Multi-objective optimization by quantum annealing (2511.01762v1)

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.

• The paper demonstrates that quantum annealing outperforms QAOA and classical algorithms by rapidly recovering full Pareto fronts with significantly fewer QPU calls.
• It details a parallel sampling methodology on D-Wave hardware that achieves superior hypervolume metrics in both three- and four-objective optimization scenarios.
• The study highlights practical implications for real-world MOO tasks, offering enhanced sample efficiency and reduced parameter tuning compared to existing methods.

Multi-objective Optimization by Quantum Annealing: Comparative Performance and Implications

Introduction

This paper investigates the application of quantum annealing (QA) to multi-objective optimization (MOO), specifically focusing on the generation of Pareto fronts for weighted maximum-cut problems. The paper directly compares QA to the quantum approximate optimization algorithm (QAOA) and classical methods, using identical problem instances and methodologies. The central claim is that QA, as implemented on D-Wave hardware, significantly outperforms both QAOA (on IBM hardware and in noise-free simulation) and classical algorithms in terms of both solution quality and computational efficiency. The results have implications for the practical deployment of quantum optimization in MOO and for the ongoing debate regarding the relative merits of QA and QAOA on current quantum hardware.

Methodology

The experimental setup targets the multi-objective weighted maximum-cut problem, formulated as an Ising optimization task. For a graph $G$ with $N=42$ vertices and $M$ objective functions, each objective is defined as:

$F_k(s) = -\sum_{u,v\in E} s_i s_j J_{i,j,k}$

where $s \in \{-1,1\}^N$ and $J_{i,j,k}$ are Gaussian-distributed edge weights. The heavy-hex connectivity of the IBM quantum processor is embedded as a subgraph within the D-Wave architecture, ensuring a fair comparison.

The Pareto front is approximated by sampling from the weighted sum of objectives for many random convex combinations of weights $\{c_k\}$, with $c_k \geq 0$ and $\sum_k c_k = 1$. For each weighting, QA draws 1000 samples, leveraging the parallelism of the D-Wave system to process up to 96 (Advantage2_system1.6) or 114 (Advantage_system4.1) weightings simultaneously. The total QPU access time per batch is approximately 0.2 seconds, including programming and readout.

The primary performance metric is the hypervolume (HV) of the non-dominated set, computed using the moocore Python package. The reference point for HV is the vector of minimum values for each objective, ensuring that all Pareto-optimal points dominate this reference.

Results: Three-objective and Four-objective Cases

For the three-objective case, QA (Advantage2_system1.6) achieves the optimal hypervolume in a median of three QPU calls, routinely recovering the full Pareto front of 2067 non-dominated points in under two seconds. This is over two orders of magnitude faster than the best-case MPS-simulated QAOA and three orders of magnitude faster than QAOA on IBM hardware. Classical algorithms (DCM, DPA-a) are also outperformed, as they are limited by discretization and do not recover the full front for continuous weights.

Figure 1: Three-objective results showing the rapid convergence of QA to the optimal hypervolume compared to QAOA and classical methods.

In the four-objective scenario, QA identifies 30,419 non-dominated points, exceeding the 30,409 points previously reported by QAOA-based methods. The QA approach achieves this in a median of 203 QPU calls (under 20 million samples), compared to 100 million samples for MPS QAOA. No other method, quantum or classical, recovers the full Pareto front in this setting.

Figure 2: Four-objective results demonstrating that QA finds a strictly larger Pareto front than all other approaches, with superior sample efficiency.

Discussion

The empirical results demonstrate that QA, as implemented on current D-Wave hardware, is substantially more effective for MOO than QAOA on IBM hardware or in idealized simulation. The speedup is attributed to both the parallel sampling capabilities of the D-Wave system and the inherent suitability of QA for Ising-type binary optimization. Notably, the QA approach does not require advanced parameter tuning or spin-reversal transformations, suggesting that further gains may be possible with more sophisticated techniques.

The findings challenge the prevailing assumption that QAOA, with its theoretical guarantees in the fault-tolerant regime, is the preferred quantum approach for combinatorial optimization. On current hardware, QA is empirically superior for the class of problems studied. The results also highlight the limitations of classical integer-based algorithms when applied to continuous-weighted MOO.

From a practical perspective, the demonstrated ability to recover large Pareto fronts rapidly and reliably positions QA as a viable tool for real-world MOO tasks, particularly in domains where solution diversity and trade-off analysis are critical. The paper's methodology, including the use of hypervolume as a performance metric and the parallel sampling strategy, provides a template for future benchmarking of quantum optimization algorithms.

Implications and Future Directions

The strong performance of QA in this context suggests several avenues for further research:

• Algorithmic Enhancements: Incorporating advanced parameter tuning, spin-reversal transformations, or hybrid quantum-classical post-processing may yield further improvements in solution quality and efficiency.
• Hardware Scaling: As quantum annealers increase in qubit count and connectivity, larger and more complex MOO instances can be addressed, potentially extending the observed performance gap.
• Generalization to Other MOO Problems: The methodology is directly applicable to other binary MOO tasks, and its extension to constrained or non-binary domains warrants investigation.
• Comparative Studies: Systematic benchmarking across a broader range of quantum and classical algorithms, including emerging variational and hybrid approaches, will further clarify the landscape of quantum MOO.

Conclusion

This paper provides compelling evidence that quantum annealing, as realized on current D-Wave hardware, outperforms both QAOA and classical methods for multi-objective weighted maximum-cut problems. The results underscore the practical utility of QA for generating Pareto fronts in MOO and suggest that, for certain classes of combinatorial optimization, QA is the preferred quantum approach on existing hardware. The findings motivate continued exploration of QA-based methods and their integration into real-world decision-making workflows.