Every NAND formula of size N can be evaluated in time N^{1/2+o(1)} on a quantum computer

TL;DR

This paper presents a quantum algorithm that evaluates any NAND formula of size N in roughly N^{1/2} time, nearly matching the optimal query complexity and addressing a longstanding open problem.

Contribution

It introduces a quantum algorithm based on quantum walks that evaluates NAND formulas in near-optimal time, improving previous bounds and nearly resolving an open complexity question.

Findings

Quantum algorithm evaluates NAND formulas in N^{1/2+o(1)} time.

Balanced NAND formulas can be evaluated in O(sqrt{N}) queries.

Almost establishes a lower bound relating formula size to quantum query complexity.

Abstract

For every NAND formula of size N, there is a bounded-error N^{1/2+o(1)}-time quantum algorithm, based on a coined quantum walk, that evaluates this formula on a black-box input. Balanced, or ``approximately balanced,'' NAND formulas can be evaluated in O(sqrt{N}) queries, which is optimal. It follows that the (2-o(1))-th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed by Laplante, Lee and Szegedy.