Combinatorial Optimization and Quantum Computers

Xiaoyuan Liu, Anthony Angone, Ruslan Shaydulin +3 more

Georg Kruse, Rodrigo Coelho, Andreas Roßkopf +2 more

Advancements in Quantum Computing (QC) and Neural Combinatorial Optimization (NCO) represent promising steps in tackling complex computational challenges. On the one hand, Variational Quantum Algorithms such as QAOA can be used to solve a wide range of combinatorial optimization problems. On the other hand, the same class of problems can be solved by NCO, a method that has shown promising results, particularly since the introduction of Graph Neural Networks. Given recent advances in both research areas, we introduce Hamiltonian-based Quantum Reinforcement Learning (QRL), an approach at the intersection of QC and NCO. We model our ansatzes directly on the combinatorial optimization problem's Hamiltonian formulation, which allows us to apply our approach to a broad class of problems. Our ansatzes show favourable trainability properties when compared to the hardware efficient ansatzes, while also not being limited to graph-based problems, unlike previous approaches. In this work, we evaluate the performance of Hamiltonian-based QRL on a diverse set of combinatorial optimization problems to demonstrate the broad applicability of our approach and compare it to QAOA.

Yunpyo Hong, Jaejoon Gill, Daeyeun Kim +3 more

Niklas Pirnay, Vincent Quentin Ulitzsch, Frederik Wilde +2 more

It is unclear to what extent quantum algorithms can outperform classical algorithms for problems of combinatorial optimization. In this work, by resorting to computational learning theory and cryptographic notions, we give a fully constructive proof that quantum computers feature a super-polynomial advantage over classical computers in approximating combinatorial optimization problems. Specifically, by building on seminal work by Kearns and Valiant, we provide special instances that are hard for classical computers to approximate up to polynomial factors. Simultaneously, we give a quantum algorithm that can efficiently approximate the optimal solution within a polynomial factor. The quantum advantage in this work is ultimately borrowed from Shor's quantum algorithm for factoring. We introduce an explicit and comprehensive end-to-end construction for the advantage bearing instances. For these instances, quantum computers have, in principle, the power to approximate combinatorial optimization solutions beyond the reach of classical efficient algorithms.

Juan Fernando Giraldo Botello

Quantum computing has the potential to surpass the capabilities of current classical computers when solving complex problems. Combinatorial optimization has emerged as a pivotal target area for quantum computers, as problems in this field are renowned for their complexity and resource-intensive nature. Moreover, these challenges play a critical role in various industrial sectors, including logistics, manufacturing, and finance. This thesis explores the integration of quantum computation into classical software tools as a means to potentially address combinatorial optimization problems more efficiently and effectively. This work introduces QPLEX, a Python software library that enables practitioners and researchers to implement the general mathematical formulation of a given combinatorial optimization problem once and execute it seamlessly on multiple quantum devices using various quantum algorithms. This software solution automatically adapts a general optimization model to the specific instructions utilized by the target quantum device’s SDK. It offers a versatile execution workflow capable of running gate-based hybrid quantum-classical algorithms for combinatorial optimization in a platform-agnostic manner. This approach reduces the programming overhead required for modeling and experimenting with combinatorial optimization solutions. Within this manuscript, we address and introduce the various aspects associated with the development of QPLEX in a clear and comprehensive manner. These aspects encompass the quantum algorithms and quantum hardware available in the library, along with QPLEX’s system design and implementation. Additionally, we provide a guide on how to use the library and conduct a thorough evaluation of the software solution within a specific use case as part of this thesis.

Hayato Goto