Variational Quantum Multi-Objective Optimization

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, mostly conflicting objectives, such as cost and quality. We present a variational quantum optimization algorithm to solve discrete multi-objective optimization problems on quantum computers. The proposed quantum multi-objective optimization (QMOO) algorithm incorporates all cost Hamiltonians representing the classical objective functions in the quantum circuit and produces a quantum state consisting of Pareto-optimal solutions in superposition. From this state we retrieve a set of solutions and utilize the widely applied hypervolume indicator to determine its quality as an 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 show the effectiveness of the proposed algorithm on several benchmark problems with up to five objectives. We investigate the influence of the classical optimizer, the circuit depth and compare to results from classical optimization algorithms. We find that the algorithm is robust to shot noise and produces good results with as low as 128 measurement shots in each iteration. These promising result open the perspective to run the algorithm on near-term quantum hardware.