Combinatorial Optimization with Quantum Computers
Francisco Chicano, Gabiel Luque, Zakaria Abdelmoiz Dahi, Rodrigo Gil-Merino
TL;DR
The paper addresses how to perform combinatorial optimization on quantum devices by converting problems into unconstrained QUBO/Ising forms and solving them with both quantum annealers and gate-based systems. It presents a full workflow from problem representation and domain encoding to penalty-based constraint handling and quadratization of higher-order terms, including practical methods like Rosenberg penalties and local transformations. The discussion covers hardware-specific approaches, such as current quantum annealers for Ising/QUBO problems and gate-based techniques like QAOA and VQE, supplemented by worked examples and code (eg D-Wave Ocean and Qiskit). It also candidly addresses challenges—penalty tuning, embedding overhead, and the current state of quantum speedups—calling for continued algorithmic innovation and hybrid strategies to realize practical quantum advantage in optimization.
Abstract
Quantum computers leverage the principles of quantum mechanics to do computation with a potential advantage over classical computers. While a single classical computer transforms one particular binary input into an output after applying one operator to the input, a quantum computer can apply the operator to a superposition of binary strings to provide a superposition of binary outputs, doing computation apparently in parallel. This feature allows quantum computers to speed up the computation compared to classical algorithms. Unsurprisingly, quantum algorithms have been proposed to solve optimization problems in quantum computers. Furthermore, a family of quantum machines called quantum annealers are specially designed to solve optimization problems. In this paper, we provide an introduction to quantum optimization from a practical point of view. We introduce the reader to the use of quantum annealers and quantum gate-based machines to solve optimization problems.