Quadratic versus Polynomial Unconstrained Binary Models for Quantum Optimization illustrated on Railway Timetabling

TL;DR

This paper compares quadratic and polynomial unconstrained binary models for quantum optimization, demonstrating that the polynomial formulation with penalty terms outperforms the quadratic one in railway timetabling problems using QAOA.

Contribution

It introduces a generic method to reformulate polynomial problems into PUBO and QUBO forms, highlighting the impact of problem formulation on quantum optimization performance.

Findings

PUBO with penalty terms outperforms QUBO in the studied problem

Reformulation techniques improve quantum optimization results

Demonstrates importance of problem encoding in QAOA performance

Abstract

Quantum Approximate Optimization Algorithm (QAOA) is one of the most short-term promising quantum-classical algorithm to solve unconstrained combinatorial optimization problems. It alternates between the execution of a parametrized quantum circuit and a classical optimization. There are numerous levers for enhancing QAOA performances, such as the choice of quantum circuit meta-parameters or the choice of the classical optimizer. In this paper, we stress on the importance of the input problem formulation by illustrating it with the resolution of an industrial railway timetabling problem. Specifically, we present a generic method to reformulate any polynomial problem into a Polynomial Unconstrained Binary Optimization (PUBO) problem, with a specific formulation imposing penalty terms to take binary values when the constraints are linear. We also provide a generic reformulation into a Quadratic Unconstrained Binary Optimization (QUBO) problem. We then conduct a numerical comparison between the PUBO with binary penalty terms and the QUBO formulations proposed on a railway timetabling problem solved with QAOA. Our results illustrate that the PUBO reformulation outperforms the QUBO one for the problem at hand.