@inproceedings {pub5813,
	title = {Variational Quantum multi-objective optimization},
	author = {Linus Ekstrom AND Hao Wang AND Sebastian Schmitt},
	year = {2024},
	abstract = {Introduction

Solving combinatorial optimization problems on near-term quantum devices has gained a lot of attraction in recent years. Currently, most works have focused on single-objective problems, whereas many real-world applications need to consider multiple conflicting objectives, such as, cost and quality.

Methods

We present a variational quantum optimization algorithm to solve discrete multi-objective optimization problems. The proposed quantum multi-objective optimization (QMOO) algorithm produces a quantum state which is a superposition of Pareto-optimal solutions, thereby solving the original multi-objective problem. This is achieved by incorporating all cost Hamiltonians representing the classical objectives in the circuit. We retrieve a set of solutions and utilize hypervolume indicator to determine the quality of the approximation to the Pareto-front. The variational parameters of the QMOO circuit are tuned by maximizing the hypervolume indicator in a quantum-classical hybrid fashion. We formulate the benchmark problems and the algorithm in terms of qudit variables and operators.

Results

We show the effectiveness of QMOO on several benchmark problems with up to five objectives. The hypervolume increases over the iterations of the algorithm indicating that the QMOO quantum state encodes increasingly better approximations to the Pareto front. We also study the influence of the number of shots in the quantum algorithm, where we find that the algorithm also works with very few shots. However, the cost function for the classical optimization becomes increasingly noisy for fewer shots, which can have an influence on the achievable performance. We compare to results of several classical multi-objective optimization algorithms where we find that QMOO performance is only slightly below the classical performance.

Discussion

Our work represents a first very promising step toward solving realistic multi-objective optimization problems natively on quantum hardware. It emphasizes a novel research area, where many aspects of single-objective variational quantum algorithms such as alternative mixing operators, influence of circuit complexity and barren plateaus or warm starting schemes still need to be investigated. However, we are convinced that future works will improve the algorithm and identify problems where quantum multi-objective approaches offer a practical quantum advantage.},
	publisher = {none},
	booktitle = {QTech Berlin Conference}
}