Variational Quantum Generative Modeling by Sampling Expectation Values of Tunable Observables

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.

💡 Research Summary

The paper addresses the limitations of Expectation‑Value Samplers (EVS), a class of quantum generative models that produce continuous‑valued data by measuring expectation values of fixed observables on states prepared by a parameterized quantum circuit (PQC). While EVS can, in principle, approximate high‑dimensional distributions with few qubits, practical implementations suffer from large measurement (shot) requirements and deep circuit depths, making them unsuitable for near‑term quantum hardware.

To overcome these issues, the authors introduce the Observable‑Tunable Expectation‑Value Sampler (OT‑EVS). In OT‑EVS the set of observables ({O_\ell}{\ell=1}^L) is combined linearly with a tunable weight matrix (\alpha\in\mathbb{R}^{M\times L}) to form the effective output observables (A_m=\sum{\ell}\alpha_{m\ell}O_\ell). This decouples the quantum part (the PQC) from the classical part (the linear combination), allowing the model to enlarge its expressive power without increasing quantum resources. The authors formalize a notion of “relative expressivity” and prove two propositions: (1) any fixed‑observable EVS (OF‑EVS) is never more expressive than an OT‑EVS built from the same circuit and a superset of observables, and (2) when the OF‑EVS is already universal, adding tunable observables does not increase expressivity. Two concrete examples (a two‑qubit rotation circuit and a Haar‑random circuit) demonstrate that OT‑EVS can be strictly more expressive than its fixed‑observable counterpart.

A key technical contribution is the “shadow‑frugal” parameterization of observables. By restricting each observable to a (k)-local Pauli string with (k\in O(\mathrm{polylog}(n))), the authors can employ the classical‑shadows measurement protocol. This protocol enables simultaneous estimation of many observables with a number of shots that scales only as (\mathcal{O}(\mathrm{poly}(n)/\epsilon^2)), a super‑polynomial improvement over conventional direct measurements. Theorem 1 quantifies the required shot count for both the shadow‑based and conventional schemes, showing that the shadow approach reduces the prefactor by a factor proportional to the total number of observables (L).

Training is performed within the Wasserstein‑1 GAN (WGAN) framework. The generator (G_{\theta,\alpha}) maps latent variables to data, while a classical critic (D_w) approximates the Kantorovich–Rubinstein dual. The standard WGAN alternates updates of the generator and critic; however, updating the quantum parameters (\theta) is costly because each gradient component requires the parameter‑shift rule, i.e., (2N_d N_s) circuit executions per update (with (N_d) the number of parameters and (N_s) the number of shots per sample). To reduce quantum overhead, the authors propose three training schedules:

1. Joint – update (\theta) and (\alpha) simultaneously after each critic update.
2. Asynchronous – perform multiple ((N_\alpha>1)) updates of (\alpha) (and the critic) before a single update of (\theta).
3. Decoupled – interleave blocks of (\alpha) updates with critic updates, and update (\theta) only after a prescribed number of blocks.

Because (\alpha) updates involve only classical linear algebra and measurement data, they are far cheaper than quantum updates, allowing the model to spend most of the training budget on refining the observable weights.

The empirical evaluation focuses on an 8‑qubit, 2‑local observable setting that generates 8‑dimensional data. The authors systematically vary (i) the three training schedules, (ii) the measurement method (classical shadows vs conventional direct measurement), and (iii) nine different shot budgets, repeating each configuration 20 times. Performance is measured by the Kullback‑Leibler (KL) divergence after 50 000 training iterations. Results show:

• The shadow‑based measurement achieves comparable KL values with at least a four‑fold reduction in required shots relative to conventional measurements.
• Both Asynchronous and Decoupled schedules consistently outperform the Joint schedule, confirming that reducing the frequency of quantum parameter updates improves sample efficiency.
• An intermediate shot budget yields the best performance; too few shots hinder convergence, while excessive shots remove a beneficial regularization effect of measurement noise. The authors hypothesize that moderate shot noise acts similarly to noise injection in classical GANs, stabilizing training (Hypothesis 3).

Finally, the paper compares OT‑EVS with OF‑EVS across multiple circuit depths and observable configurations, confirming that tunable observables broaden the reachable distribution family without additional quantum depth.

In summary, the work makes three intertwined contributions: (1) a theoretical proof that observable tunability never harms and often enhances expressivity, (2) a practical measurement scheme based on classical shadows that dramatically lowers shot complexity, and (3) a resource‑aware adversarial training protocol that prioritizes cheap classical updates over expensive quantum ones. Together, these advances bring continuous‑variable quantum generative modeling closer to feasibility on near‑term quantum devices, opening avenues for further research on hardware‑aware observable design, noise‑robust training, and scaling to higher‑dimensional real‑world datasets.