  We present a scalable protocol for measuring full counting statistics (FCS)
in experiments or tensor-network simulations. In this method, an ancilla in the
middle of the system acts as a turnstile, with its phase keeping track of the
time-integrated particle flux. Unlike quantum gas microscopy, the turnstile
protocol faithfully captures FCS starting from number-indefinite initial states
or in the presence of noisy dynamics. In addition, by mapping the FCS onto a
single-body observable, it allows for stable numerical calculations of FCS
using approximate tensor-network methods. We demonstrate the wide-ranging
utility of this approach by computing the FCS of the transferred magnetization
in a Floquet Heisenberg spin chain, as studied in a recent experiment with
superconducting qubits, as well as the FCS of charge transfer in random
circuits.
