Nonstabilizerness determining the hardness of direct fidelity estimation

Abstract: In this work, we show how the resource theory of nonstabilizerness quantifies the hardness of direct fidelity estimation protocols. In particular, the resources needed for a direct fidelity estimation conducted on generic states, such as Pauli fidelity estimation and shadow fidelity estimation protocols, grow exponentially with the stabilizer R\'enyi entropy. Remarkably, these protocols are shown to be feasible only for those states that are useless to attain any quantum speedup or advantage. This result suggests the impossibility of estimating efficiently fidelity for generic states and, at the same time, leaves the window open to those protocols specialized at directly estimating the fidelity of particular states. We then extend our results to quantum evolutions, showing that the resources needed to certify the quality of the implementation of a given unitary $U$ are governed by the nonstabilizerness in the Choi state associated with $U$, which is shown to possess a profound connection with out-of-time order correlators.