Analysis of the Non-variational Quantum Walk-based Optimisation Algorithm

TL;DR

This paper presents a non-variational quantum algorithm leveraging continuous-time quantum walks to efficiently solve diverse combinatorial optimization problems, demonstrating its effectiveness through simulations on multiple problem types.

Contribution

It introduces a novel non-variational quantum optimization algorithm using CTQWs, applicable to constrained and non-binary problems, with detailed circuit implementation and penalty optimization methods.

Findings

Achieves globally optimal solutions in few iterations

Demonstrates versatility across multiple problem types

Provides efficient circuit implementations for CTQWs

Abstract

This paper introduces in detail a non-variational quantum algorithm designed to solve a wide range of combinatorial optimisation problems, including constrained problems and problems with non-binary variables. The algorithm returns optimal and near-optimal solutions from repeated preparation and measurement of an amplified state. The amplified state is prepared via repeated application of two unitaries; one which phase-shifts solution states dependent on objective function values, and the other which mixes phase-shifted probability amplitudes via a continuous-time quantum walk (CTQW) on a problem-specific mixing graph. The general interference process responsible for amplifying optimal solutions is derived in part from statistical analysis of objective function values as distributed over the mixing graph. The algorithm's versatility is demonstrated through its application to various problems: weighted maxcut, k-means clustering, quadratic assignment, maximum independent set and capacitated facility location. In all cases, efficient circuit implementations of the CTQWs are discussed. A penalty function approach for constrained problems is also introduced, including a method for optimising the penalty function. For each of the considered problems, the algorithm's performance is simulated for a randomly generated problem instance, and in each case, the amplified state produces a globally optimal solution within a small number of iterations.