Near invariance of the hypercube

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 $P_{x \in C_n } \left( {M_x \in C_n } \right) \geqslant n^{ - C}$, 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.