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Explicit orthogonal and unitary designs

Published 20 Oct 2023 in cs.CC | (2310.13597v1)

Abstract: We give a strongly explicit construction of $\varepsilon$-approximate $k$-designs for the orthogonal group $\mathrm{O}(N)$ and the unitary group $\mathrm{U}(N)$, for $N=2n$. Our designs are of cardinality $\mathrm{poly}(Nk/\varepsilon)$ (equivalently, they have seed length $O(nk + \log(1/\varepsilon)))$; up to the polynomial, this matches the number of design elements used by the construction consisting of completely random matrices.

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