Equivariant QAOA and the Duel of the Mixers

TL;DR

This paper introduces a systematic method for constructing symmetry-aligned mixer Hamiltonians for QAOA, demonstrating improved performance in combinatorial optimization problems through theoretical validation and empirical comparison.

Contribution

The authors develop a novel construction method for symmetry-preserving mixer Hamiltonians tailored to the symmetric group, validated through formulas, circuits, and performance evaluations.

Findings

New mixer Hamiltonian outperforms conventional in experiments

Classical mixer commutes with a smaller subgroup of symmetries

Analysis explains poor performance in some warm-start QAOA variants

Abstract

Constructing an optimal mixer for Quantum Approximate Optimization Algorithm (QAOA) Hamiltonian is crucial for enhancing the performance of QAOA in solving combinatorial optimization problems. We present a systematic methodology for constructing the QAOA tailored mixer Hamiltonian, ensuring alignment with the inherent symmetries of classical optimization problem objectives. The key to our approach is to identify an operator that commutes with the action of the group of symmetries on the QAOA underlying Hilbert space and meets the essential technical criteria for effective mixer Hamiltonian functionality. We offer a construction method specifically tailored to the symmetric group $S_d$, prevalent in a variety of combinatorial optimization problems. By rigorously validating the required properties, providing a concrete formula and corresponding quantum circuit for implementation, we establish the viability of the proposed mixer Hamiltonian. Furthermore, we demonstrate that the classical mixer $B$ commutes only with a subgroup of $S_d$ of significantly smaller order than the group itself, enhancing the efficiency of the proposed approach. To evaluate the effectiveness of our methodology, we compare two QAOA variants utilizing different mixer Hamiltonians: conventional $B=\sum X_i$ and the newly proposed $H_M$ in edge coloring and graph partitioning problems across various graphs. We observe statistically significant differences in mean values, with the new variant consistently demonstrating superior performance across multiple independent simulations. Additionally, we analyze the phenomenon of poor performance in alternative warm-start QAOA variants, providing a conceptual explanation supported by recent literature findings.