论文标题
布朗动力学的电路复杂性的线性生长
Linear Growth of Circuit Complexity from Brownian Dynamics
论文作者
论文摘要
我们计算出$ n $ spins或Fermions的布朗群集的框架潜力，并具有时间依赖于所有的全部相互作用。在这两种情况下，问题都可以映射到有效的统计力学问题，我们使用路径积分方法研究。我们认为，$ k $ th框架电势在订单$ t \ sim k n + k \ log k + k +logε^{ - 1} $之后，在HAAR值的$ε$之内。使用对钻石规范的界限，这意味着此类电路能够在订单$ t \ sim k n $之后非常接近统一的$ k $ -Design。我们还考虑了与时间无关的哈密顿式系统的系统相同的问题，并认为少数时间依赖的随机性足以在线性时间内产生$ k $ - 设计，前提是基础的汉密尔顿人是量子混乱的。这些模型提供了线性复杂性生长的明确示例，在分析上也可以进行分析。
We calculate the frame potential for Brownian clusters of $N$ spins or fermions with time-dependent all-to-all interactions. In both cases the problem can be mapped to an effective statistical mechanics problem which we study using a path integral approach. We argue that the $k$th frame potential comes within $ε$ of the Haar value after a time of order $t \sim k N + k \log k + \log ε^{-1}$. Using a bound on the diamond norm, this implies that such circuits are capable of coming very close to a unitary $k$-design after a time of order $t \sim k N$. We also consider the same question for systems with a time-independent Hamiltonian and argue that a small amount of time-dependent randomness is sufficient to generate a $k$-design in linear time provided the underlying Hamiltonian is quantum chaotic. These models provide explicit examples of linear complexity growth that are also analytically tractable.
