  The classical shadow estimation protocol is a noise-resilient and
sample-efficient quantum algorithm for learning the properties of quantum
systems. Its performance depends on the choice of a unitary ensemble, which
must be chosen by a user in advance. What is the weakest assumption that can be
made on the chosen unitary ensemble that would still yield meaningful and
interesting results? To address this question, we consider the class of
Pauli-invariant unitary ensembles, i.e. unitary ensembles that are invariant
under multiplication by a Pauli operator. This class includes many previously
studied ensembles like the local and global Clifford ensembles as well as
locally scrambled unitary ensembles. For this class of ensembles, we provide an
explicit formula for the reconstruction map corresponding to the shadow channel
and give explicit sample complexity bounds. In addition, we provide two
applications of our results. Our first application is to locally scrambled
unitary ensembles, where we give explicit formulas for the reconstruction map
and sample complexity bounds that circumvent the need to solve an
exponential-sized linear system. Our second application is to the classical
shadow tomography of quantum channels with Pauli-invariant unitary ensembles.
Our results pave the way for more efficient or robust protocols for predicting
important properties of quantum states, such as their fidelity, entanglement
entropy, and quantum Fisher information.
