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

TL;DR

This paper introduces a quantum algorithm leveraging Laplace transforms and linear combination of Hamiltonian simulation to perform eigenvalue transformations, enabling efficient solutions to matrix inverse and differential equations without explicit inversion.

Contribution

It extends the linear combination of Hamiltonian simulation method to a broader class of eigenvalue transformations using Laplace transforms, including matrix inverse powers and exponentials.

Findings

Enables eigenvalue transformations like $A^{-k}$ and $e^{-A^{-1}}$ on quantum computers.

Reduces computational complexity by avoiding explicit matrix inversion.

Applies to solving mass-matrix differential equations efficiently.

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.