Operator backflow and the classical simulation of quantum transport
Operator backflow and the classical simulation of quantum transport
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DOI：
10.1103/physrevb.105.245101
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发表时间：
2021-11
影响因子：
3.7
通讯作者：
C. von Keyserlingk;F. Pollmann;Tibor Rakovszky
中科院分区：
文献类型：
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作者：
C. von Keyserlingk;F. Pollmann;Tibor Rakovszky
张量积态在模拟多体系统的面积律纠缠态，如一维带隙哈密顿算符的基态时，已经被证明是非常强大的。这种方法对多体系统的\“动力学\”的适用性还不太清楚：在大多数情况下，所需的内存会随着时间呈指数级增长，很快就变得难以管理。新方法通过选择性地丢弃/耗散多体波函数的部分来减少所需的内存，这些部分预计对通常感兴趣的流体动力学观测值几乎没有影响：例如，一些方法丢弃与$n$点函数相关的细粒度相关性，其中$n$超过某些截止值$\ell_*$。在这项工作中，我们提出了一个理论的大小“回流校正”，即，由于丢弃这种细粒度信息而导致的系统错误。特别是，我们专注于它们对传输系数的影响。我们的研究结果表明，回流校正指数抑制的大小的截止$\ell_*$。此外，回流误差本身有一个流体动力学的扩展，我们阐明。我们测试我们的预测对随机酉电路和遍历自旋链上运行的数值模拟。这些结果导致的猜想，遍历扩散系统中的输运系数可以捕获到一个给定的精度$\exp $的内存缩放量为$\exp[\mathcal{O}（\log（\poly ^{-1}））]$，显着优于更暴力的方法所需的内存$\exp[\mathcal{O}（\mathrm{poly}（\poly ^{-1}））]$的朴素估计。
Tensor product states have proved extremely powerful for simulating the area-law entangled states of many-body systems, such as the ground states of gapped Hamiltonians in one dimension. The applicability of such methods to the \emph{dynamics} of many-body systems is less clear: the memory required grows exponentially in time in most cases, quickly becoming unmanageable. New methods reduce the memory required by selectively discarding/dissipating parts of the many-body wavefunction which are expected to have little effect on the hydrodynamic observables typically of interest: for example, some methods discard fine-grained correlations associated with $n$-point functions, with $n$ exceeding some cutoff $\ell_*$. In this work, we present a theory for the sizes of `backflow corrections', i.e., systematic errors due to discarding this fine-grained information. In particular, we focus on their effect on transport coefficients. Our results suggest that backflow corrections are exponentially suppressed in the size of the cutoff $\ell_*$. Moreover, the backflow errors themselves have a hydrodynamical expansion, which we elucidate. We test our predictions against numerical simulations run on random unitary circuits and ergodic spin-chains. These results lead to the conjecture that transport coefficients in ergodic diffusive systems can be captured to a given precision $\epsilon$ with an amount of memory scaling as $\exp[\mathcal{O}(\log(\epsilon)^2)]$, significantly better than the naive estimate of memory $\exp[\mathcal{O}(\mathrm{poly}(\epsilon^{-1}))]$ required by more brute-force methods.