Exponential Quantum Advantage for Pathfinding in Regular Sunflower Graphs

TL;DR

This paper demonstrates an exponential quantum speedup for the pathfinding problem in regular sunflower graphs, a class of graphs that are mild expanders, highlighting potential implications for cryptography.

Contribution

It introduces regular sunflower graphs as a new class allowing exponential quantum advantage in pathfinding, with an efficient quantum algorithm and proof of classical hardness.

Findings

Quantum algorithm finds s-t path efficiently in regular sunflower graphs.

Classical algorithms require exponential time for the same problem.

Regular sunflower graphs are proven to be mild expanders with high probability.

Abstract

Finding problems that allow for superpolynomial quantum speedup is one of the most important tasks in quantum computation. A key challenge is identifying problem structures that can only be exploited by quantum mechanics. In this paper, we find a class of graphs that allows for exponential quantum-classical separation for the pathfinding problem with the adjacency list oracle, and this class of graphs is named regular sunflower graphs. We prove that, with high probability, a regular sunflower graph of degree at least $7$ is a mild expander graph, that is, the spectral gap of the graph Laplacian is at least inverse polylogarithmic in the graph size. We provide an efficient quantum algorithm to find an $s$-$t$ path in the regular sunflower graph while any classical algorithm takes exponential time. This quantum advantage is achieved by efficiently preparing a $0$-eigenstate of the adjacency matrix of the regular sunflower graph as a quantum superposition state over the vertices, and this quantum state contains enough information to help us efficiently find an $s$-$t$ path in the regular sunflower graph. Because the security of an isogeny-based cryptosystem depends on the hardness of finding an $s$-$t$ path in an expander graph \cite{Charles2009}, a quantum speedup of the pathfinding problem on an expander graph is of significance. Our result represents a step towards this goal as the first provable exponential speedup for pathfinding in a mild expander graph.