Quantum Approximate Multi-Objective Optimization

TL;DR

This work introduces a quantum approach to multi-objective optimization (MOO) by applying low-depth QAOA to MO-MAXCUT and leveraging parameter transfer to avoid re-optimizing QAOA angles for every scalarized objective. By training parameters on small representative instances offline and sampling randomized weight vectors $c$, the method can efficiently approximate the Pareto front, including non-supported solutions, using both MPS simulations and NISQ hardware. Experimental results on 42-node MO-MAXCUT graphs show competitive hypervolume performance against classical methods for three objectives, and favorable trends for four objectives when incorporating fidelity-based hardware scaling. The findings suggest quantum heuristics, combined with transfer learning and fair sampling concepts, can forecast and potentially surpass classical performance as quantum devices improve, while also offering a new paradigm for constrained optimization via Pareto-front sampling.

Abstract

The goal of multi-objective optimization is to understand optimal trade-offs between competing objective functions by finding the Pareto front, i.e., the set of all Pareto optimal solutions, where no objective can be improved without degrading another one. Multi-objective optimization can be challenging classically, even if the corresponding single-objective optimization problems are efficiently solvable. Thus, multi-objective optimization represents a compelling problem class to analyze with quantum computers. In this work, we use low-depth Quantum Approximate Optimization Algorithm to approximate the optimal Pareto front of certain multi-objective weighted maximum cut problems. We demonstrate its performance on an IBM Quantum computer, as well as with Matrix Product State numerical simulation, and show its potential to outperform classical approaches.

Figures (12)

• Figure 1: Pareto front & HV for two objective functions: The red diamond represents the reference point $r$. The green points on the green solid line correspond to the optimal Pareto front, the green shaded area represents the optimal HV with respect to $r$, and the blue dashed line corresponds to the convex hull of the Pareto front. The green point in the middle of the Pareto front does not lie on the convex hull and illustrates a non-supported Pareto optimal solution, the four other green points lie on the convex hull and illustrate supported Pareto optimal solutions. The orange diamonds represent examples of dominated solutions approximating the Pareto front, with the orange area illustrating their corresponding HV.
• Figure 2: The $\epsilon$-CM: Given the lower bounds $\epsilon_i$ on the objective function values, the resulting feasible set (green area) only contains the non-supported Pareto optimal solution, which -- in this simplified example -- will be found by maximizing any of the two objective functions or convex combinations of them.
• Figure 3: Graph topologies: The whole graph shows the coupling map of the 156-qubit ibm_fez device. The green-shaded areas show the 27-qubit graph of the training problem (top) and the 42-qubit graph of the considered MO-MAXCUT problem (bottom) embedded in the ibm_fez coupling map.
• Figure 4: Comparison of quantum and classical results for MO-MAXCUT with $m=3$ objective functions: Average progress of $\text{HV}_t$ of the MPS simulation ($\chi = 50$, solid line), ibm_fez (dashed line), and the classical algorithms DCM, DPA-a, and $\epsilon$-CM (dotted-dashed lines). The shaded areas denote the minimum and maximum performance over the five repetitions of the MPS simulation and results from ibm_fez. The best found solution $\text{HV}_{\text{max}}$ is indicated by the red dotted horizontal line.
• Figure 5: Number of non-dominated points: The solid lines show the average number of unique non-dominated points approximating the Pareto front resulting from the five repetitions using MPS simulation, with the shaded areas denoting the corresponding minimum and maximum. The largest value reached is 2772.0. For $\chi = 1$ or $p=1$, the number of points was only between 170 and 255 and is not shown here. The dashed line and corresponding shaded area shows the results from the runs on ibm_fez reaching a maximum of 2726.0. For both, the MPS simulation and the results from ibm_fez 25 million shots were taken in total. The dotted-dashed lines show the result for DPA-a and DCM (both achieve the same value) and the $\epsilon$-CM.