The superiority of variational quantum algorithms (VQAs) such as quantum neural networks (QNNs) and variational quantum eigensolvers (VQEs) heavily depends on the expressivity of the employed Ansätze. Namely, a simple Ansatz is insufficient to capture the optimal solution, while an intricate Ansatz leads to the hardness of trainability. Despite its fundamental importance, an effective strategy of measuring the expressivity of VQAs remains largely unknown. Here, we exploit an advanced tool in statistical learning theory, i.e., covering number, to study the expressivity of VQAs. Particularly, we first exhibit how the expressivity of VQAs with an arbitrary Ansätze is upper bounded by the number of quantum gates and the measurement observable. We next explore the expressivity of VQAs on near-term quantum chips, where the system noise is considered. We observe an exponential decay of the expressivity with increasing circuit depth. We also utilize the achieved expressivity to analyze the generalization of QNNs and the accuracy of VQE. We numerically verify our theory employing VQAs with different levels of expressivity. Our Letter opens the avenue for quantitative understanding of the expressivity of VQAs.
Abstract In the context of Variational Quantum Algorithms (VQAs), selecting an appropriate ansatz is crucial for efficient problem-solving. Hamiltonian expressibility has been introduced as a metric to quantify a circuit’s ability to uniformly explore the energy landscape associated with a Hamiltonian ground state search problem. However, its influence on solution quality remains largely unexplored. In this work, we estimate the Hamiltonian expressibility of a well-defined set of circuits applied to various Hamiltonians using a Monte Carlo-based approach. We analyse how ansatz depth influences expressibility and identify the most and least expressive circuits across different problem types. We then train each ansatz using the Variational Quantum Eigensolver (VQE) and analyse the correlation between solution quality and expressibility. Our results indicate that, under ideal or low-noise conditions and particularly for small-scale problems, ansätze with high Hamiltonian expressibility yield better solution quality for problems with non-diagonal Hamiltonians and superposition state solutions. Conversely, circuits with low expressibility are more effective for problems whose solutions are basis states, including those defined by diagonal Hamiltonians. Under noisy conditions, low-expressibility circuits remain preferable for problems with solutions in a computational basis state, while intermediate expressibility yields better results for some problems involving superposition state solutions.
The variational quantum eigensolver (VQE) is a promising algorithm for near-term quantum chemistry applications, but selecting optimal ansatz circuits remains challenging. Expressibility, a metric quantifying a circuit's ability to explore the Hilbert space, has been proposed as a guide for ansatz selection, but recent work showed it inconsistently predicts VQE performance under realistic noise for
H2. We extend this investigation to cover both
H2 and
H3+ under four execution scenarios: ideal, noisy, and noisy with zero-noise extrapolation (ZNE) or probabilistic error cancellation (PEC). We find that error mitigation does not reliably restore expressibility's predictive power. ZNE reduces error for only 4 of 12
H2 circuits and 4 of 6
H3+ circuits, while PEC actually increases error in 11 of 12
H2 circuits and all 6
H3+ circuits. We reproduce and extend Saib et al.'s key finding that circuit rankings scramble under noise (Spearman
ρ≈−0.1 between ideal and noisy rankings), and identify a new result: ZNE largely preserves noisy rankings (
ρ=+0.80 for
H2) while PEC actively reorders them (
ρ=−0.22). Noisy expressibility, computed from density matrix simulations, strongly predicts unmitigated performance for
H3+ (Pearson
r=+0.91,
p=0.01), but this metric is computationally intractable at scale. We demonstrate that zero-cost circuit topology metrics such as two-qubit gate count provide comparable or superior predictive power for PEC degradation (
r=+0.96 for
H3+), while standard expressibility best predicts noisy and ZNE performance for
H2 (
r=+0.74 and
r=+0.77).
This study analyzes the expressibility characteristics and gradient distribution in the variational quantum eigensolver (VQE) algorithm using nine variations of hardware-efficient ansatz (HEA) in the hydrogen molecular system. The evaluation was conducted by measuring Kullback–Leibler divergence ( D KL ) value as an indicator of expressibility ansatz and the gradient variance as an indicator of the barren plateau phenomenon. The system was optimized using the L-BFGS algorithm, and each ansatz was tested to identify the extent to its parametric structure affected optimization performance and the potential for barren plateaus. The analysis results show differences among the nine ansatzes. Variations such as SR y, TR y, R y R x, and R y R z tend to produce high expressibility but are followed by a drastic decrease in gradient variance, thus showing symptoms of a barren plateau. Conversely, more complex ansatzes involving the mixing of rotation bases, such as HR y, R y, and XR y provide a more stable gradient distribution and more consistent optimization performance. These findings are consistent with the proposed linear mixing expressibility–trainability model, states that increased expressibility is not always directly proportional to the ease of the optimization process; homogenization of the state space actually reduces the directional information needed by the optimizer. These results provide an important basis for selecting more effective ansatzes in future implementations of variational quantum algorithms.
Whether parameterized quantum circuits (PQCs) can be systematically constructed to be both trainable and expressive remains an open question. Highly expressive PQCs often exhibit barren plateaus, while several trainable alternatives admit efficient classical simulation. We address this question by deriving a finite-sample, dimension-independent concentration bound for estimating the variance of a PQC cost function, yielding explicit trainability guarantees. Across commonly used ansätze, we observe an anticorrelation between trainability and expressibility, consistent with theoretical insights. Building on this observation, we propose a property-based ansatz-search framework for identifying circuits that combine trainability and expressibility. We demonstrate its practical viability on a real quantum computer and apply it to variational quantum algorithms. We identify quantum neural network ansätze with improved effective dimension using over
6× fewer parameters, and for VQE on
H2 we achieve UCCSD-like accuracy at substantially reduced circuit complexity.
Abstract Quantum computing is an exciting area, which has grown at an astonishing rate in the last decade. It is specially promising for the computational and theoretical chemistry area. One algorithm has received a lot of attention lately, the variational quantum eigensolver (VQE). Is is used to solve electronic structure problems and it is suitable to the noisy intermediate-scale quantum (NISQ) hardware. VQE calculations require ansatze and one of the most known is the unitary coupled-cluster (UCC). It uses the chosen basis set to generate a circuit which will be iteratively minimized. The present work investigates the circuit depth as a function of basis sets and molecular size. It has been shown that for the current quantum devices, only the smallest molecules and basis sets are tractable. The H2 molecule with the cc-pVTZ and aug-cc-pVTZ basis sets have circuit depths in the order of 106 to 107 gates and the C2H6 molecule with 3-21G basis set has a circuit depth of 2.2 ×108 gates. At the same time the analysis demonstrate that the H2 molecule with STO-3G basis set, requires at least 500 shots to reduce the error and that, although error mitigation schemes can diminish the error, they were not able to completely negate it.