We give a strongly explicit construction of ε-approximate k-designs for the orthogonal group O(N) and the unitary group U(N), for N=2n. Our designs are of cardinality poly(Nk/ε) (equivalently, they have seed length O(nk+log(1/ε))); up to the polynomial, this matches the number of design elements used by the construction consisting of completely random matrices.
Cite
@article{arxiv.2310.13597,
title = {Explicit orthogonal and unitary designs},
author = {Ryan O'Donnell and Rocco A. Servedio and Pedro Paredes},
journal= {arXiv preprint arXiv:2310.13597},
year = {2023}
}