The Clifford group forms a unitary 3-design

@article{Webb2016TheCG,
  title={The Clifford group forms a unitary 3-design},
  author={Zak Webb},
  journal={Quantum Inf. Comput.},
  year={2016},
  volume={16},
  pages={1379-1400}
}
  • Z. Webb
  • Published 9 October 2015
  • Mathematics
  • Quantum Inf. Comput.
Unitary $k$-designs are finite ensembles of unitary matrices that approximate the Haar distribution over unitary matrices. Several ensembles are known to be 2-designs, including the uniform distribution over the Clifford group, but no family of ensembles was previously known to form a 3-design. We prove that the Clifford group is a 3-design, showing that it is a better approximation to Haar-random unitaries than previously expected. Our proof strategy works for any distribution of unitaries… 
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