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arXiv · 2404.17677

Stabilizer operators and Barnes-Wall lattices

Vadym Kliuchnikov·Sebastian Schönnenbeck
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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.

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Vadym Kliuchnikov, Sebastian Schönnenbeck. 2024-04-26. Stabilizer operators and Barnes-Wall lattices. https://arxiv.org/abs/2404.17677

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