OT-EVS (f) for the MNIST dataset. Samples generated by a $12$-qubit $1$-local Pauli-Z observable OT-EVS (g), a $12$-input MLP (h), a $12$-qubit $1$-local general observable OT-EVS (i), and a $12$-qubit $2$-local general observable OT-EVS (j) for the Fashion-MNIST dataset.
  • Figure 5: (a) Noise-perturbed and ideal empirical densities of an example $2D$ synthetic dataset with estimated KL divergence (and measurement budget) labelled. (b) Generated distributions at selected training steps (ignoring shot noise), with estimated KL divergence labelled. (c) Example training curves (generator loss, critic loss, and estimated KL divergence) for a training using the Asynchronous method.
  • ...and 3 more figures

Theorems & Definitions (17)

  • Definition 1: Expectation Value Sampler, adapted from barthe_expressivity_2024
  • Definition 2: Observable-Tunable Expectation Value Sampler (OT-EVS)
  • Definition 3: Relative Expressivity
  • Proposition 1: Expressivity never decreases using tunable observables
  • Proposition 2: Expressivity of a universal generative model cannot increase further
  • Example 1
  • Example 2
  • Definition 4: Kantorovich-Rubenstein duality kantorovich_mathematical_1960
  • Definition 5: Shadow-Frugal Parameterization
  • Theorem 1: Sample Complexity
  • ...and 7 more