Discrete quadratic model QUBO solution landscapes

TL;DR

This paper studies how different encoding methods and penalty parameters affect the solution landscapes of QUBO models derived from discrete quadratic problems, which are relevant for quantum computing applications.

Contribution

It provides an analysis of the impact of encoding choices and penalty strengths on QUBO solution landscapes for discrete quadratic models, aiding better problem encoding strategies.

Findings

Encoding choice influences landscape structure and optimization difficulty.

Penalty strength affects the validity of solutions and landscape topology.

One-hot and domain-wall encodings exhibit distinct landscape characteristics.

Abstract

Many computational problems involve optimization over discrete variables with quadratic interactions. Known as discrete quadratic models (DQMs), these problems in general are NP-hard. Accordingly, there is increasing interest in encoding DQMs as quadratic unconstrained binary optimization (QUBO) models to allow their solution by quantum and quantum-inspired hardware with architectures and solution methods designed specifically for such problem types. However, converting DQMs to QUBO models often introduces invalid solutions to the solution space of the QUBO models. These solutions must be penalized by introducing appropriate constraints to the QUBO objective function that are weighted by a tunable penalty parameter to ensure that the global optimum is valid. However, selecting the strength of this parameter is non-trivial, given its influence on solution landscape structure. Here, we investigate the effects of choice of encoding and penalty strength on the structure of QUBO DQM solution landscapes and their optimization, focusing specifically on one-hot and domain-wall encodings.