The Quantum Query Complexity of Algebraic Properties

Sebastian Dörn and Thomas Thierauf

We present quantum query complexity bounds for testing algebraic properties. For a set S and a binary operation on S, we consider the decision problem whether S is a semigroup or has an identity element. If S is a monoid, we want to decide whether S is a group.

We present quantum algorithms for these problems that improve the best known classical complexity bounds. In particular, we give the first application of the new quantum random walk technique by Magniez, Nayak, Roland, and Santha 06 that improves the previous bounds by Ambainis 04 and Szegedy 04. We also present several lower bounds for testing algebraic properties.