Generalising quantum imaginary time evolution to solve linear partial differential equations

TL;DR

This paper extends quantum imaginary time evolution (QITE) to develop a quantum algorithm capable of solving linear partial differential equations, demonstrated through heat equation simulations on small quantum systems.

Contribution

It introduces a novel QITE-based method for solving PDEs by tracking state vector scale, expanding QITE's application beyond ground state approximation.

Findings

Successfully solved 1D and 2D heat equations with small quantum systems.

Demonstrated feasibility of quantum PDE solver via numerical simulations.

Showed potential for quantum algorithms in numerical analysis.

Abstract

The quantum imaginary time evolution (QITE) methodology was developed to overcome a critical issue as regards non-unitarity in the implementation of imaginary time evolution on a quantum computer. QITE has since been used to approximate ground states of various physical systems. In this paper, we demonstrate a practical application of QITE as a quantum numerical solver for linear partial differential equations. Our algorithm takes inspiration from QITE in that the quantum state follows the same normalised trajectory in both algorithms. However, it is our QITE methodology's ability to track the scale of the state vector over time that allows our algorithm to solve differential equations. We demonstrate our methodology with numerical simulations and use it to solve the heat equation in one and two dimensions using six and ten qubits, respectively.