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Entanglement Theory Quantum Correlations Quantum Simulation

Clifford groups are not always 2-designs

arXiv

Authors: Matthew A. Graydon, Joshua Skanes-Norman, Joel J. Wallman

Year

2021

Paper ID

62502

Status

Preprint

Abstract Read

~2 min

Abstract Words

116

Citations

N/A

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 Uoverline{U} representation is sqrt{2}.

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The Clifford group is the quotient of the normalizer of the Weyl-Heisenberg group in dimension d by its centre.

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References & Citation Signals

[1] DOI https://doi.org/arXiv:2108.04200 [2] arXiv https://arxiv.org/abs/2108.04200 [3] Publisher https://arxiv.org/abs/2108.04200

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