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

Abstract

The paper is devoted to projective Clifford groups of quantum N-dimensional systems (with configuration space ). 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 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.