Operator Spreading in Random Unitary Circuits
Operator Spreading in Random Unitary Circuits
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DOI：
10.1103/physrevx.8.021014
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发表时间：
2018-04-11
影响因子：
12.5
通讯作者：
Haah, Jeongwan
中科院分区：
文献类型：
--
作者：
Nahum, Adam;Vijay, Sagar;Haah, Jeongwan
随机量子电路产生混沌量子动力学的最小结构模型，其能够捕获例如纠缠增长的普遍性质。我们提供了精确的结果和粗粒度的模型，由Haar随机幺正组成的量子电路的运营商的扩展。我们研究了1 + 1D和更高的维度，并认为粗粒度的图片结转到运营商在通用的多体系统的传播。在1 + 1维中，我们证明了无序相关器（OTOC）满足有偏扩散方程，该方程给出了OTOC空间分布的精确结果，并确定了蝶形速度v（B）。我们发现，在1 + 1D中，OTOC的“前沿”扩散加宽，宽度随时间的变化为t（1/2）。我们解决的随机电路的不同实现之间的OTOC的波动，认为它们是可以忽略不计的实现内的前端的扩大相比。转向更高维度，我们表明平均OTOC可以通过与纯粹经典的液滴生长问题的显着对应来准确理解。这意味着平均OTOC的前沿宽度在2 + 1D中为t（1/3），在3 + 1D中为t（0.240）（Kardar-Parisi-Zhang普适类的指数）。我们支持我们的分析论点与模拟2 + 1D。我们指出，在两个或更高的空间维度，在后期的扩展运营商的形状是受潜在的晶格对称性，在一般情况下，不是球形的。然而，当全空间旋转对称存在于2 + 1D，我们的映射意味着一个确切的渐近形式的OTOC，在Tracy-Widom分布。对于1 + 1D中的OTOC的另一种观点，我们将其映射到Ising类统计力学模型的配分函数。由于幺正性引起的特殊结构，这种配分函数简化为可以精确执行的随机游走计算。我们也使用这个映射给出1 + 1D电路中纠缠增长的精确结果。
Random quantum circuits yield minimally structured models for chaotic quantum dynamics, which are able to capture, for example, universal properties of entanglement growth. We provide exact results and coarse-grained models for the spreading of operators by quantum circuits made of Haar-random unitaries. We study both 1 + 1D and higher dimensions and argue that the coarse-grained pictures carry over to operator spreading in generic many-body systems. In 1 + 1D, we demonstrate that the out-of-time-order correlator (OTOC) satisfies a biased diffusion equation, which gives exact results for the spatial profile of the OTOC and determines the butterfly speed v(B). We find that in 1 + 1D, the "front" of the OTOC broadens diffusively, with a width scaling in time as t(1/2). We address fluctuations in the OTOC between different realizations of the random circuit, arguing that they are negligible in comparison to the broadening of the front within a realization. Turning to higher dimensions, we show that the averaged OTOC can be understood exactly via a remarkable correspondence with a purely classical droplet growth problem. This implies that the width of the front of the averaged OTOC scales as t(1/3) in 2 + 1D and as t(0.240) in 3 + 1D (exponents of the Kardar-Parisi-Zhang universality class). We support our analytic argument with simulations in 2 + 1D. We point out that, in two or higher spatial dimensions, the shape of the spreading operator at late times is affected by underlying lattice symmetries and, in general, is not spherical. However, when full spatial rotational symmetry is present in 2 + 1D, our mapping implies an exact asymptotic form for the OTOC, in terms of the Tracy-Widom distribution. For an alternative perspective on the OTOC in 1 + 1D, we map it to the partition function of an Ising-like statistical mechanics model. As a result of special structure arising from unitarity, this partition function reduces to a random walk calculation which can be performed exactly. We also use this mapping to give exact results for entanglement growth in 1 + 1D circuits.