Performance analysis of multi-shot shadow estimation

Quantum 7, 1044 (2023).

https://doi.org/10.22331/q-2023-06-29-1044

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 $|O |_{mathrm{shadow}}$ introduced in [1], the variance is also related to another norm, and we denote it as the cross-shadow-norm $|O |_{mathrm{Xshadow}}$. For both random Pauli and Clifford measurements, we analyze and show the upper bounds of $|O |_{mathrm{Xshadow}}$. 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.