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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 [20, 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 [13, 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 .

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