Stabilizer operators and Barnes-Wall lattices

2024/04/26 by Vadym Kliuchnikov, Kliuchnikov, Vadym, Sebastian Schonnenbeck +1
Mathematics · #FOS: Physical sciences #Mathematical Dynamics and Fractals #Quantum Physics (quant-ph)

paper · pdf · doi:10.48550/arxiv.2404.17677

openalex publication_date 2024/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ℚ(exp(i(2π)/(2^m))), we show that it can be implemented by a post-selected stabilizer circuit if it has entries in ℤ[exp(i(2π)/(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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