LAUSR

The paper is devoted to projective Clifford groups of quantum N-dimensional systems. 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 don’t concern the semidirect structure. Using appropriate group presentation of SL(2, ZN ) it is proved that for even N projective Clifford groups are not natural semidirect products if and only if N is divisible by four.