Multi-objective optimization by quantum annealing

TL;DR

The paper tackles multi-objective optimization over binary variables and the challenge of constructing the Pareto front for $M$ objectives, comparing quantum annealing (QA) and quantum approximate optimization (QAOA) as hardware solvers. It maps the MOO problems to an Ising formulation with $F_k(s) = -\sum_{(i,j)\in E} s_i s_j J_{i,j,k}$ on a heavy-hex graph with $N=42$ and $|E|=46$, enabling a weighted maximum-cut interpretation solvable by QA on a D-Wave processor. Results show QA dramatically outperforms both QAOA and classical baselines, achieving near-optimal hypervolume $HV_{max}$ with far fewer samples and QPU calls; for four objectives, QA identified $30{,}419$ non-dominated points and reached optimal front performance in as few as a few hundred QPU calls. This provides strong evidence for the practical utility of quantum annealing in multi-objective optimization on current hardware and is accompanied by open-source code and data for reproducibility.

Abstract

An important task in multi-objective optimization is generating the Pareto front -- the set of all Pareto-optimal compromises among multiple objective functions applied to the same set of variables. Since this task can be computationally intensive even for small problems, it is a natural target for quantum optimization. Indeed, this problem was recently approached using the quantum approximate optimization algorithm (QAOA) on an IBM gate-model processor. Here we compare these QAOA results with quantum annealing on the same two input problems, using the same methodology. We find that quantum annealing vastly outperforms not just QAOA run on the IBM processor, but all classical and quantum methods analyzed in the previous study. On the harder problem, quantum annealing improves upon the best known Pareto front. This small study reinforces the promise of quantum annealing in multi-objective optimization.

Figures (2)

• Figure 1: Three-objective results. The figure of merit, $\text{HV}_\text{max}-\text{HV}+1$, is plotted for various classical and quantum approaches. For sampling-based approaches, i.e. QA and QAOA, experiments are run five times, with the mean indicated by a line and the shaded region indicating the best and worst among five runs. Both QA systems found the same optimal solution as Ref. kotil_quantum_2025, with the Pareto front made up of 2067 non-dominated points. All results other than those for QA are taken from Ref. kotil_quantum_2025.
• Figure 2: Four-objective results. Data is as in Fig. \ref{['fig:1']}, but no results are available for ibm_fez. Estimates are derived from MPS results assuming circuit fidelities of 53% and 3.71% (the latter being an estimate for current systems). QA found a better Pareto front than all other approaches, consisting of 30,419 non-dominated points. All results other than those for QA are taken from Ref. kotil_quantum_2025.