Solving generalized eigenvalue problems by ordinary differential equations on a quantum computer

Shao, Changpeng; Liu, Jin-Peng

doi:10.1098/rspa.2021.0797

Solving generalized eigenvalue problems by ordinary differential equations on a quantum computer

Journal Article · 01 June 2022 · Proceedings of the Royal Society. A. Mathematical, Physical and Engineering Sciences

DOI:https://doi.org/10.1098/rspa.2021.0797· OSTI ID:1982741

^[1]; Liu, Jin-Peng ^[2]

School of Mathematics, University of Bristol, Bristol BS8 1UG, UK
Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742, USA; Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742, USA; Department of Mathematics, University of Maryland, College Park, MD 20742, USA

Many eigenvalue problems arising in practice are often of the generalized form $A x = λ B x$ . One particularly important case is symmetric, namely $A, B$ are Hermitian and $B$ is positive definite. The standard algorithm for solving this class of eigenvalue problems is to reduce them to Hermitian eigenvalue problems. For a quantum computer, quantum phase estimation is a useful technique to solve Hermitian eigenvalue problems. In this work, we propose a new quantum algorithm for symmetric generalized eigenvalue problems using ordinary differential equations. The algorithm has lower complexity than the standard one based on quantum phase estimation. Moreover, it works for a wider case than symmetric: $B$ is invertible, $B^{- 1} A$ is diagonalizable and all the eigenvalues are real.

Research Organization:: US Department of Energy (USDOE), Washington, DC (United States). Office of Science, Advanced Scientific Computing Research (ASCR)

Sponsoring Organization:: USDOE

OSTI ID:: 1982741

Journal Information:: Proceedings of the Royal Society. A. Mathematical, Physical and Engineering Sciences, Vol. 478, Issue 2262; ISSN 1364-5021

Publisher:: The Royal Society Publishing

Country of Publication:: United States

Language:: English

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