Clifford groups are not always 2-designs

A group 2-design is a unitary 2-design arising via the image of a suitable compact group under a projective unitary representation in dimension d.  The Clifford group in dimension d is the quotient of the normalizer of the Weyl-Heisenberg group in dimension d, by its centre: namely U(1).  In this talk, we prove that the Clifford group is not a group 2-design when d is not prime. Our main proofs rely, primarily, on elementary representation theory, and so we review the essentials. We also discuss the general structure of group 2-designs. In particular, we show that the adjoint action induced by a group 2-design splits into exactly two irreducible components; moreover, a group is a group 2-design if and only if the norm of the character of its so-called U-Ubar representation is the square root of two. Finally, as a corollary, we see that the multipartite Clifford group (on some finite number of quantum systems) also often fails to be a group 2-design.

This talk is based on joint work with Joshua Skanes-Norman and Joel J. Wallman; arXiv:2108.04200 [quant-ph].