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 v, outputs the children of v. 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 inÕ( √ 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 anÕ( √ 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.
In REF quantum algorithms for the problem are studied.
Quantum algorithm for tree size estimation, with applications to backtracking and 2-player games
STOC 2017
Mathematics Physics Computer Science