Quantum optimization with linear Ising penalty functions for customer data science

TL;DR

This paper explores the use of linear Ising penalty functions in quantum optimization to improve resource efficiency and performance for customer data science problems, compared to traditional quadratic penalties.

Contribution

It introduces linear Ising penalties as an alternative to quadratic penalties for quantum constraints, with theoretical and numerical evidence of improved performance.

Findings

Linear Ising penalties can outperform quadratic penalties in quantum optimization.

The linear method is more resource-efficient and better suited for near-term quantum devices.

Combining linear and quadratic penalties can address limitations of the linear approach.

Abstract

Constrained combinatorial optimization problems, which are ubiquitous in industry, can be solved by quantum algorithms such as quantum annealing (QA) and the quantum approximate optimization algorithm (QAOA). In these quantum algorithms, constraints are typically implemented with quadratic penalty functions. This penalty method can introduce large energy scales and make interaction graphs much more dense. These effects can result in worse performance of quantum optimization, particularly on near-term devices that have sparse hardware graphs and other physical limitations. In this work, we consider linear Ising penalty functions, which are applied with local fields in the Ising model, as an alternative method for implementing constraints that makes more efficient use of physical resources. We study the behaviour of the penalty method in the context of quantum optimization for customer data science problems. Our theoretical analysis and numerical simulations of QA and the QAOA indicate that this penalty method can lead to better performance in quantum optimization than the quadratic method. However, the linear Ising penalty method is not suitable for all problems as it cannot always exactly implement the desired constraint. In cases where the linear method is not successful in implementing all constraints, we propose that schemes involving both quadratic and linear Ising penalties can be effective.