Clifford groups are not always 2-designs

2021/08/09 by Matthew A. Graydon, Graydon, Matthew A., Joshua Skanes-Norman +3 · 1 citation
Engineering · Materials Science · Mathematics · Physics and Astronomy · #FOS: Physical sciences #Finite Group Theory Research #Quantum Physics (quant-ph) #Quasicrystal Structures and Properties #graph theory and CDMA systems #quant-ph

paper · pdf · doi:10.48550/arxiv.2108.04200

5 pages

arxiv created 2021/08/09 · openalex publication_date 2021/08/09 · arxiv updated 2021/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

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 UU representation is √(2).

Citations

• Efficient Simulation of Random Quantum States and Operators

Cited by

• Quantum Universality in Composite Systems: A Trichotomy of Clifford Resources

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