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-1Ax=C-1b. Moreover, different from the preconditioned linear system considered in [Phys. Rev. Lett. 110, 250504 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.250504], 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-1A, as well as the Frobenius norm AF.