ASSOCIATION RULES IN QUBO SAMPLES AND WHERE TO FIND THEM

Abstract

There are sometimes strong associations between variables in the samples to a Quadratic Unconstrained Binary Optimization (QUBO) problem. A natural question arises to us: Are there any value in these association? We study max-cut problem and observe that association can be represented as rules to simplify QUBO problem. Classical and quantum annealers work better when the problem size is smaller. To effectively and efficiently find associations between variables, we adapt traditional association rule mining in the case of QUBO samples and propose a Fast Association Rule Mining algorithm (FARM) specifically for mining QUBO samples. We also propose strategies and a workflow to select and apply promising rules and simplify QUBO problems. We evaluate our method on D-Wave Quantum Annealer as well as Fujitsu Digital Annealer. The experiments demonstrate the utility of FARM as a visualisation tool for understanding associations in QUBO samples. The results also demonstrate the potential of our method in closing the gap between samples and ground truth. The source code will be disclosed to the public if the manuscript is accepted.

1. INTRODUCTION

Many combinatorial optimization problems can be formulated as a quadratic unconstrained binary optimization problem (QUBO) Lucas (2014), Glover et al. (2018) . QUBO corresponds naturally to the transverse Ising model and benefits from the speed up by quantum annealing Kadowaki & Nishimori (1998) Annealing, e.g., simulated annealingKirkpatrick et al. (1983) , is a family of probabilistic methods for optimising the variables of a system (e.g. minimising a function). In annealing, we heat the system to a high-temperature level and gradually bring down the temperature. Variables in the system gradually lose their energy and eventually sit in a low-energy state. One round of heating and cooling, i.e., an annealing process, produces one sample, which is a configuration of all variables. The annealing process is random. The energy distribution of the samples follows a Boltzmann distributionNelson et al. ( 2022), i.e., the system is more likely to be in a lower energy state. Quantum annealing works similarly, except that it makes use of quantum mechanics. We can use annealing for optimisation in a sampling manner. While we cannot find optimal solution in one shot, one possible way to obtain a promising solution is to collect more samples. However, the marginal profit of "more samples" decreases as we collect more samples. Besides, access to quantum devices is expensive. We are looking for a more efficient method to make additional samples more productive. The main idea of the paper is that we examine existing samples to discover interesting rules and simplify the original QUBO problem. More specifically, if most of the variables agrees on certain decision, then most likely these variables have been solved correctly. We should then solve the remaining variables that we are less certain of. There are some earlier works in this domain, with their own limitations. Lewis & Glover (2017) invented a few rules to transform the underlying QUBO structure into an equivalent graph with fewer nodes and edges. However, such rules are hand-crafted. A question of interest is whether we can discover such rules that could be specific to a particular application. Chardaire et al. (1995) ; Karimi 1

