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Volume 11 (2015) Article 16 pp. 403-412 [Note]
Quantum Algorithm for Monotonicity Testing on the Hypercube
Received: March 24, 2015
Revised: July 16, 2015
Published: December 29, 2015
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Keywords: Boolean functions, quantum query complexity, property testing, monotonicity
ACM Classification: F.2.2, F.1.3, G.1.6
AMS Classification: 68Q17, 52C07, 11H06, 11H31, 05B40

Abstract: [Plain Text Version]

In this note, we develop a bounded-error quantum algorithm that makes $\tilde{O}(n^{1/4}\varepsilon^{-1/2})$ queries to a function $f\colon \{0,1\}^n \to\{0,1\}$, accepts when $f$ is monotone, and rejects when $f$ is $\varepsilon$-far from being monotone. This result gives a super-quadratic improvement compared to the best known randomized algorithm for all $\varepsilon = o(1)$. The improvement is cubic when $\varepsilon = 1/\sqrt{n}$.