We prove that random quantum circuits on any geometry, including a 1D line, can form approximate unitary designs over $n$ qubits in $\log n$ depth. In a similar manner, we construct pseudorandom unitaries (PRUs) in 1D circuits in $\text{poly} \log n $ depth, and in all-to-all-connected circuits in $\text{poly} \log \log n $ depth. In all three cases, the $n$ dependence is optimal and improves exponentially over known results. These shallow quantum circuits have low complexity and create only short-range entanglement, yet are indistinguishable from unitaries with exponential complexity. Our construction glues local random unitaries on $\log n$-sized or $\text{poly} \log n$-sized patches of qubits to form a global random unitary on all $n$ qubits. In the case of designs, the local unitaries are drawn from existing constructions of approximate unitary $k$-designs, and hence also inherit an optimal scaling in $k$. In the case of PRUs, the local unitaries are drawn from existing unitary ensembles conjectured to form PRUs. Applications of our results include proving that classical shadows with 1D log-depth Clifford circuits are as powerful as those with deep circuits, demonstrating superpolynomial quantum advantage in learning low-complexity physical systems, and establishing quantum hardness for recognizing phases of matter with topological order.

翻译：我们证明，任意几何结构（包括一维链）上的随机量子电路，均可在$\log n$深度内形成$n$个量子比特上的近似酉设计。类似地，我们在一维电路中以$\text{poly} \log n$深度构建伪随机酉矩阵（PRU），并在全连接电路中以$\text{poly} \log \log n$深度构建PRU。在所有三种情形中，$n$依赖关系均达到最优，且较已知结果呈指数级改进。这些浅层量子电路具有低复杂度且仅产生短程纠缠，却与具有指数级复杂度的酉矩阵不可区分。我们的构造方法通过粘合$\log n$尺度或$\text{poly} \log n$尺度量子比特块上的局部随机酉矩阵，形成所有$n$个量子比特上的全局随机酉矩阵。在设计情形中，局部酉矩阵选自现有近似酉$k$设计构造，因此同样继承了$k$的最优标度律。在PRU情形中，局部酉矩阵选自被推测可形成PRU的现有酉矩阵系综。我们结果的应用包括：证明采用一维对数深度Clifford电路的经典阴影与采用深电路的经典阴影具有同等能力，展示学习低复杂度物理系统时的超多项式量子优势，以及为识别具有拓扑序的物质相建立量子计算困难性。