Advances in quantum algorithms for the shortest path problem

Given an undirected, weighted graph, with $n$ vertices and $m$ edges, and two special vertices $s$ and $t$, the problem is to find the shortest path between them. We give two bounded-error quantum algorithms with improved runtime in the adjacency list model that solve the problem on special classes of graphs defined via pathfinding probabilities of classical random walks and the electrical network framework. Firstly, we give a simple quantum algorithm based on sampling edges from a graph via the quantum flow state and running a classical algorithm on the sampled edges. It runs in $\tilde{O}(l^2\sqrt{m})$ expected time and uses $O(\log{n})$ space on graphs where the shortest $s$-$t$ path is also a minimum resistance $s$-$t$ subgraph. Our main algorithm can be thought of as a divide and conquer version of this approach and works on a special class of graphs where classical loop-erased random walk has a probability $q>0.537$ of finding the shortest $s$-$t$ path. In such cases the quantum algorithm outputs the shortest $s$-$t$ path with high probability in $\widetilde{O}(\ell\sqrt{m})$ expected time and $O(\log{n})$ space, where $l$ is the length (or total weight, in case of weighted graphs) of the shortest $s$-$t$ path. This algorithm can be parallelised to $\tilde{O}(\sqrt{lm})$ circuit depth when using $O(l\log{n})$ space. With the latter we partially resolve with an affirmative answer the open problem of whether a path between two vertices can be found in the number of steps required to detect it.