On good encodings for quantum annealer and digital optimization solvers

Abstract

Several optimization solvers inspired by quantum annealing have been recently developed, either running on actual quantum hardware or simulating it on traditional digital computers. Industry and academics look at their potential in solving hard combinatorial optimization problems. Formally, they provide heuristic solutions for Ising models, which are equivalent to quadratic unconstrained binary optimization (QUBO). Constraints on solutions feasibility need to be properly encoded. We experiment on different ways of performing such an encoding. As benchmark we consider the cardinality constrained quadratic knapsack problem (CQKP), a minimal extension of QUBO with one inequality and one equality constraint. We consider different strategies of constraints penalization and variables encoding. We compare three QUBO solvers: quantum annealing on quantum hardware (D-Wave Advantage), probabilistic algorithms on digital hardware and mathematical programming solvers. We analyze their QUBO resolution quality and time, and the persistence values extracted in the quantum annealing sampling process. Our results show that a linear penalization of CQKP inequality improves current best practice. Furthermore, using such a linear penalization, persistence values produced by quantum hardware in a generic way allow to match a specific CQKP metric from literature. They are therefore suitable for general purpose variable fixing in core algorithms for combinatorial optimization.

Benchmarking quantum annealing with maximum cardinality matching problems

“…Studies were performed for academic instances such as random spin glasses (Rønnow et al, 2014), specially crafted problems with or without planted solutions (Hen et al, 2015;King et al, 2015;Albash and Lidar, 2018;Vert et al, 2020;McLeod and Sasdelli, 2022), a variety of problems with different level of difficulty (Jünger et al, 2021;McGeoch and Farre, 2023) and problems with industrial application such as the multi-car paint shop problem (Yarkoni et al, 2021), job shop scheduling problem (Carugno et al, 2022), and Earth-observation satellite mission planning problem (Stollenwerk et al, 2021). Studies benchmarking QA against classical algorithms comprise annealing-like algorithms such as SA (Rønnow et al, 2014;Hen et al, 2015;King et al, 2015;Albash and Lidar, 2018;Vert et al, 2020;Yarkoni et al, 2021;Carugno et al, 2022;McLeod and Sasdelli, 2022;Ceselli and Premoli, 2023;McGeoch and Farre, 2023), parallel tempering (McGeoch and Farre, 2023), simulated QA and SVMC (Hen et al, 2015;Albash and Lidar, 2018), and heuristic algorithms such as Tabu search (McGeoch and Wang, 2013;Yarkoni et al, 2021;Carugno et al, 2022), Hamze-de Freitas-Selby algorithm (Hen et al, 2015;King et al, 2015), or greedy algorithms (Yarkoni et al, 2021;Carugno et al, 2022;McGeoch and Farre, 2023), as well as exact solvers (McGeoch and Wang, 2013;Jünger et al, 2021;…”