A Performance Comparison of Variable Encoding Techniques for QUIO and QUBO Problems

Pedro Maciel Xavier, Yirang Park, Diego Noble, Italo Santana, Miguel Paredes, David E. Bernal Neira

Keywords: QIP, QUIO, QUBO, variable encoding comparison

Abstract:

In this paper, we explore the encoding Quadratic Integer Optimization (QIP) problems into Quadratic Unconstrained Integer Optimization (QUIO) formulations with a target integer basis.
These formulations are suitable for solving on quantum computers based on qudits, which natively extend the integer representation of their qubit-based counterparts.
For qubit-based computers, the usual framework for representing discrete optimization problems is the Quadratic Unconstrained Binary Optimization (QUBO) formulation.
Decision variables can, for QUBOs, assume only binary values, while for QUIOs, they can represent integer values from zero up to a machine-dependent maximum.
One advantage of working with a larger domain of decision variables is that it enables QIP problems to be cast as QUIO formulations.
As our results highlight, these formulations use fewer decision variables than, for example, Quadratic Unconstrained Binary Optimization (QUBO)-based formulations.
We selected various candidate problems to empirically verify our reformulations: Quadratic Facility Location Problem, Quadratic Inventory Management Problem, Quadratic Vehicle Routing Problem, and Quadratic Knapsack Problem.
Problem instances for these problems are diverse in data distribution and size and are reformulated into QUBO and QUIO.
We compare these formulations, characterizing them using metrics associated with solution performance.
Moreover, using qubit- and qudit-based entropy quantum machines, we compared the performance of the resulting formulations for instances amenable to these architectures.
Our primary goal is to conduct computational experiments to verify the impacts of each encoding type on these problems, aiming to find insights that could potentially generalize to similar optimization problems.
Moreover, we also indicate guidelines to accelerate the encoding process, ensuring that the potential quantum advantage using qudit quantum computers is not lost during the classical pre-processing and during problem reformulation.
Finally, we provide open-source software to perform the reformulations and communicate them with qudit-based entropy quantum devices, allowing others to map and solve QIP problems using QUIO reformulations.

Submission Number: 4