Quantum Algorithms for Algebraic Problems ∗

In this paper we present quantum query and time complexity bounds for several group testing problems. For a set S and a binary operation on S, we consider the decision problems whether a given structure with the promise of being a groupoid, semigroup, monoid or quasigroup is in fact a semigroup, monoid, quasigroup or a group. In particular, we give the first application of the new quantum random walk technique by Magniez, Nayak, Roland, and Santha [MNRS07] that improves the previous bounds by Ambainis [Amb04] and Szegedy [Sze04]. Our quantum algorithms for these problems improve the best known classical complexity bounds. We also present upper and lower bounds for testing distributivity and commutativity.

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