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Clifford groups are not always 2-designs
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Clifford groups are not always 2-designs
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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 $\sqrt{2}$.
Forward citations
Cited by 2 Pith papers
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Quantum Universality in Composite Systems: A Trichotomy of Clifford Resources
Single-qudit universality for Clifford gate sets plus one non-Clifford gate follows a trichotomy determined by the prime factorization of the local dimension d.
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Simple slow operators and quantum thermalization
Absence of simple slow operators implies that typical low-complexity states thermalize in quantum systems.
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