Provable Classical and Quantum Local Algorithms for Max-$k$-Cut and Quantum Advantage at Moderate Girth

2026-09-30

AI Summary

This study investigates the provable performance gap between classical and quantum algorithms for the Max-k-Cut problem on regular graphs. Methodologically, it introduces a local vector algorithm based on tree probability matching (TPM) to establish a lower bound on the optimal classical approximation ratio. Concurrently, it develops a tensor network framework that enables exact evaluation of the Quantum Approximate Optimization Algorithm (QAOA) in the infinite-degree limit via an equivalent mapping of coupled qudit-boson systems. The results demonstrate that at circuit depths p ≥ 9, the QAOA approximation ratio strictly surpasses the best guaranteed classical bound. This provides the first theoretical confirmation of quantum advantage for this combinatorial optimization problem under moderate girth conditions.

Abstract

Broadening the study of quantum optimization algorithms from binary to $k$-element alphabets has been shown to open new avenues for potential quantum advantage. A quantum advantage claim for approximate optimization requires showing that, under the same assumptions, an efficient quantum algorithm provably achieves a better performance than can be proven for the best known efficient classical algorithms. We study local classical and quantum algorithms for Max-$k$-Cut on $d$-regular graphs of girth $g$. We advance classical algorithms for this problem by developing a local vector algorithm based on the explicit vector construction of Thompson, Parekh, and Marwaha (TPM). Our algorithm gives the best provable cut fraction guarantee among known efficient classical algorithms on regular graphs for $k\geq3$. To evaluate the performance of QAOA under identical assumptions of girth and regularity, we develop tensor network techniques for general $d$ and $k \geq 2$. In addition, using an equivalence to a coupled qudit--boson system, we compute the QAOA performance in the infinite-degree limit. Together, these techniques give provable guarantees on QAOA performance on large graphs. Despite the improvements we introduce to the classical algorithm, QAOA achieves a better cut fraction guarantee for depths $p\geq 9$, corresponding to girth $g\geq 20$, for both finite- and infinite-degree regimes. Thus, we obtain an apparent quantum advantage from applying QAOA to the Max-$k$-Cut problem.

Problem

• Max-k-Cut
• Quantum Advantage
• QAOA
• Local Algorithms
• Regular Graphs

Innovation

• Max-k-Cut
• QAOA
• quantum advantage
• tensor network
• local vector algorithm