Variational Quantum Linear Solver

C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subaşı, L. Cincio, P. J. Coles · 2019 · arXiv

| 编号 QC016 | 主题 量子算法工具箱 | 链接 arXiv:1909.05820 |

摘要

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Previously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the required circuit depth. Here, we propose a hybrid quantum-classical algorithm, called Variational Quantum Linear Solver (VQLS), for solving linear systems on near-term quantum computers. VQLS seeks to variationally prepare |x⟩ such that $A|x⟩\propto |b⟩$ . We derive an operationally meaningful termination condition for VQLS that allows one to guarantee that a desired solution precision ϵ is achieved. Specifically, we prove that $C⩾{ϵ}^2/{\kappa }^2,$ , where C is the VQLS cost function and κ is the condition number of A. We present efficient quantum circuits to estimate $C,$ while providing evidence for the classical hardness of its estimation. Using Rigetti’s quantum computer, we successfully implement VQLS up to a problem size of 1024 × 1024. Finally, we numerically solve nontrivial problems of size up to $2^{50}\times 2^{50}$ . For the specific examples that we consider, we heuristically find that the time complexity of VQLS scales efficiently in $ϵ,\kappa ,$ and the system size N.

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