Quadratic Quantum Speedup for Finding Independent Set of a Graph

TL;DR

This paper proves analytically and confirms numerically that a quantum adiabatic algorithm can achieve a quadratic speedup over classical algorithms in finding large independent sets in graphs, with implications for quantum computing and Rydberg atom experiments.

Contribution

The paper introduces an analytical framework based on the Magnus expansion in the interaction picture to demonstrate quadratic quantum speedup for the independent set problem.

Findings

Quantum algorithm achieves O(n) complexity for size-2 independent sets.

Numerical results confirm quantum speedup over classical greedy algorithms.

Analysis links quantum performance to graph spectral structure and experimental parameters.

Abstract

A quadratic speedup of the quantum adiabatic algorithm (QAA) for finding independent sets (ISs) in a graph is proven analytically. In comparison to the best classical algorithm with $O(n^2)$ scaling, where $n$ is the number of vertexes, our quantum algorithm achieves a time complexity of $O(n^2)$ for finding a large IS, which reduces to $O(n)$ for identifying a size-2 IS. The complexity bounds we obtain are confirmed numerically for a specific case with the $O(n^2)$ quantum algorithm outperforming the classical greedy algorithm, that also runs in $O(n^2)$. The definitive analytical and numerical evidence for the quadratic quantum speedup benefited from an analytical framework based on the Magnus expansion in the interaction picture (MEIP), which overcomes the dependence on the ground state degeneracy encountered in conventional energy gap analysis. In addition, our analysis links the performance of QAA to the spectral structure of the median graph, bridging algorithmic complexity, graph theory, and experimentally realizable Rydberg Hamiltonians. The understanding gained provides practical guidance for optimizing near-term Rydberg atom experiments by revealing the significant impact of detuning on blockade violations.