Contour-integral based quantum eigenvalue transformation: analysis and applications

TL;DR

This paper analyzes and develops quantum algorithms based on contour integral representations for eigenvalue transformations, demonstrating their efficiency and practical advantages in Hamiltonian simulation and differential equations.

Contribution

It provides a complete complexity analysis of contour integral methods and introduces a new quantum algorithm using minimal qubits for eigenvalue transformations.

Findings

Contour integral algorithms can outperform existing quantum methods for stable differential equations.

The proposed algorithms efficiently estimate eigenvalue transformation observables.

A new quantum algorithm uses only 3 additional qubits for eigenvalue estimation.

Abstract

Eigenvalue transformations appear ubiquitously in scientific computation, ranging from matrix polynomials to differential equations, and are beyond the reach of the quantum singular value transformation framework. In this work, we study the efficiency of quantum algorithms based on contour integral representation for eigenvalue transformations from both theoretical and practical aspects. Theoretically, we establish a complete complexity analysis of the contour integral approach proposed in [Takahira, Ohashi, Sogabe, and Usuda. Quant. Inf. Comput., 22, 11\&12, 965--979 (2021)]. Moreover, we combine the contour integral approach and the sampling-based linear combination of unitaries to propose a quantum algorithm for estimating observables of eigenvalue transformations using only $3$ additional qubits. Practically, we design contour integral based quantum algorithms for Hamiltonian simulation, matrix polynomials, and solving linear ordinary differential equations, and show that the contour integral algorithm can outperform all the existing quantum algorithms in the case of solving asymptotically stable differential equations.