Advances in quantum algorithms for the shortest path problem

Given an undirected, weighted graph, and two special vertices s and t, the problem is to find the shortest path between them. We give two bounded-error quantum algorithms in the adjacency list model that solve the problem on structured instances. The first approach is based on sparsifying the original graph via sampling the quantum flow state and running a classical algorithm on the smaller problem. It has time complexity of Õ(l²√m) and uses O(logl) space, where l is the length (or total weight, in case of weighted graphs) of the shortest s-t path. The main result is the second approach which is based on a divide and conquer procedure that outputs the shortest path in Õ(l√m) steps when using O(logl) space, and can be parallelised to Õ(√lm) circuit depth when using O(llogl) 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 time required to detect it.