Clifford group is not a semidirect product in dimensions N divisible by four

Abstract

Abstract The paper is devoted to projective Clifford groups of quantum N-dimensional systems (with configuration space Z N ). Clearly, Clifford gates allow only the simplest quantum computations which can be simulated on a classical computer (Gottesmann–Knill theorem). However, it may serve as a cornerstone of full quantum computation. As to its group structure it is well-known that—in N-dimensional quantum mechanics—the Clifford group is a natural semidirect product provided the dimension N is an odd number. For even N special results on the Clifford groups are scattered in the mathematical literature, but they mostly do not concern the semidirect structure. Using appropriate group presentation of S L ( 2 , Z N ) it is proved that for even N the projective Clifford groups are not natural semidirect products if and only if N is divisible by four.