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

arXiv:2611.05479v1 Announce Type: new 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.