Quantum Algorithm for Monotonicity Testing on the Hypercube

TL;DR

This paper introduces a quantum algorithm that significantly improves the efficiency of testing whether a Boolean function is monotone, outperforming classical methods especially for small error parameters.

Contribution

The authors present a quantum algorithm with query complexity better than classical algorithms for monotonicity testing on the hypercube.

Findings

Quantum algorithm achieves $ ilde O(n^{1/4} ext{epsilon}^{-1/2})$ query complexity.

Super-quadratic improvement over classical algorithms for small epsilon.

Cubic improvement when epsilon equals $1/\sqrt{n}$.

Abstract

In this note, we develop a bounded-error quantum algorithm that makes $\tilde O(n^{1/4}\varepsilon^{-1/2})$ queries to a Boolean function $f$, accepts a monotone function, and rejects a function that is $\varepsilon$-far from being monotone. This gives a super-quadratic improvement compared to the best known randomized algorithm for all $\varepsilon = o(1)$. The improvement is cubic when $\varepsilon = 1/\sqrt{n}$.