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 $C^{-1}Ax=C^{-1}b$. 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 $C^{-1}A$, as well as the Frobenius norm ${\parallel A\parallel }_F$.

• Received 13 July 2018
• Revised 18 November 2018

DOI:https://doi.org/10.1103/PhysRevA.98.062321