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

• URL: http://arxiv.org/abs/2010.15027v2
• Date: Tue, 19 Oct 2021 08:22:47 GMT
• Title: Solving generalized eigenvalue problems by ordinary differential equations on a quantum computer
• Authors: Changpeng Shao and Jin-Peng Liu
• Abstract summary: 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.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Many eigenvalue problems arising in practice are often of the generalized form $A\x=\lambda B\x$. 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.

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