Quantum block Krylov subspace projector algorithm for computing low-lying eigenenergies

Abstract

Computing eigenvalues is a computationally intensive task central to numerous applications in the natural sciences. Toward this end, we investigate the quantum block Krylov subspace projector (QBKSP) algorithm - a multireference quantum Lanczos method designed to accurately compute low-lying eigenenergies, including degenerate ones, of quantum systems. We present three compact quantum circuits tailored to different problem settings for evaluating the required expectation values. To assess the impact of the number and fidelity of initial reference states, as well as time evolution duration, we perform error-free and limited-precision numerical simulations and quantum circuit simulations. Our results show that using multiple reference states significantly enhances convergence, particularly in precision-limited scenarios and in cases where a single reference state fails to capture all target eigenvalues. Additionally, the QBKSP algorithm allows for the determination of degenerate eigenstates and their multiplicities through appropriate convergence conditions.