Variational Quantum algorithm for Poisson equation

The Poisson equation has wide applications in many areas of science and engineering. Although there are some quantum algorithms that can efficiently solve the Poisson equation, they generally require a fault-tolerant quantum computer which is beyond the current technology. In this paper, we propose...

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Bibliographic Details
Published in:arXiv.org 2020-12
Main Authors: Liu, Hailing, Wu, Yusen, Wan, Linchun, Pan, Shijie, Qin, Sujuan, Gao, Fei, Wen, Qiaoyan
Format: Article
Language:English
Subjects:
Algorithms
Fault tolerance
Finite difference method
Mathematical analysis
Operators
Poisson equation
Quantum computers
Tensors
Online Access:Get full text
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Summary:The Poisson equation has wide applications in many areas of science and engineering. Although there are some quantum algorithms that can efficiently solve the Poisson equation, they generally require a fault-tolerant quantum computer which is beyond the current technology. In this paper, we propose a Variational Quantum Algorithm (VQA) to solve the Poisson equation, which can be executed on Noise Intermediate-Scale Quantum (NISQ) devices. In detail, we first adopt the finite difference method to transform the Poisson equation into a linear system. Then, according to the special structure of the linear system, we find an explicit tensor product decomposition, with only \(2\log n+1\) items, of its coefficient matrix under a specific set of simple operators, where \(n\) is the dimension of the coefficient matrix. This implies that the proposed VQA only needs \(O(\log n)\) measurements, which dramatically reduce quantum resources. Additionally, we perform quantum Bell measurements to efficiently evaluate the expectation values of simple operators. Numerical experiments demonstrate that our algorithm can effectively solve the Poisson equation.
ISSN:2331-8422
DOI:10.48550/arxiv.2012.07014