We investigate the full counting statistics of charge transport in U(1)-symmetric random unitary circuits. We consider an initial mixed state prepared with a chemical potential imbalance between the left and right halves of the system and study the fluctuations of the charge transferred across the central bond in typical circuits. Using an effective replica statistical mechanics model and a mapping onto an emergent classical stochastic process valid at large on-site Hilbert space dimension, we show that charge transfer fluctuations approach those of the symmetric exclusion process at long times, with subleading $t^{-1/2}$ quantum corrections. We discuss our results in the context of fluctuating hydrodynamics and macroscopic fluctuation theory of classical nonequilibrium systems and check our predictions against direct matrix-product state calculations.

• Received 10 March 2023
• Revised 28 June 2023
• Accepted 31 October 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.210402