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

We analyze generalizations of algorithms based on the short-path framework first proposed by Hastings [Quantum 2, 78 (2018)], which has been extended and shown by Dalzell et al.[STOC'22] 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, the Ising Model, and the Sherrington Kirkpatrick Model, whose runtimes are asymptotically faster than can be established using previous short path techniques. For random regular graphs of sufficiently high degree, our algorithm is super-quadratically faster than the best rigorously proven classical runtimes for regular graphs. Our results also shed light on the quantum nature of short path algorithms, by identifying a setting where our algorithm is super-quadratically faster than any polynomial time Gibbs sampler, unless NP = RP. We conclude the with a numerical analysis that guides the choice of parameters for short path algorithms and raises the possibility of super-quadratic speedups in a setttings that is currently beyond our theoretical methods.