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

Computing eigenvalues is a computationally intensive task central to many applications in the natural sciences, particularly in quantum physics and chemistry, where determining low-lying energy spectra is key to characterizing physical and chemical properties. Toward this end, we present and investigate the quantum block Krylov subspace projector (QBKSP) algorithm---a multireference quantum Lanczos method---to accurately compute low-lying eigenenergies, including degenerate ones. We propose three compact quantum circuits, each suited to different problem settings, for evaluating the necessary expectation values. In contrast to previous multireference quantum Krylov methods, QBKSP achieves linear scaling with respect to the number of Krylov iterations instead of a quadratic one, and furthermore reduces the scaling with respect to the number of references by half for real Hamiltonians and reference states. To assess the impact of the number and fidelity of the initial reference states, as well as the QBKSP performance in different parameter regimes, we perform both error-free and sampling noise-limited simulations. Our results demonstrate that employing multiple reference states significantly improves convergence, particularly in noisy scenarios or when a single reference fails to capture all target eigenstates. Moreover, the QBKSP algorithm allows for the determination of degenerate eigenstates and their multiplicities through appropriate convergence conditions.