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. 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 $SL(2,Z_N)$ it is proved that for even $N$ projective Clifford groups are not natural semidirect products if and only if $N$ is divisible by four.