Unified Framework for Matchgate Classical Shadows

Estimating quantum fermionic properties is a computationally difficult yet crucial task for the study of electronic systems. Recent developments have begun to address this challenge by introducing classical shadows protocols relying on sampling of Fermionic Gaussian Unitaries (FGUs): a class of transformations in fermionic space which can be conveniently mapped to matchgates circuits. The different protocols proposed in the literature use different sub-ensembles of the orthogonal group $O(2n)$ to which FGUs can be associated. We propose an approach that unifies these different protocols, proving their equivalence, and deriving from it an optimal sampling scheme. We begin by demonstrating that the first three moments of the FGU ensemble associated with $SO(2n)$ and of its intersection with the Clifford group are equal, generalizing a result known for $O(2n)$ and addressing a question raised in previous works. Building on this proof, we establish the equivalence between the shadows protocols resulting from FGU ensembles analyzed in the literature. Finally, from our results, we propose a sampling scheme for a small sub-ensemble of matchgates circuits that is optimal in terms of number of gates and that inherits the performances guarantees of the previous ensembles.