AI 中文总结
本文证明随机酉量子电路的常数误差复杂度随时间近似线性增长（下界 $\Omega(T/\log T)$），改进了先前下界，方法基于电路对单个门变化的响应。
AI 中文摘要
随机酉量子电路在指数长的时间内预期是不可压缩的。我们证明，随机酉量子电路的常数误差电路复杂度随时间近似线性增长，下界为 $\Omega(T/\log T)$。该下界对所有 $2\leq T\leq 4^n$ 成立，其中 $n$ 为系统大小，且不涉及其他 $n$ 依赖。这比先前基于谱间隙和酉设计的下界改进了 $\mathrm{poly}(n)$ 因子。借鉴随机微积分、几何泛函分析和随机线性代数的见解，我们的方法利用了电路对单个门变化的响应，且无需控制对高阶酉设计的收敛。
英文摘要
A random unitary quantum circuit is expected to be incompressible for exponentially long times. We show that the constant-error circuit complexity of a random unitary circuit grows almost linearly with time as $Ω(T/\log T)$. The bound holds for all $2\leq T\leq 4^n$ where $n$ is the system size, and involves no other $n$-dependence. This improves previous lower bounds derived from spectral gaps and unitary designs by a factor of $\mathrm{poly}(n)$. Drawing on insights from stochastic calculus, geometric functional analysis, and randomized linear algebra, our approach exploits the circuit's response to variations of individual gates and requires no control over convergence to high-order unitary designs.
