Quantum Rational Transformation Using Linear Combinations of Hamiltonian Simulations

TL;DR

This paper introduces efficient quantum algorithms for rational transformations of operators using Hamiltonian simulations with linear combinations of unitaries, enabling improved spectral analysis of many-body systems.

Contribution

It develops two novel LCU-based methods for implementing rational transformations on quantum hardware, enhancing quantum spectral estimation techniques.

Findings

Demonstrates accurate low-lying energy estimation in spin systems

Develops real-time quantum spectral analysis framework

Shows efficiency of rational transformations via Hamiltonian simulation

Abstract

Rational functions are exceptionally powerful tools in scientific computing, yet their abilities to advance quantum algorithms remain largely untapped. In this paper, we introduce effective implementations of rational transformations of a target operator on quantum hardware. By leveraging suitable integral representations of the operator resolvent, we show that rational transformations can be performed efficiently with Hamiltonian simulations using a linear-combination-of-unitaries (LCU). We formulate two complementary LCU approaches, discrete-time and continuous-time LCU, each providing unique strategies to decomposing the exact integral representations of a resolvent. We consider quantum rational transformation for the ubiquitous task of approximating functions of a Hermitian operator, with particular emphasis on the elementary signum function. For illustration, we discuss its application to the ground and excited state problems. Combining rational transformations with observable dynamic mode decomposition (ODMD), our recently developed noise-resilient quantum eigensolver, we design a fully real-time approach for resolving many-body spectra. Our numerical demonstration on spin systems indicates that our real-time framework is compact and achieves accurate estimation of the low-lying energies.