aiqbench analysis
This research establishes the emergence of fermionic Porter-Thomas statistics in random matchgate circuits, demonstrating that free-fermion states exhibit only polynomial anticoncentration rather than the exponential spreading seen in Haar-random states. The study rigorously proves that achieving collision probability convergence requires circuit depths scaling almost quadratically with system size, while also proposing a specific log-based benchmark for evaluating ideal free-fermion sampling performance.
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. 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.
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.
arXiv:2611.05479 [quant-ph]
(or arXiv:2611.05479v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2611.05479 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 270 words