Stabilizer operators and Barnes-Wall lattices

TL;DR

This paper explores the connection between stabilizer operators, Barnes-Wall lattices, and Clifford groups, providing a framework for implementing certain matrices via post-selected stabilizer circuits and revealing that minimal vectors of Barnes-Wall lattices are stabilizer states.

Contribution

It introduces a new basis related to Barnes-Wall lattices for implementing matrices with stabilizer circuits and generalizes the link between Clifford groups and Barnes-Wall lattices.

Findings

Matrices with entries in specific cyclotomic fields can be implemented by stabilizer circuits.

Minimal vectors of Barnes-Wall lattices are stabilizer states.

The work extends the connection between Clifford groups and Barnes-Wall lattices.

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.