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.

arXiv comment28 pages

References in corpus (13)

• A Theory of Fault-Tolerant Quantum Computation
• Hudson's Theorem for finite-dimensional quantum systems
• SIC-POVMs and the Extended Clifford Group
• Stabilizer states and Clifford operations for systems of arbitrary dimensions, and modular arithmetic
• Automorphisms of the fine grading of sl(n,C) associated with the generalized Pauli matrices
• Feynman's path integral and mutually unbiased bases
• On Clifford groups in quantum computing
• The monomial representations of the Clifford group
• The role of cohomology in quantum computation with magic states
• Hopf Algebras and Congruence Subgroups
• Symmetries of the finite Heisenberg group for composite systems
• Symmetries of finite Heisenberg groups for multipartite systems
• Combinatorics of finite abelian groups and Weil representations