Fermionic Porter-Thomas statistics and their emergence in random matchgate circuits

Abstract

For Haar-random quantum states, measurement probabilities follow the universal Porter--Thomas distribution. This implies anticoncentration, i.e.\ the spreading of probability over exponentially many outcomes, a key property underlying random-circuit-sampling experiments. We derive the exact analogue of the Porter--Thomas law for random fermionic Gaussian states, both with and without particle-number conservation. We also consider Clifford Gaussian states, for which the problem maps onto a simple model of arcs and loops. For large systems the probabilities become log-normally distributed, with moments growing polynomially in system size. Free-fermion states thus anticoncentrate only up to polynomial factors. We then show how this fermionic Porter--Thomas law emerges in random one-dimensional brickwork matchgate circuits. A simple single-arc approximation accurately describes the full distribution of probabilities, revealing an early diffusive regime followed by a slow relaxation on times scaling as the square of the system size. We rigorously prove that depths growing almost quadratically with system size are required for the collision probability to converge. Finally, we show that ideal free-fermion sampling yields large linear cross-entropy scores, and propose a log-based benchmark with an exact fermionic normalization.