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 families 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 practical guidelines for designing resource aware, efficient and trainable quantum circuits, moving beyond heuristic methods for near term applications.
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 families 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 practical guidelines for designing resource aware, efficient and trainable quantum circuits, moving beyond heuristic methods for near term applications.
Ansatz selection is a key factor in the performance of variational quantum algorithms (VQAs). While much of the state-of-the-art still relies on heuristic choices, an inadequate circuit structure can compromise both the expressive power and the trainability of the resulting model. Recent results have also established theoretical connections between expressibility and the onset of barren plateaus, highlighting the need for systematic criteria for ansatz selection. In this work, the ansatz is treated as a design feature to be optimized rather than a fixed block, and a design space exploration (DSE) is performed over a diverse set of parametrized quantum circuits (PQCs). Three complementary metrics -- expressibility, trainability, and resource cost -- are evaluated and used to analyze the trade-offs that emerge across different PQCs. Beyond identifying Pareto-optimal candidates, this multi-objective perspective helps clarify the interplay between these metrics and contributes quantitative evidence toward understanding the expressibility--trainability tension in variational circuits.
Variational quantum circuits (VQCs) -- parameterised quantum circuits trained by classical optimisation of a quantum cost function -- underpin most near-term quantum machine learning, yet systematic guidance on design choices (ansatz structure, encoding, optimiser, gradient estimation) and their interaction with specific ML task requirements remains fragmented. This paper proposes the Variational Quantum Circuit Design Space (VQCDS) framework, characterising 48 VQC configurations (3 ansatz families x 3 encoding strategies x 4 optimisers) across 12 ML benchmark tasks -- regression, classification, generative modelling, and reinforcement learning -- on IBM Quantum Eagle and IonQ Forte. VQCDS finds: LSAD-structured ansatz (Silva et al., 2024) outperforms hardware-efficient ansatz on 10 of 12 tasks (+8.0pp mean AUC, 2.1x lower variance); amplitude encoding outperforms angle encoding for p > 16 features (+12.4% R2); quantum natural gradient descent (QNG) reduces regression convergence by 48.4% vs. Adam; and task-ansatz alignment via mutual information-guided entanglement provides +8.4pp AUC and +14.4% R2 across task types. VQCDS provides an evidence-based VQC design guide covering 48 configuration evaluations across 12 tasks.
Variational quantum algorithms (VQAs) rely on parameterized quantum circuits (PQCs), whose performance is governed by expressibility and trainability. Existing studies typically evaluate these properties at the logical circuit level, implicitly assuming that designed PQCs remain unchanged during hardware execution. In practice, however, hardware-aware transpilation modifies circuit structure through qubit mapping, routing, and basis decomposition, potentially altering PQC behavior. In this paper, we perform a systematic hardware-aware analysis of expressibility and trainability by comparing logical and transpiled PQCs across multiple ansatz families, qubit counts, and circuit depths. Expressibility is measured using fidelity-based KL divergence, while trainability is quantified through gradient variance. Our results show that transpilation acts as an implicit architectural perturbation, producing strongly ansatz-dependent effects. Expressibility deviations exceed upto 125% in some cases, while trainability variations reach up to 25%. Structured ansatzes are generally more robust, whereas highly entangled architectures are more sensitive to transpilation-induced transformations. We further show that transpilation can alter the commonly assumed expressibility-trainability trade-off, demonstrating that logical-level analyses may not reliably predict hardware-level behavior. These findings highlight the importance of hardware-aware evaluation for accurate characterization of VQAs.
Variational Quantum Algorithms have emerged as promising tools for solving optimization problems on quantum computers. These algorithms leverage a parametric quantum circuit called ansatz, where its parameters are adjusted by a classical optimizer with the goal of optimizing a certain cost function. However, a significant challenge lies in designing effective circuits for addressing specific problems. In this study, we leverage the powerful and flexible Reinforcement Learning paradigm to train an agent capable of autonomously generating quantum circuits that can be used as ansatzes in variational algorithms to solve optimization problems. The agent is trained on diverse problem instances, including Maximum Cut, Maximum Clique and Minimum Vertex Cover, built from different graph topologies and sizes. Our analysis of the circuits generated by the agent and the corresponding solutions shows that the proposed method is able to generate effective ansatzes. While our goal is not to propose any new specific ansatz, we observe how the agent has discovered a novel family of ansatzes effective for Maximum Cut problems, which we call Ryz-connected. We study the characteristics of one of these ansatzes by comparing it against state-of-the-art quantum algorithms across instances of varying graph topologies, sizes, and problem types. Our results indicate that the Ryz-connected circuit achieves high approximation ratios for Maximum Cut problems, further validating our proposed agent. In conclusion, our study highlights the potential of Reinforcement Learning techniques in assisting researchers to design effective quantum circuits which could have applications in a wide number of tasks.
Variational Quantum Circuits (VQCs) have emerged as a cornerstone of hybrid quantum–classical algorithms designed to harness the computational potential of near-term quantum devices. By combining parameterized quantum gates with classical optimization, VQCs provide a flexible framework for tackling machine learning, chemistry, and optimization problems intractable for classical methods. This review comprehensively overviews VQC design principles, ansatz structures, optimization strategies, and real-world applications. Furthermore, we discuss fundamental challenges such as barren plateaus, the expressibility–trainability trade-off, and current noisy intermediate-scale quantum (NISQ) hardware limitations. Finally, we highlight emerging directions that could enable scalable, noise-resilient, and physically interpretable variational quantum models for future quantum computing applications
Variational quantum algorithms are a leading approach to near-term quantum computing, but their scalability can be limited by barren plateaus and the sampling cost of resolving small changes in the loss landscape. Here, we study the trainability of passive linear-optical quantum circuits and introduce a framework based on the ratio of sample variance to circuit variance. This ratio determines the number of circuit samples required to resolve local loss differences and gradients to proportional accuracy. We apply this framework to photon-number observables and identify both trainable and non-trainable regimes. Supported by analytic results and a numerically observed polynomial decay of the circuit variance, we find that fixed-order photon-number polynomials require only polynomially many samples as the system size grows, whereas high-order polynomials and observables based on output probabilities generally require exponentially many samples. Within the trainable regime, we further identify classes of observables in which quantum estimation achieves a polynomial speed-up over multiple classical methods. Within this family, neural network observables provide one practical construction that allow measurement outcomes to be efficiently processed into the desired polynomial. These results establish photonic variational quantum computing as a promising platform for near-term applications.