Multiqubit Clifford groups are unitary 3-designs

@article{Zhu2017MultiqubitCG,
  title={Multiqubit Clifford groups are unitary 3-designs},
  author={Huangjun Zhu},
  journal={Physical Review A},
  year={2017},
  volume={96},
  pages={062336}
}
  • Huangjun Zhu
  • Published 9 October 2015
  • Mathematics
  • Physical Review A
Unitary $t$-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 $t$-designs with $t\geq3$ 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… 
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It is proved that the Clifford group is a 3-design, showing that it is a better approximation to Haar-random unitaries than previously expected and characterizing how well random Clifford elements approximateHaar- random unitaries.
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