Classical shadows provide a noise-resilient and sample-efficient method for learning quantum system properties, relying on a user-specified unitary ensemble. What is the weakest assumption on this ensemble that can still yield meaningful results? To address this, we focus on Pauli-invariant unitary ensembles—those invariant under multiplication by Pauli operators. For these ensembles, we present explicit formulas for the reconstruction map and sample complexity bounds and extend our results to the case when noise impacts the protocol implementation. Two applications are explored: one for locally scrambled unitary ensembles, where we present formulas for the reconstruction map and sample complexity bounds that circumvent the need to solve an exponential-sized linear system, and another for the classical shadows of quantum channels. Our results establish a unified framework for classical shadows with Pauli-invariant unitary ensembles, applicable to both noisy and noiseless scenarios for states and channels and primed for implementation on near-term quantum devices.

经典阴影依赖于用户指定的酉系综，提供了一种抗噪声且样本高效的方法来学习量子系统属性。这个集合中仍然可以产生有意义的结果的最弱假设是什么？为了解决这个问题，我们关注泡利不变量酉系综——泡利算子乘法下的不变量。对于这些集成，我们提出了重建图和样本复杂度界限的明确公式，并将我们的结果扩展到噪声影响协议实现的情况。探索了两种应用：一种用于局部置乱的酉系综，其中我们提出了重构图和样本复杂度界限的公式，从而避免了求解指数大小的线性系统的需要，另一种用于量子通道的经典阴影。我们的结果为具有泡利不变酉系综的经典阴影建立了一个统一的框架，适用于状态和通道的噪声和无噪声场景，并为近期量子设备上的实现做好了准备。