Non-Local Magic Resources for Fermionic Gaussian States

TL;DR

This paper introduces a polynomial-time method to evaluate non-local magic in fermionic Gaussian states using eigenvalues of the covariance matrix, enabling scalable analysis of quantum complexity.

Contribution

It provides a closed-form expression for non-local stabilizer entropies in fermionic Gaussian states, facilitating efficient computation from correlation functions.

Findings

Derived an exact Page-like curve for random states.

Revealed logarithmic scaling at the XY model's quantum critical point.

Established a quasiparticle picture for magic during quantum quenches.

Abstract

Entanglement and magic are fundamental resources that capture the complexity of quantum many-body systems. Non-local magic isolates the irreducible nonstabilizerness intrinsically tied to entanglement. However, evaluating this quantity generally requires a prohibitive minimization over the full Hilbert space, making it computationally inaccessible beyond a few qubits. Here, we overcome this bottleneck by suggesting a closed-form expression for the non-local stabilizer entropies of fermionic Gaussian states over local Gaussian unitaries, which can be evaluated in polynomial time directly from the eigenvalues of the reduced Majorana covariance matrix. We apply this framework to characterize fermionic non-local magic across diverse physical regimes: we derive an exact Page-like curve for typical random states, reveal logarithmic scaling at the quantum critical point of the XY model, and establish a quasiparticle picture for magic generation during out-of-equilibrium quantum quenches. Crucially, because our result relies solely on two-point correlation functions, it provides a scalable route for the experimental estimation of fermionic non-local magic in large-scale quantum processors via fermionic shadow tomography.