PIRSA:24050008 Conference Quantum Information

DOI10.48660/24050008

Conference/School: Foundations of Quantum Computational Advantage

Kliuchnikov, V. (2024). Stabilizer operators and Barnes-Wall lattices. Perimeter Institute. https://pirsa.org/24050008

Kliuchnikov, Vadym. Stabilizer operators and Barnes-Wall lattices. Perimeter Institute, May. 01, 2024, https://pirsa.org/24050008 

@misc{ pirsa_PIRSA:24050008,
  doi = {10.48660/24050008},
  url = {https://pirsa.org/24050008},
  author = {Kliuchnikov, Vadym},
  keywords = {Quantum Information},
  language = {en},
  title = {Stabilizer operators and Barnes-Wall lattices},
  publisher = {Perimeter Institute},
  year = {2024},
  month = {may},
  note = {PIRSA:24050008 see, \url{https://pirsa.org}}
}

Abstract

We give a simple description of rectangular matrices that can be implemented by a post-selected stabilizer circuit. Given a matrix with entries in dyadic cyclotomic number fields $\mathbb{Q}(\exp(i\frac{2\pi}{2^m}))$, we show that it can be implemented by a post-selected stabilizer circuit if it has entries in $\mathbb{Z}[\exp(i\frac{2\pi}{2^m})]$ when expressed in a certain non-orthogonal basis. This basis is related to Barnes-Wall lattices. Our result is a generalization to a well-known connection between Clifford groups and Barnes-Wall lattices. We also show that minimal vectors of Barnes-Wall lattices are stabilizer states, which may be of independent interest. Finally, we provide a few examples of generalizations beyond standard Clifford groups. Joint work with Sebastian Schonnenbeck

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