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arXiv · 1807.04563

Quantum Circulant Preconditioner for Linear System of Equations

Changpeng Shao·Hua Xiang
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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 $\|A\|_F$.

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Changpeng Shao, Hua Xiang. 2018-07-12. Quantum Circulant Preconditioner for Linear System of Equations. https://doi.org/10.1103/physreva.98.062321

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