Generalized Short Path Algorithms: Towards Super-Quadratic Speedup over Markov Chain Search for Combinatorial Optimization

TL;DR

This paper introduces generalized quantum short path algorithms that achieve super-quadratic speedups over classical Markov chain-based optimization methods for various combinatorial problems.

Contribution

It extends existing quantum short path frameworks to attain super-quadratic speedups over classical algorithms for combinatorial optimization problems.

Findings

Algorithms outperform classical counterparts in runtime.

Super-quadratic speedups achieved for specific problems.

Rigorously established runtimes for new quantum algorithms.

Abstract

We analyze generalizations of quantum algorithms based on the short path framework first proposed by Hastings~[\textit{Quantum} 2, 78 (2018)], which has been extended and shown by Dalzell~et~al.~[STOC~'23] to achieve super-Grover speedups for certain binary optimization problems. We demonstrate that, under some commonly satisfied technical conditions, an appropriate generalization can achieve super-quadratic speedups not only over unstructured search but also over a classical optimization algorithm that searches for the optimum by drawing samples from the stationary distribution of a Markov chain. We employ this framework to obtain algorithms for problems including variants of Max Bisection, Max Independent Set, and finding the ground states of the Antiferromagnetic Ising Model and the Sherrington-Kirkpatrick Model, whose runtimes are asymptotically faster than those obtainable with previous short path techniques. In certain cases, our algorithms achieve super-quadratic speedups compared to the best known classical algorithms with rigorously established runtimes.