Understanding how fast physical systems can resemble Haar-random unitaries is a fundamental question in physics. Many experiments of interest in quantum gravity and many-body physics, including the butterfly effect in quantum information scrambling and the Hayden-Preskill thought experiment, involve queries to a random unitary $U$ alongside its inverse $U^\dagger$, conjugate $U^*$, and transpose $U^T$. However, conventional notions of approximate unitary designs and pseudorandom unitaries (PRUs) fail to capture these experiments. In this work, we introduce and construct strong unitary designs and strong PRUs that remain robust under all such queries. Our constructions achieve the optimal circuit depth of $O(\log n)$ for systems of $n$ qubits. We further show that strong unitary designs can form in circuit depth $O(\log^2 n)$ in circuits composed of independent two-qubit Haar-random gates, and that strong PRUs can form in circuit depth $\text{poly}(\log n)$ in circuits with no ancilla qubits. Our results provide an operational proof of the fast scrambling conjecture from black hole physics: every observable feature of the fastest scrambling quantum systems reproduces Haar-random behavior at logarithmic times.

翻译：理解物理系统能以多快的速度逼近哈尔随机酉矩阵是物理学中的一个基本问题。量子引力与多体物理中的许多重要实验——包括量子信息置乱中的蝴蝶效应以及Hayden-Preskill思想实验——都涉及对随机酉矩阵$U$及其逆$U^\dagger$、共轭$U^*$和转置$U^T$的联合查询。然而，传统的近似酉设计及伪随机酉矩阵概念均无法刻画此类实验。本工作首次提出并构建了能在所有此类查询下保持鲁棒性的强酉设计与强伪随机酉矩阵。对于$n$量子比特系统，我们的构造达到了$O(\log n)$的最优电路深度。进一步证明：在由独立双量子比特哈尔随机门构成的电路中，强酉设计可在$O(\log^2 n)$电路深度内形成；而在无需辅助量子比特的电路中，强伪随机酉矩阵可在$\text{poly}(\log n)$电路深度内形成。我们的结果为黑洞物理中的快速置乱猜想提供了操作性证明：最快置乱量子系统的所有可观测特征均能在对数时间内复现哈尔随机行为。