论文标题
随机量子电路是近似统一$ t $ -DESIGNS，深入$ o \ left（nt^{5+o（1）} \ right）$
Random quantum circuits are approximate unitary $t$-designs in depth $O\left(nt^{5+o(1)}\right)$
论文作者
论文摘要
随机量子电路的应用范围从量子计算和量子多体系统到黑洞的物理。这些应用中的许多应用与量子伪随机性的产生有关：已知随机量子电路近似于单一$ t $ - 设计。统一$ t $ - 设计是模仿$ t $ th时刻的概率分布。在一份开创性的论文中，布兰德·哈罗（Brandão），哈罗（Harrow）和霍德基（Horodecki）证明，在深度$ o（n t^{10.5}）$的砖瓦架构中，随机量子电路是近似统一的$ t $ designs。在这项工作中，我们重新审视了该参数，该参数将局部随机量子电路的矩量差降低了$ω（n^{ - 1} t^{ - 9.5}）$。我们将此下限提高到$ω（n^{ - 1} t^{ - 4-o（1）}）$，其中$ o（1）$项为$ 0 $ as as $ t \ to \ to \ infty $。此缩放的直接结果是，随机量子电路会在深度$ o（nt^{5+o（1）}）$中生成近似统一的$ t $ designs。我们的技术涉及GAO的量子联盟界限和Clifford集团的不合理效力。作为辅助结果，我们证明了与HAAR随机单量子单位型单位交织的随机Clifford Unitaries的快速收敛。
The applications of random quantum circuits range from quantum computing and quantum many-body systems to the physics of black holes. Many of these applications are related to the generation of quantum pseudorandomness: Random quantum circuits are known to approximate unitary $t$-designs. Unitary $t$-designs are probability distributions that mimic Haar randomness up to $t$th moments. In a seminal paper, Brandão, Harrow and Horodecki prove that random quantum circuits on qubits in a brickwork architecture of depth $O(n t^{10.5})$ are approximate unitary $t$-designs. In this work, we revisit this argument, which lower bounds the spectral gap of moment operators for local random quantum circuits by $Ω(n^{-1}t^{-9.5})$. We improve this lower bound to $Ω(n^{-1}t^{-4-o(1)})$, where the $o(1)$ term goes to $0$ as $t\to\infty$. A direct consequence of this scaling is that random quantum circuits generate approximate unitary $t$-designs in depth $O(nt^{5+o(1)})$. Our techniques involve Gao's quantum union bound and the unreasonable effectiveness of the Clifford group. As an auxiliary result, we prove fast convergence to the Haar measure for random Clifford unitaries interleaved with Haar random single qubit unitaries.
