The non-stabilizerness of fermionic Gaussian states

TL;DR

This paper presents an efficient method to quantify nonstabilizerness in fermionic Gaussian states, enabling analysis of large systems and revealing extensive nonstabilizerness behavior similar to Haar random states, with applications in quantum phase transitions.

Contribution

The authors develop a novel sampling scheme to compute Stabilizer Renyi Entropies for large fermionic Gaussian states, overcoming previous computational challenges and enabling phase transition detection.

Findings

Fermionic Gaussian states exhibit extensive nonstabilizerness similar to Haar random states.

The nonstabilizerness converges logarithmically with system size during time evolution.

A sharp transition in nonstabilizerness is observed at phase boundaries in a 2D topological model.

Abstract

We introduce an efficient method to quantify nonstabilizerness in fermionic Gaussian states, overcoming the long-standing challenge posed by their extensive entanglement. Using a perfect sampling scheme based on an underlying determinantal point process, we compute the Stabilizer Renyi Entropies (SREs) for systems with hundreds of qubits. Benchmarking on random Gaussian states with and without particle conservation, we reveal an extensive leading behavior equal to that of Haar random states, with logarithmic subleading corrections. We support these findings with analytical calculations for a set of related quantities, the participation entropies in the computational (or Fock) basis, for which we derive an exact formula. We also investigate the time evolution of non-stabilizerness in a random unitary circuit with Gaussian gates, observing that it converges in a time that scales logarithmically with the system size. Applying the sampling algorithm to a two-dimensional free-fermionic topological model, we uncover a sharp transition in non-stabilizerness at the phase boundaries, highlighting the power of our approach in exploring different phases of quantum many-body systems, even in higher dimensions.