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arXiv:2507.22550 · Quantum Machine Intelligence

Hamiltonian expressibility for ansatz selection in variational quantum algorithms

Filippo Brozzi, Gloria Turati, Maurizio Ferrari Dacrema, Filippo Caruso, Paolo Cremonesi

Published Jul 3, 2026• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1007/s42484-026-00407-3

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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 expressibility of an ansatz used in a variational quantum algorithm is defined as the uniformity with which it can explore the space of unitary matrices, i.e., its covering number. The expressibility of a particular ansatz has a well-defined upper bound [1]. In this work, we show that the expressibility also has a well-defined lower bound in the hypothesis space. We provide an analytical expression for the lower bound of the covering number, which is directly related to expressibility. Further, we provide numerical analysis to support our claim. By calculating the bond length of hydrogen molecule (H2H_2H2​) using different ansatzes in a variational quantum eigensolver (VQE) setting, we study the variation of equilibrium energy error with circuit depths. We show that in each ansatz template, a plateau exists for a range of circuit depths, which we call the set of acceptable points, and the corresponding expressibility as the best expressive region. We report that the width of this best expressive region in the hypothesis space is inversely proportional to the average error. Our analysis reveals that alongside trainability, the lower bound of expressibility also plays a crucial role in selecting variational quantum ansatzes
Variational quantum algorithms are suitable for use on noisy quantum systems. One of the most important use-cases is the quantum simulation of materials, using the variational quantum eigensolver (VQE). To optimize VQE performance, a suitable parameterized quantum circuit (ansatz) must be selected. We investigate a class of ansatze that incorporates knowledge of the quantum hardware, namely the hardware efficient ansatze. The performance of hardware efficient ansatze is affected differently by noise, and our goal is to study the effect of noise on evaluating which ansatz gives more accurate results in practice. First, we study the effect of noise on the different hardware efficient ansatze by benchmarking and ranking the performance of each ansatz family (i) on a chemistry application using VQE and (ii) by the recently established metric of "expressibility". The results demonstrate the ranking of optimal circuits does not remain constant in the presence of noise. Second, we evaluate the suitability of the expressibility measure in this context by performing a correlation study between expressibility and the performance of the same circuits on a chemistry application using VQE. Our simulations reveal a weak correlation and therefore demonstrate that expressibility is not an adequate measure to quantify the effectiveness of parameterized quantum circuits for quantum chemistry. Third, we evaluate the effect of different quantum device noise models on the ordering of which ansatz family is best. Interestingly, we see that to decide which ansatz is optimal for use, one needs to consider the specific hardware used even within the same family of quantum hardware.
Quantum computers promise a great computational advantage over classical computers, which might help solve various computational challenges such as the simulation of complicated quantum systems, finding optimum in large optimization problems, and solving large-scale linear algebra problems. Current available quantum devices have only a limited amount of qubits and a high level of noise, limiting the size of problems that can be solved accurately with those devices. Variational quantum algorithms (VQAs) have emerged as a leading strategy to address these limitations by optimizing cost function based on measurement results of shallow depth circuits. Recently, various pulse engineering methods were suggested in order to improve VQA results, including optimizing pulse parameters instead of gate angles as part of the VQA optimization process. In this paper, we suggest a novel pulse-based ansatz, which is parameterized mainly by pulses’ duration of pre-defined pulse structures. This ansatz structure provides relatively low amounts of optimization parameters while maintaining high expressibility, allowing fast convergence. In addition, the ansatz has structured adaptivity to the entanglement level required by the problem, allowing low noise and accurate results. We tested this ansatz against quantum chemistry problems. Specifically, finding the ground-state energy associated with the electron configuration problem, using the variational quantum eigensolver (VQE) algorithm for several different molecules. We manage to achieve chemical accuracy both in simulation for several molecules and on one of IBM’s NISQ devices for the H2 molecule in the STO-3G basis, without the need for extensive error mitigation. Our results are compared to a common gate-based ansatz and show better accuracy and significant latency reduction—up to 7× shorter ansatz schedules.
Variational quantum algorithms have emerged as a leading paradigm that extracts practical computation from near-term intermediate-scale quantum devices, enabling advances in quantum chemistry simulations, combinatorial optimization, and quantum machine learning. However, the performance of variational quantum algorithms is highly sensitive to the design of the ansatze. To be effective, ansatze must be expressive enough to capture target states but shallow enough to be trainable. We propose a genetic algorithm-inspired framework for designing ansatze that achieve high expressibility while maintaining shallow depth and low parameter count. Our approach evolves ansatze through mutation and selection based on an expressibility metric. The circuit generated by our framework consistently demonstrates high expressibility at any target depth and performs comparably to traditional ansatz design approaches. This work presents a problem-agnostic, scalable solution for ansatz design, producing expressive, low-depth circuits that need to be designed only once and can serve a wide range of applications.
Abstract With the exponentially faster computation for certain problems, quantum computing has garnered significant attention in recent years. Variational quantum algorithms are crucial methods to implement quantum computing, and an appropriate task‐specific quantum circuit ansatz can effectively enhance the quantum advantage of VQAs. However, the vast search space makes it challenging to find the optimal task‐specific ansatz. Expressibility, quantifying the diversity of quantum circuit ansatz states to explore the Hilbert space effectively, can be used to evaluate whether one ansatz is superior to another. In this work, using a transformer model to predict the expressibility of quantum circuit ansatze is proposed. A dataset containing random PQCs generated by the gatewise pipeline, with varying numbers of qubits and gates is constructed. The expressibility of the circuits is calculated using three measures: KL divergence, relative KL divergence, and maximum mean discrepancy. A transformer model is trained on the dataset to capture the intricate relationships between circuit characteristics and expressibility. Four evaluation metrics are employed to assess the performance of the transformer. Numerical results demonstrate that the trained model achieves high performance and robustness across various expressibility measures. This research can enhance the understanding of the expressibility of quantum circuit ansatze and advance quantum architecture search algorithms.
The variational quantum eigensolver (VQE), which has attracted attention as a promising application of noisy intermediate-scale quantum devices, finds a ground state of a given Hamiltonian by variationally optimizing the parameters of quantum circuits called Ans\"atze. Since the difficulty of the optimization depends on the complexity of the problem Hamiltonian and the structure of the Ans\"atze, it has been difficult to systematically analyze the performance of optimizers for the VQE. To resolve this problem, we propose a technique to construct a benchmark problem whose ground state is guaranteed to be achievable with a given Ansatz by using the idea of a parent Hamiltonian of a low-depth parametrized quantum circuit. We compare the convergence of several optimizers by varying the distance of the initial parameters from the solution and find that the converged energies showed a thresholdlike behavior depending on the distance. This work provides a systematic way to analyze optimizers for the VQE and contribute to the design of the Ansatz and its initial parameters.