Time complexity analysis of quantum difference methods for linear high dimensional and multiscale partial differential equations

We investigate time complexities of finite difference methods for solving the high-dimensional linear heat equation, the high-dimensional linear hyperbolic equation and the multiscale hyperbolic heat system with quantum algorithms (hence referred to as the “quantum difference methods”). For the heat...

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Bibliographic Details
Published in:Journal of computational physics Vol. 471; p. 111641
Main Authors: Jin, Shi, Liu, Nana, Yu, Yue
Format: Journal Article
Language:English
Published: Elsevier Inc 15.12.2022
Subjects:
Asymptotic-preserving schemes
HHL algorithm
Quantum difference methods
Quantum linear systems algorithms
Time complexity
Quantum linear systems algorithms
Asymptotic-preserving schemes
Time complexity
Quantum difference methods
HHL algorithm
ISSN:0021-9991, 1090-2716
Online Access:Get full text
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Summary:We investigate time complexities of finite difference methods for solving the high-dimensional linear heat equation, the high-dimensional linear hyperbolic equation and the multiscale hyperbolic heat system with quantum algorithms (hence referred to as the “quantum difference methods”). For the heat and linear hyperbolic equations we study the impact of explicit and implicit time discretizations on quantum advantages over the classical difference method. For the multiscale problem, we find the time complexity of both the classical treatment and quantum treatment for the explicit scheme scales as O(1/ε), where ε is the scaling parameter, while the scaling for the multiscale Asymptotic-Preserving (AP) schemes does not depend on ε. This indicates that it is still of great importance to develop AP schemes for multiscale problems in quantum computing. •New quantum algorithm is proposed for a prototype multiscale problem.•Asymptotic-preserving schemes are shown to be important in quantum computation.•There is no difference in the cost of the quantum algorithm for the heat equation when using the forward-Euler compared to the Crank-Nicolson scheme.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2022.111641