Exploring the Role of Hamiltonian Expressibility in Ansatz Selection for Variational Quantum Algorithms

A key challenge in Variational Quantum Algorithms (VQAs) consists in designing effective parametric quantum circuits, or ansatzes, that account for both the specific problem and the limitations of quantum hardware. While Machine Learning-based methods for circuit construction have been developed, selecting the appropriate metrics to include in the cost function to ensure the circuit has the desired properties remains a non-trivial task. Hamiltonian expressibility has been proposed as a metric to quantify a circuit ability to explore the problem energy landscape. However, its potential to enhance the likelihood of obtaining high-quality solutions remains underexplored. In this work, we investigate the correlation between an ansatz Hamiltonian expressibility and the quality of solutions found using the Variational Quantum Eigensolver (VQE), aiming to determine whether such metric could enhance Machine Learning methods for ansatz design. Specifically, we propose a protocol to estimate the Hamiltonian expressibility of a given ansatz and Hamiltonian, and employ VQE to solve the associated optimization problem. By conducting a correlation analysis, we assess the relationship between Hamiltonian expressibility and solution quality. Our results suggest that Hamiltonian expressibility could serve as a valuable metric to enhance Machine Learning techniques used for ansatz design.