Quantum Algorithm for Approximating Maximum Independent Sets

TL;DR

This paper introduces a quantum algorithm utilizing non-Abelian adiabatic mixing to efficiently approximate maximum independent sets in graphs, outperforming classical algorithms in terms of approximation quality.

Contribution

The paper presents a novel quantum algorithm based on adiabatic mixing for approximating maximum independent sets, achieving closer results to the maximum than classical methods.

Findings

Quantum algorithm finds independent sets close to the maximum size.

Time complexity of the quantum algorithm is approximately O(n^2).

Quantum approach outperforms classical approximation algorithms.

Abstract

We present a quantum algorithm for approximating maximum independent sets of a graph based on quantum non-Abelian adiabatic mixing in the sub-Hilbert space of degenerate ground states, which generates quantum annealing in a secondary Hamiltonian. For both sparse and dense graphs, our quantum algorithm on average can find an independent set of size very close to $\alpha(G)$, which is the size of the maximum independent set of a given graph $G$. Numerical results indicate that an $O(n^2)$ time complexity quantum algorithm is sufficient for finding an independent set of size $(1-\epsilon)\alpha(G)$. The best classical approximation algorithm can produce in polynomial time an independent set of size about half of $\alpha(G)$.