Laplace transform based quantum eigenvalue transformation via linear combination of Hamiltonian simulation

• URL: http://arxiv.org/abs/2411.04010v1
• Date: Wed, 06 Nov 2024 15:47:48 GMT
• Title: Laplace transform based quantum eigenvalue transformation via linear combination of Hamiltonian simulation
• Authors: Dong An, Andrew M. Childs, Lin Lin, Lexing Ying,
• Abstract summary: We propose an efficient quantum algorithm for performing a class of eigenvalue transformations that can be expressed as a certain type of matrix Laplace transformation. We demonstrate that our eigenvalue transformation approach can solve this problem without explicitly inverting $A$.
• Score: 13.96848357202551
• License:
• Abstract: Eigenvalue transformations, which include solving time-dependent differential equations as a special case, have a wide range of applications in scientific and engineering computation. While quantum algorithms for singular value transformations are well studied, eigenvalue transformations are distinct, especially for non-normal matrices. We propose an efficient quantum algorithm for performing a class of eigenvalue transformations that can be expressed as a certain type of matrix Laplace transformation. This allows us to significantly extend the recently developed linear combination of Hamiltonian simulation (LCHS) method [An, Liu, Lin, Phys. Rev. Lett. 131, 150603, 2023; An, Childs, Lin, arXiv:2312.03916] to represent a wider class of eigenvalue transformations, such as powers of the matrix inverse, $A^{-k}$, and the exponential of the matrix inverse, $e^{-A^{-1}}$. The latter can be interpreted as the solution of a mass-matrix differential equation of the form $A u'(t)=-u(t)$. We demonstrate that our eigenvalue transformation approach can solve this problem without explicitly inverting $A$, reducing the computational complexity.

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