Improving Quantum Multi-Objective Optimization with Archiving and Substitution

TL;DR

This work enhances quantum variational multi-objective optimization (QMOO) by introducing a Pareto archive and dominated substitution, and establishes RMNK-landscapes as a flexible classical-to-quantum benchmarking platform. By mapping RMNK landscapes to cost Hamiltonians and tuning QMOO hyperparameters, the authors demonstrate improved hypervolume convergence and competitive performance against NSGA-II/III on small instances, with potential advantages on more complex problems. The study also highlights practical considerations for balancing quantum and classical resources, and outlines pathways for scaling and robust performance under hardware constraints. Overall, the paper provides a structured approach to benchmarking and improving QMOO, contributing a step toward realizing quantum advantages in multi-objective optimization.

Abstract

Finding optimal solutions of conflicting objectives is a daily matter in many industrial applications, with multi-objective optimization trying to find the best solutions to them. The advent of quantum computing has led to researchers wondering if the promised exponential advantage can be obtained for these problems by variational quantum multi-objective optimization (QMOO) algorithm. Here, we improve it by introducing a Pareto Archive and dominated solutions substitution, clearly improving in hyper-volume convergence at additional quantum and classical cost. We propose the use of RMNK-landscapes as a unifying testbed for benchmarking QMOO, as it is common in classical multi-objective field. By devising a generic classical-to-quantum mapping of these landscapes, we perform a numerical hyperparameter tuning of QMOO, significantly enhancing its performance. Finally, we compare QMOO against well-known classical solvers for multi-objective tasks, NSGA-II/III, showing comparable results in small instances. Our results demonstrate that QMOO, when carefully tuned for the task at hand, might be advantageous on harder problems than its classical counterparts.

Figures (5)

• Figure 1: (Left) NK-landscape examples for two different values of $K$. We see the same general structure, but increasing $K$ increases the ruggedness. (Right) Median number of bit-flip connected components in the Pareto front as a function of $K$ for ten different seeds per parameter setting. In general, the number of connected regions grows with $M$, $N$, and $K$ while varying with $\rho$.
• Figure 2: Improvement of normalized hypervolume HV/HV$_{ideal}$ of QMOO variants as function of iterations (left) and cost function evaluations (right) for one specific RMNK-landscape model with $M=5$ uncorrelated ($\rho=0$) objectives, $N=12$ variables and $K=0$.
• Figure 3: HV/HV$_{ideal}$ convergence of the QMOO with archiving and Pareto solution substitution for a selection of $N_{\text{most prob}}$ and $N_{\text{shots}}$ values. Note here the different starting points of the first iteration value as we observe the substitution can attempt to replace solutions up to $2\times N_{\text{most prob}}$ times resulting in a better average starting hypervolume. All cases use a fixed budget of $300$ QMOO iterations
• Figure 4: Hyperparameter tuning results for QMOO with archiving and Pareto solution substitution. For each problem setting, the numbers are the median number of iterations and function evaluations required to reach the threshold HV/HV$_\mathrm{ideal} \geq 0.95$. For empty cells no runs were performed as a larger $N_{\text{most prob}}$ setting supercedes it.
• Figure 5: Normalized hypervolume values, using QMOO with archive and substitution, and NSGA-II/III with archive under different $(M, N, K)$. For the problem settings where hyperparameter tuning was conducted, the optimal set of hyperparameters was applied to each algorithm.

Theorems & Definitions (2)

• definition 1: Pareto dominance
• definition 2: Hypervolume