Stabilizer operators and Barnes-Wall lattices

          @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