Scrambling, a process in which quantum information spreads over a complex quantum system, becoming inaccessible to simple probes, occurs in generic chaotic quantum many-body systems, ranging from spin chains to metals and even to black holes. Scrambling can be measured using out-of-time-ordered correlators (OTOCs), which are closely tied to the growth of Heisenberg operators. We present a general method to calculate OTOCs of local operators in one-dimensional systems based on approximating Heisenberg operators as matrix product operators (MPOs). Contrary to the common belief that such tensor network methods work only at early times, we show that the entire early growth region of the OTOC can be captured using an MPO approximation with modest bond dimension. We analytically establish the goodness of the approximation by showing that, if an appropriate OTOC is close to its initial value, then the associated Heisenberg operator has low entanglement across a given cut. We use the method to study scrambling in a chaotic spin chain with \(201\) sites. On the basis of these data and previous results, we conjecture a universal form for the dynamics of the OTOC near the wavefront. We show that this form collapses the chaotic spin chain data over more than 15 orders of magnitude.

加扰是一种在复杂的量子多体系统中发生的过程，在该过程中，量子信息会散布在复杂的量子系统上，而简单的探针则无法访问它，其范围从自旋链到金属，甚至到黑洞。可以使用乱序的相关器（OTOC）来测量加扰，后者与海森堡算子的增长紧密相关。我们提出一种基于将海森堡算子近似为矩阵乘积算子（MPO）的一维系统来计算本地算子的OTOC的通用方法。与通常认为这种张量网络方法仅在早期起作用的通常看法相反，我们表明可以使用具有适度键维的MPO近似来捕获OTOC的整个早期增长区域。通过证明，我们通过分析确定了近似的优度，如果合适的OTOC接近其初始值，则关联的Heisenberg算子在给定的切口处具有低纠缠度。我们使用该方法来研究混沌自旋链中的加扰\（201 \）网站。根据这些数据和先前的结果，我们推测OTOC在波前附近的动力学的通用形式。我们表明，这种形式使超过15个数量级的混沌自旋链数据崩溃。