(Almost) quadruply optimal unitary designs in 1D

We construct $n$-qubit approximate unitary $k$-designs in 1D systems, achieving circuit depth $O(\log (n\mathrm{/}\epsilon )+k\log k)$ with relative error $\epsilon$ and requiring $O(nk\log k)$ magic gates. This matches existing lower bounds $\mathrm{\Omega }(\log (n\mathrm{/}\epsilon )+k)$ for circuit depth, and $\tilde{\mathrm{\Omega }}(nk)$ for the required number of $T$ gates, up to a $\log k$ factor, achieving simultaneous near-optimality in all parameters. Our construction is based on a combination and refinement of two existing results. We reduce the required magic block size for breaking Clifford symmetries in the magic-augmented circuit construction of Zhang et al. from $O(k\log k)$ to $O(\log k)$. We also improve the breakthrough construction of Chen et al. to construct a generating set of 1D local constant-depth circuits for the unitary group with a constant spectral gap, making $O(\log k)$-local random unitaries realizable in depth $O(k\log k)$. As a by-product, we provide a constant-size 1D-local generating set for the Clifford group, which we expect to be of independent interest. Combining the two results with the gluing lemma, we prove the final result.