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Multiqubit Clifford groups are unitary 3-designs
Unitary -designs are a ubiquitous tool in many research areas, including
randomized benchmarking, quantum process tomography, and scrambling. Despite
the intensive efforts of many researchers, little is known about unitary
-designs with in the literature. We show that the multiqubit
Clifford group in any even prime-power dimension is not only a unitary
2-design, but also a 3-design. Moreover, it is a minimal 3-design except for
dimension~4. As an immediate consequence, any orbit of pure states of the
multiqubit Clifford group forms a complex projective 3-design; in particular,
the set of stabilizer states forms a 3-design. In addition, our study is
helpful to studying higher moments of the Clifford group, which are useful in
many research areas ranging from quantum information science to signal
processing. Furthermore, we reveal a surprising connection between unitary
3-designs and the physics of discrete phase spaces and thereby offer a simple
explanation of why no discrete Wigner function is covariant with respect to the
multiqubit Clifford group, which is of intrinsic interest to studying quantum
computation.Comment: 7 pages, published in Phys. Rev.
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