Qantum algorithm for tree size estimation, with applications to backtracking and 2-player games

We study quantum algorithms on search trees of unknown structure, in a model where the tree can be discovered by local exploration. That is, we are given the root of the tree and access to a black box which, given a vertex ν, outputs the children of ν. We construct a quantum algorithm which, given such access to a search tree of depth at most n, estimates the size of the tree T within a factor of 1 ± δ? in Õ(√nT) steps. More generally, the same algorithm can be used to estimate size of directed acyclic graphs (DAGs) in a similar model. We then show two applications of this result: a) We show how to transform a classical backtracking search algorithm which examines T nodes of a search tree into an Õ(√Tn^3/2) time quantum algorithm, improving over an earlier quantum backtracking algorithm of Montanaro (arXiv:1509.02374). b) We give a quantum algorithm for evaluating AND-OR formulas in a model where the formula can be discovered by local exploration (modeling position trees in 2-player games) which evaluates formulas of size T and depth T^o(1) in time O(T^1/2+o(1)). Thus, the quantum speedup is essentially the same as in the case when the formula is known in advance.