Quantum Krylov-Subspace Method Based Linear Solver

TL;DR

This paper introduces a hybrid quantum-classical Krylov-subspace method that reduces redundancies and improves efficiency and accuracy in solving large linear systems, demonstrated through extensive numerical experiments.

Contribution

The paper proposes the quantum Krylov-subspace method (QKSM), a novel hybrid algorithm that refines existing quantum linear solvers by reducing redundancies and enhancing performance.

Findings

Significant reduction in computational resources needed.

Improved accuracy in linear system solutions.

Effective for systems up to 2^10 by 2^10 dimensions.

Abstract

Despite the successful enhancement to the Harrow-Hassidim-Lloyd algorithm by Childs et al., who introduced the Fourier approach leveraging linear combinations of unitary operators, our research has identified non-trivial redundancies within this method. This finding points to a considerable potential for refinement. In this paper, we propose the quantum Krylov-subspace method (QKSM), which is a hybrid classical-quantum algorithm, to mitigate such redundancies. By integrating QKSM as a subroutine, we introduce the quantum Krylov-subspace method based linear solver that not only reduces computational redundancy but also enhances efficiency and accuracy. Extensive numerical experiments, conducted on systems with dimensions up to $2^{10} \times 2^{10}$, have demonstrated a significant reduction in computational resources and have led to more precise approximations.