Generating probability distributions using variational quantum circuits

TL;DR

The paper addresses how to efficiently generate probability distributions with variational quantum circuits on near-term devices by dissecting the roles of expressibility and entangling capability in trainability. It benchmarks three ansatz families across multiple cost functions and optimizers, using a fidelity-based Jensen–Shannon divergence to quantify expressibility and the Meyer–Wallach measure to quantify entangling power, including noise models. The key finding is that high expressibility alone is insufficient; optimal performance arises from combining high expressibility with moderate entanglement and appropriate resource usage, leading to the development of expressibility-enhanced circuits that perform comparably to highly expressive predefined circuits but with far fewer resources. This work provides practical guidelines for resource-aware design of variational algorithms for sampling and quantum machine learning on noisy devices, with implications for near-term quantum advantage in probabilistic modeling tasks.

Abstract

Sampling from a probability distribution is a core task in many quantum and classical algorithms. Variational quantum circuits provide a natural approach to generating such distributions, as measurement outcomes directly define the probability values. However, designing circuits that train reliably while utilizing limited quantum resources remains largely a heuristic approach. In particular, the roles of expressibility, entanglement capability, and quantum resources in training performance and scalability are not well understood. In this work we present a systematic study of variational quantum circuits where we compare different ansatze family across multiple cost functions and classical optimization methods. We use expressibility and entanglement capability as circuit descriptors to explain convergence behaviors, optimizer sensitivity and robustness to noise. Our results provide a practical guidelines for designing resource aware, efficient and trainable quantum circuits, moving beyond heuristic methods for near term applications.

Figures (7)

• Figure 1: Circuit template used throughout this work: stacked parameterized single-qubit layers (random, expressible, or predefined) followed by a fixed entangling block $U_{\mathrm{ent}}$. This template is used to construct all three ansatz families.
• Figure 2: Left: Training convergence showing Jensen–Shannon divergence (JSD) loss versus optimization step for an 8-qubit high-expressibility predefined ansatz trained with the Adam optimizer on an asymmetric bimodal target distribution. Right: final variational quantum ansatz (VQA) output distribution (bars) compared to the target distribution (dashed).
• Figure 3: Relationship between expressibility, entanglement capability, and final training performance across all evaluated circuits. Each point corresponds to a circuit instance from the predefined, random, or expressibility-enhanced families. High accuracy is achieved primarily by circuits combining high expressibility with moderate entanglement.
• Figure 4: Convergence analysis using metropolis optimizer and JSD metric. The plots compare predefined, random, and expressibility enhanced 4-qubit architectures' performance over the normal distribution. The metropolis optimizer ran for 50000 iterations. Enhanced architectures achieve lower orders of magnitude outperforming random architectures.
• Figure 5: Optimizer set comparison for circuit 3 in the predefined architectures for the bimodal GMM distribution. Solid lines: 4-qubits, dashed: 8-qubits. The size of the cost function landscape affects the performance of the optimizers, as the order drops significantly when scaled from 4 to 8-qubits. The steep descent of the better performing optimizers is due to the convergence protocols once a certain threshold is achieved.