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Simple Groups - Abstract Algebra

This theorem is proved in the appendix. It offers a simple explanation of the distinction between multiqubit systems and systems of odd local dimensions, which is of intrinsic interest to studying quantum computation. Moreover, it reveals a surprising connection between unitary t -designs and the physics of discrete phase spaces, which may have profound implications for the cross fertilization of the two active research fields. We showed that the multiqubit Clifford group is a unitary 3-design.

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It is also a minimal 3-design except for dimension 4. As a consequence, any orbit of pure states of the multiqubit Clifford group is a 3-design; in particular, the set of multiqubit stabilizer states is a 3-design. The methods and conclusions presented here are also useful in studying higher moments of the Clifford group. These results are of interest to many research areas both within and beyond quantum information science. Moreover, we offered a simple explanation of why no discrete Wigner function is covariant with respect to the multiqubit Clifford group by proving that no operator basis is covariant with respect to any group that forms a unitary 3-design.

This result reveals a surprising connection between unitary designs and the physics of discrete phase spaces, which is of interest to studying quantum computation and a number of nonclassical phenomena, such as negativity and contextuality. Note added. Upon completion of this work, we noticed a comprehensive math paper by Robert M. However, this paper mentions neither t -designs nor the Clifford group explicitly. Our proofs are completely independent of the CFSG and are thus simpler and more transparent.

According to a similar argument used in the proof of Theorem 2.

In view of Eq. Here S m and A m denote the symmetric group and alternating group on m letters. To prove this theorem, we need to introduce a few auxiliary concepts and results. A subgroup of S p 2 n , 2 is primitive if it acts transitively on nonzero vectors in F 2 n 2 and preserves no nontrivial partition. This contradiction shows that R is primitive. So R is antiflag transitive. According to Lemma 1 in Ref. Zhu cf. However, these triple products cannot all be real since, otherwise, the basis operators would commute with each other and thus cannot form an operator basis.

Abstract Unitary t -designs are a ubiquitous tool in many research areas, including randomized benchmarking, quantum process tomography, and scrambling. Theorem 1. Corollary 1. Lemma 1. Lemma 2. Remark 1. Theorem 2.


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MR 53 Zbl Aschbacher, On the maximal subgroups of the finite classical groups , Invent. MR 86a Aschbacher, Finite group theory , Cambridge Stud. Press, Cambridge, MR 89b Dye, Symmetric groups as maximal subgroups of orthogonal and symplectic group over the field of two elements , J. London Math. MR 80m MR 81j GAP program Jansen, The minimal degrees of faithful representations of the sporadic simple groups and their covering groups , LMS J.

MR e Kantor, Homogeneous designs and geometric lattices , Journal of combinatorial theory. MR 87c Kantor, Linear groups containing a Singer cycle , J. MR 81g King, On some maximal subgroups of the classical groups , J.