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

TL;DR

An efficient quantum algorithm is proposed for performing a class of eigenvalue transformations that can be expressed as a certain type of matrix Laplace transformation, which allows the recently developed linear combination of Hamiltonian simulation (LCHS) method to be extended.

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.

Theorems & Definitions (25)

• Theorem 3
• Theorem 4
• proof
• Lemma 5
• Theorem 6
• proof
• Corollary 7
• proof
• Corollary 8
• Corollary 9