QUBO formulations for numerical quantum computing

TL;DR

This paper develops QUBO formulations for solving linear systems on quantum computers, validates them on D-Wave hardware, and provides practical Python code for implementation.

Contribution

It introduces new QUBO models for linear system solving using binary representations and demonstrates their application on real quantum hardware.

Findings

QUBO models successfully solve simple linear systems on D-Wave

The paper provides practical Python code for implementation

Validation shows potential for quantum linear system solving

Abstract

With the advent of quantum computers, many quantum computing algorithms are being developed. Solving linear systems is one of the most fundamental problems in almost all science and engineering. The Harrow-Hassidim-Lloyd algorithm, a monumental quantum algorithm for solving linear systems on gate model quantum computers, was invented and several advanced variations have been developed. For a given n by n matrix A and a vector b, we will find unconstrained binary optimization (QUBO) models for a vector x that satisfies Ax=b. To formulate QUBO models for a linear system solving problem, we make use of a linear least-square problem with binary representation of the solution. We validate those QUBO models on the D-Wave system and discuss the results. For a simple system, we provide a Python code to calculate the matrix characterizing the relationship between the variables and to print the test code that can be used directly in the D-Wave system.