We present a scalable protocol for measuring full counting statistics (FCS) in experiments or tensor network simulations. In this method, an auxiliary member in the center of the system acts as a turnstile whose phase tracks the time-integrated particle flux. Unlike quantum gas microscopy, the turnstile protocol faithfully captures he FCS starting from an indeterminate number of initial states or in the presence of noisy dynamics. Moreover, mapping the FCS to a single observation enables stable numerical computation of his FCS using approximate tensor network methods. We demonstrate the broad utility of this approach by calculating the FCS of the transition magnetization in the Flockett-Heisenberg spin chain studied in recent experiments with superconducting qubits and the FCS of the charge transition in random circuits. Demonstrate.
