Clifford groups are not always 2-designs

Matthew A. Graydon; Joshua Skanes-Norman; Joel J. Wallman

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arXiv · 2108.04200

Clifford groups are not always 2-designs

Matthew A. Graydon·Joshua Skanes-Norman·Joel J. Wallman

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Abstract

The Clifford group is the quotient of the normalizer of the Weyl-Heisenberg group in dimension $d$ by its centre. We prove that when $d$ is not prime the Clifford group is not a group unitary $2$-design. Furthermore, we prove that the multipartite Clifford group is not a group unitary 2-design except for the known cases wherein the local Hilbert space dimensions are a constant prime number. We also clarify the structure of projective group unitary $2$-designs. We show that the adjoint action induced by a group unitary $2$-design decomposes into exactly two irreducible components; moreover, a group is a unitary 2-design if and only if the character of its so-called $U\overline{U}$ representation is $\sqrt{2}$.

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Matthew A. Graydon, Joshua Skanes-Norman, Joel J. Wallman. 2021-08-09. Clifford groups are not always 2-designs. https://arxiv.org/abs/2108.04200

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