Quantum circulant preconditioner for a linear system of equations

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 [Kerenidis and Prakash, Quantum recommendation system, in ITCS (2017)]. However, the SVE should be modified to solve the preconditioned linear system ${C}^{\ensuremath{-}1}Ax={C}^{\ensuremath{-}1}b$. Moreover, different from the preconditioned linear system considered in [Phys. Rev. Lett. 110, 250504 (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 ${C}^{\ensuremath{-}1}A$, as well as the Frobenius norm ${\ensuremath{\parallel}A\ensuremath{\parallel}}_{F}$.