Implementation of spectral methods on Ising machines: toward flow simulations on quantum annealer

TL;DR

This paper explores using Ising machines, including quantum annealers, to solve flow problems like the advection-diffusion equation, comparing spectral and finite difference methods, and assessing their current limitations.

Contribution

It formulates flow problems as QUBO problems suitable for Ising machines and evaluates their performance on classical and quantum annealers, highlighting spectral method advantages and current quantum challenges.

Findings

Spectral method requires fewer variables and offers better accuracy.

Accuracy depends on the condition number of the coefficient matrix.

Quantum annealer results are limited due to structural differences from classical annealers.

Abstract

We investigate the possibility and current limitations of flow computations using quantum annealers by solving a fundamental flow problem on Ising machines. As a fundamental problem, we consider the one-dimensional advection-diffusion equation. We formulate it in a form suited to Ising machines (i.e., both classical and quantum annealers), perform extensive numerical tests on a classical annealer, and finally test it on an actual quantum annealer. To make it possible to process with an Ising machine, the problem is formulated as a minimization problem of the residual of the governing equation discretized using either the spectral method or the finite difference method. The resulting system equation is then converted to the Quadratic Unconstrained Binary Optimization (QUBO) form though quantization of variables. It is found in the numerical tests using a classical annealer that the spectral method requiring smaller number of variables has a particular merit over the finite difference method because the accuracy deteriorates with the increase of the number of variables. We also found that the computational error varies depending on the condition number of the coefficient matrix. In addition, we extended it to a two-dimensional problem and confirmed its fundamental applicability. From the numerical test using a quantum annealer, however, it turns out that the computation using a quantum annealer is still challenging due largely to the structural difference from the classical annealer, which leaves a number of issues toward its practical use.