AI 中文总结
该研究构造了一维n量子比特近似酉k-设计，通过改进现有结果，在电路深度和魔法门数量上实现近最优性，还给出克利福德群的一维局部生成集。
AI 中文摘要
我们在一维系统中构造了n量子比特的近似酉k-设计，其电路深度为O(log(n/ε) + k log k)，相对误差为ε，所需的魔法门数量为O(nk log k)。这与现有的电路深度下界Ω(log(n/ε) + k)以及所需T门数量的Ω̃(nk)相比，仅相差一个log k因子，在所有参数上同时实现了近最优性。我们的构造基于对两个现有结果的结合与改进：在Zhang等人的魔法增强电路构造中，我们将打破克利福德对称性所需的魔法块大小从O(k log k)减小到O(log k)；同时，我们改进了Chen等人的突破性构造，以构造酉群的一维局部常深度电路生成集，使其具有恒定谱隙，从而能在O(k log k)深度内实现O(log k)局部随机酉算子。作为副产品，我们提供了克利福德群的恒定大小一维局部生成集，我们认为这一结果具有独立的研究价值。我们将这两个结果与粘合引理相结合，最终证明了我们的结论。
英文摘要
We construct $n$-qubit approximate unitary $k$-designs in 1D systems, achieving circuit depth $O(\log(n/\varepsilon) + k\log k)$ with relative error $\varepsilon$ and requiring $O(nk\log k)$ magic gates. This matches existing lower bounds $Ω(\log(n/\varepsilon) + k)$ for circuit depth, and $\widetildeΩ(nk)$ 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.
Comments50 pages, 2 figures. Comments are welcome!
