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

Introduction

The Porter‑Thomas distribution is a hallmark of Haar‑random quantum states: measurement probabilities follow an exponential law, guaranteeing strong anticoncentration—a spread of probability across exponentially many outcomes. This property underpins the hardness arguments for random‑circuit‑sampling experiments. In the recent work Fermionic Porter‑Thomas statistics and their emergence in random matchgate circuits, Guglielmo Lami extends the concept to the realm of free‑fermion systems. By analysing random fermionic Gaussian states, both with and without particle‑number conservation, the study uncovers a fermionic analogue of the Porter‑Thomas law, explores its statistical features, and demonstrates how it naturally arises in one‑dimensional brickwork matchgate circuits.

Fermionic Porter‑Thomas Law

For a system of \(n\) fermionic modes, a random Gaussian state is generated by applying a Haar‑random orthogonal transformation to the mode operators. Lami derives the exact probability distribution of measurement outcomes in the occupation‑number basis. The resulting law resembles the classic Porter‑Thomas form but with key differences:

• Log‑normal tails – In the large‑\(n\) limit the probabilities become log‑normally distributed rather than purely exponential.
• Polynomial moments – The moments of the distribution grow only polynomially with system size, indicating that free‑fermion states anticoncentrate only up to polynomial factors, unlike the exponential anticoncentration of Haar‑random states.
• Particle‑number sectors – When particle‑number conservation is imposed, the distribution acquires an additional combinatorial factor reflecting the fixed‑particle‑number subspace.

The analysis also covers Clifford Gaussian states, which map to a simple arcs‑and‑loops model. This mapping provides an intuitive picture of how correlations between modes shape the probability landscape.

Emergence in Random Matchgate Circuits

Matchgate circuits are a class of quantum circuits that can be efficiently simulated classically because they correspond to free‑fermion dynamics. Lami studies a random brickwork architecture where matchgates act on neighboring qubits in a staggered pattern. By tracking the evolution of the underlying orthogonal matrix, the paper shows:

• Single‑arc approximation – A tractable analytical approximation that captures the full probability distribution with surprising accuracy.
• Diffusive regime – At early depths the system exhibits diffusive spreading of amplitudes, reflected in a broadening of the probability distribution.
• Quadratic relaxation time – Full convergence to the fermionic Porter‑Thomas law requires circuit depths scaling almost quadratically with \(n\). A rigorous bound is proved for the collision probability, confirming that shallow circuits do not achieve full anticoncentration.

These results illuminate how even classically simulable circuits can display rich statistical behavior reminiscent of chaotic quantum dynamics.

Benchmarking Implications

The paper connects the derived statistics to practical benchmarking of quantum devices:

• Linear cross‑entropy – Ideal free‑fermion sampling yields large linear cross‑entropy scores, suggesting that naive cross‑entropy benchmarking may overestimate quantum advantage when applied to matchgate circuits.
• Log‑based benchmark – Lami proposes a benchmark based on the logarithm of measured probabilities, which incorporates the exact fermionic normalization and provides a more faithful assessment of sampling quality.

The benchmark is designed to be robust against the polynomial anticoncentration characteristic of free‑fermion states, offering a useful tool for experimentalists working with fermionic simulators or matchgate‑based architectures.

Conclusion

By establishing a fermionic counterpart to the Porter‑Thomas distribution and tracing its emergence in random matchgate circuits, the work bridges concepts from quantum chaos, statistical physics, and classical simulation. It clarifies the limits of anticoncentration for free‑fermion systems, highlights the nuanced role of circuit depth, and proposes concrete benchmarking strategies. These insights are valuable for both theorists exploring the foundations of quantum complexity and experimentalists seeking reliable metrics for near‑term quantum devices.