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

• URL: http://arxiv.org/abs/2411.10062v1
• Date: Fri, 15 Nov 2024 09:23:52 GMT
• Title: Quadratic versus Polynomial Unconstrained Binary Models for Quantum Optimization illustrated on Railway Timetabling
• Authors: Camille Grange, Marion Lavignac, Valentina Pozzoli, Eric Bourreau,
• Abstract summary: We present a generic method to reformulate any problem into a Polynomial Unconstrained Binary Optimization (PUBO) problem. We also provide a generic reformulation into a Quadratic Unconstrained Binary Optimization (QUBO) problem. Our results illustrate that the PUBO reformulation outperforms the QUBO one for the problem at hand.
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• 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.

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