Near invariance of the hypercube

TL;DR

This paper characterizes orthogonal matrices that nearly preserve the Boolean hypercube, showing they are close to permutation and reflection matrices, advancing understanding of matrices with large permanents relevant to quantum computing.

Contribution

It provides a near-complete description of orthogonal matrices that significantly rotate the hypercube onto itself, linking to matrices with large permanents and quantum computing applications.

Findings

Matrices close to permutations and reflections preserve the hypercube.

Such matrices must be nearly orthogonal to permutation/reflection matrices.

Results contribute to understanding matrices with large permanents.

Abstract

We give an almost-complete description of orthogonal matrices $M$ of order $n$ that "rotate a non-negligible fraction of the Boolean hypercube $C_n=\{-1,1\}^n$ onto itself," in the sense that $$P_{x\in C_n}(Mx\in C_n) \ge n^{-C},\mbox{ for some positive constant } C,$$ where $x$ is sampled uniformly over $C_n$. In particular, we show that such matrices $M$ must be very close to products of permutation and reflection matrices. This result is a step toward characterizing those orthogonal and unitary matrices with large permanents, a question with applications to linear-optical quantum computing.