Discrete-query quantum algorithm for NAND trees

Abstract

Recently, Farhi, Goldstone, and Gutmann gave a quantum algorithm for evaluating nand trees that runs in time O ( √ N log N) in the Hamiltonian query model. In this note, we point out that their algorithm can be converted into an algorithm using O(N 1/2+ǫ) queries in the conventional quantum query model, for any fixed ǫ> 0. A nand tree of depth n is a balanced binary tree whose internal vertices represent nand gates. Placing bits x1,...,x2 n at the leaves, the root of the nand tree evaluates to the function fn(x1,...,x2 n), where fn: {0, 1} 2n → {0, 1} is defined recursively as follows. For n = 0, f0(x) = x, and for n> 0, fn(x1,..., x2 n) = ¬ ( fn−1(x1,..., x 2 n−1) ∧ fn−1(x 2 n−1 +1,...,x2 n)). (1) The goal of the nand tree problem is to evaluate fn(x1,..., x2 n), making as few queries to the bits x1,..., x2 n as possible. The optimal classical randomized algorithm for this problem makes Θ(N 0.753) queries, where N = 2 n [7, 8, 9]. Until now, no better quantum algorithm was known, whereas the best known quantum lower bound is only Ω ( √ N) [1]. Here we show that for any fixed ǫ> 0, the quantum query complexity of evaluating nand trees is O(N 1/2+ǫ).