Quantum Circulant Preconditioner for Linear System of Equations

TL;DR

This paper introduces a quantum linear solver utilizing a circulant preconditioner, improving efficiency for dense non-Hermitian systems by adapting singular value estimation techniques.

Contribution

It develops a quantum algorithm with circulant preconditioning for general dense non-Hermitian systems, enhancing applicability and efficiency over previous methods.

Findings

Efficient quantum solver for dense non-Hermitian systems.

Time complexity depends on condition numbers and Frobenius norm.

Circulant preconditioner is easy to construct and apply.

Abstract

We consider the quantum linear solver for $Ax=b$ with the circulant preconditioner $C$. The main technique is the singular value estimation (SVE) introduced in [I. Kerenidis and A. Prakash, Quantum recommendation system, in ITCS 2017]. However, some modifications of SVE should be made to solve the preconditioned linear system $C^{-1} Ax = C^{-1} b$. Moreover, different from the preconditioned linear system considered in [B. D. Clader, B. C. Jacobs, C. R. Sprouse, Preconditioned quantum linear system algorithm, Phys. Rev. Lett., 2013], the circulant preconditioner is easy to construct and can be directly applied to general dense non-Hermitian cases. The time complexity depends on the condition numbers of $C$ and $C^{-1} A$, as well as the Frobenius norm $\|A\|_F$.