Classical shadows are a powerful method for learning many properties of quantum states in a sample-efficient manner, by making use of randomized measurements. Here we study the sample complexity of learning the expectation value of Pauli operators via “shallow shadows,” a recently proposed version of classical shadows in which the randomization step is effected by a local unitary circuit of variable depth $t$. We show that the shadow norm (the quantity controlling the sample complexity) is expressed in terms of properties of the Heisenberg time evolution of operators under the randomizing (“twirling”) circuit—namely the evolution of the weight distribution characterizing the number of sites on which an operator acts nontrivially. For spatially contiguous Pauli operators of weight $k$, this entails a competition between two processes: operator spreading (whereby the support of an operator grows over time, increasing its weight) and operator relaxation (whereby the bulk of the operator develops an equilibrium density of identity operators, decreasing its weight). From this simple picture we derive (i) an upper bound on the shadow norm which, for depth $t\sim \log (k)$, guarantees an exponential gain in sample complexity over the $t=0$ protocol in any spatial dimension, and (ii) quantitative results in one dimension within a mean-field approximation, including a universal subleading correction to the optimal depth, found to be in excellent agreement with infinite matrix product state numerical simulations. Our Letter connects fundamental ideas in quantum many-body dynamics to applications in quantum information science, and paves the way to highly optimized protocols for learning different properties of quantum states.

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操作员放松和经典阴影的最佳深度

经典阴影是一种强大的方法，通过利用随机测量，以样本有效的方式学习量子态的许多属性。在这里，我们研究通过“浅阴影”学习泡利算子期望值的样本复杂性，“浅阴影”是最近提出的经典阴影版本，其中随机化步骤受到可变深度的局部酉电路的影响$t$。我们证明，影子范数（控制样本复杂性的数量）是用随机化（“旋转”）电路下算子的海森堡时间演化的属性来表达的，即表征上位点数量的权重分布的演化操作员的行为并不平凡。对于空间连续的权重泡利算子$k$，这需要两个过程之间的竞争：算子扩散（算子的支持随着时间的推移而增长，增加其权重）和算子松弛（算子的大部分形成恒等算子的平衡密度，减少其权重）。从这张简单的图片中，我们得出（i）阴影范数的上限，对于深度$t～\mathrm{日志}（k）$，保证样本复杂度呈指数增长$t=0$任何空间维度的协议，以及（ii）平均场近似内一维的定量结果，包括对最佳深度的通用次领先校正，发现与无限矩阵产品状态数值模拟非常一致。我们的信将量子多体动力学的基本思想与量子信息科学的应用联系起来，并为学习量子态不同属性的高度优化协议铺平了道路。