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

TL;DR

This paper introduces a quantum algorithm for estimating the size of unknown search trees and DAGs efficiently, with applications to improving backtracking algorithms and evaluating game-related formulas.

Contribution

It presents a novel quantum algorithm for size estimation of unknown trees and DAGs, enabling faster backtracking and game evaluation in unknown structures.

Findings

Quantum size estimation runs in O(\u221a{nT}) steps.

Backtracking algorithms are improved to O(T^{3/2}) time.

Formulas of size T and shallow depth can be evaluated in O(T^{1/2+o(1)}) quantum time.

Abstract

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 $1\pm \delta$ in $\tilde{O}(\sqrt{nT})$ 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 $\tilde{O}(\sqrt{T}n^{3/2})$ 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). We show that, in this setting, formulas of size $T$ and depth $T^{o(1)}$ can be evaluated in quantum 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.