Abstract
Shadow estimation is an efficient method for predicting many observables of a quantum state with a statistical guarantee. In the multi-shot scenario, one performs projective measurement on the sequentially prepared state for K times after the same unitary evolution, and repeats this procedure for M rounds of random sampled unitary. As a result, there are MK times measurements in total. Here we analyze the performance of shadow estimation in this multi-shot scenario, which is characterized by the variance of estimating the expectation value of some observable O. We find that in addition to the shadow-norm O _shadow introduced in [Huang et.al.~Nat.~Phys.~2020\cite{huang2020predicting}], the variance is also related to another norm, and we denote it as the cross-shadow-norm O _Xshadow. For both random Pauli and Clifford measurements, we analyze and show the upper bounds of O _Xshadow. In particular, we figure out the exact variance formula for Pauli observable under random Pauli measurements. Our work gives theoretical guidance for the application of multi-shot shadow estimation.