High order schemes for solving partial differential equations on a quantum computer

TL;DR

This paper investigates higher-order discretization methods for solving PDEs on quantum computers, demonstrating they can reduce qubit requirements without increasing Trotter steps, thus improving quantum algorithm efficiency.

Contribution

It introduces an efficient decomposition of diagonal matrices into Pauli strings and applies higher-order schemes to quantum PDE solutions, highlighting their practical benefits.

Findings

Higher-order methods reduce qubit count for discretization.

Number of Trotter steps remains unchanged with higher-order methods.

Efficient decomposition of matrices into Pauli strings enhances quantum simulation.

Abstract

We explore the utilization of higher-order discretization techniques in optimizing the gate count needed for quantum computer based solutions of partial differential equations. To accomplish this, we present an efficient approach for decomposing $d$-band diagonal matrices into Pauli strings that are grouped into mutually commuting sets. Using numerical simulations of the one-dimensional wave equation, we show that higher-order methods can reduce the number of qubits necessary for discretization, similar to the classical case, although they do not decrease the number of Trotter steps needed to preserve solution accuracy. This result has important consequences for the practical application of quantum algorithms based on Hamiltonian evolution.