Exponential Quantum Speedup on Structured Hard Instances of Maximum Independent Set

TL;DR

This paper demonstrates an exponential quantum speedup for a specific class of structured maximum independent set problems using a non-stoquastic quantum algorithm that leverages quantum interference to bypass tunneling, supported by analytical and numerical evidence.

Contribution

The work introduces a non-stoquastic adiabatic quantum algorithm tailored for structured MIS instances, revealing a quantum speedup mechanism based on sign-structured subspaces and interference effects.

Findings

Polynomial runtime of the quantum algorithm on structured instances

Exponential speedup over classical solvers and quantum annealing

Scalable models capturing the essential quantum dynamics

Abstract

Establishing quantum speedup for computationally hard problems of practical relevance, particularly combinatorial optimization problems, remains a central challenge in quantum computation. In this work, we identify a structurally defined family of classically hard maximum independent set (MIS) instances, and design and analyze a non-stoquastic adiabatic quantum optimization algorithm that exploits this structure. The algorithm runs in polynomial time and achieves an exponential speedup over both transverse-field quantum annealing and state-of-the-art classical solvers on these instances, under assumptions supported by analytical and numerical evidence. We identify the essential quantum mechanism enabling the speedup as the use of a non-stoquastic XX-driver to access a larger sign-structured admissible subspace beyond the stoquastic regime, which allows sign-generating quantum interference to create smooth evolution paths that bypass tunneling. This identifies a distinctive quantum mechanism underlying the speedup and explains why no efficient classical analogue is likely to exist. In addition, our analysis produces scalable small-scale models, derived from our structural reduction, that capture the essential dynamics of the algorithm. These models provide a concrete opportunity for verification of the quantum advantage mechanism on currently available universal quantum computers.