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 $N=2^n$. Our designs are of cardinality $\mathrm{poly}(N^k/\varepsilon)$ (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.

翻译：我们给出了正交群 $\mathrm{O}(N)$ 和酉群 $\mathrm{U}(N)$（其中 $N=2^n$）的 $\varepsilon$-近似 $k$-设计的强显式构造。我们的设计基数为 $\mathrm{poly}(N^k/\varepsilon)$（等价地，其种子长度为 $O(nk + \log(1/\varepsilon))$）；在多项式意义上，这与完全随机矩阵构造所使用的设计元素数量相匹配。