The non-stabilizerness of fermionic Gaussian states

TL;DR

This work introduces a scalable Majorana-sampling method to quantify nonstabilizerness in fermionic Gaussian states through Stabilizer Rényi Entropies, bypassing entanglement barriers that thwart traditional approaches. By exploiting a Determinantal Point Process kernel given by the covariance matrix, the authors sample Majorana monomials exactly and connect SREs to Participation Rényi Entropies and Inverse Participation Ratios, deriving exact average formulas. They demonstrate Haar-like leading behavior of SREs in random Gaussian states with subleading logarithmic corrections, analyze dynamical generation of magic in random Gaussian circuits, and reveal phase-boundary signatures of nonstabilizerness in a 2D topological superconductor. Overall, the method provides a high-fidelity, scalable avenue to probe quantum magic across dimensions and dynamical regimes, with potential applications to phase characterization and non-equilibrium phenomena.

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 Rényi 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.

Figures (4)

• Figure 1: Average filtered SREs $\tilde{M}_{\alpha}$ ($\alpha=1,2,3$) for random free-fermionic states, as a function of the system size $L$(particle-number is not conserved). The black line represents the leading extensive term for generic (non-Gaussian) Haar states, which is $L \log 2$. We averaged over $200$ realizations of random states in ${\mathcal{G}}_{L}^{\text{pure}}$, with $5 \cdot 10^3$ Majorana samples for each state. Inset: the difference $L \log 2 - M_{\alpha}$.
• Figure 2: Average SRE density $M_{2}/L$ (left) and PRE density $S_{2}/L$ (right) for random gaussian states with fixed number of particles $N$. Different system sizes $L$ are explored (see color bar), with $n$ representing the ratio $N/L$. The participation entropy $S_{2}$ is computed analytically with Eq.\ref{['eq:ipr']}, while the SRE is estimated with Majorana sampling ($200$ realizations of random states in ${\mathcal{G}}_{L,N}^{\text{pure}}$, with $5 \cdot 10^3$ Majorana samples for each state). The black line represents the leading extensive contribution, which is the same for both Haar-random states and fermionic Gaussian states and corresponds to the first term in Eq.\ref{['eq:participation_entropies_final']}.
• Figure 3: Top: average filtered SREs $\tilde{M}_{\alpha}$ ($\alpha=1,2$) for random free-fermionic brick wall circuits (Eq. \ref{['eq:brickwall']}) as a function of time $t$(particle-number is not conserved). Dotted horizontal lines represent saturation values $\tilde{M}_{\alpha}^{\text{sat}}$, extracted from data in Fig. \ref{['fig:random_gaussian_1']}. Bottom: the difference $\Delta \tilde{M}_{\alpha}/L = (\tilde{M}_{\alpha}^{\text{sat}} - \tilde{M}_{\alpha})/L$ approaches exponential decay.
• Figure 4: Average SRE density $M_{1}/\ell^2$ in the ground state of the 2D Hamiltonian in Eq.(\ref{['eq:2D_kitaev_H']}) as a function of the chemical potential $\mu$, at $\Delta = 0$ (left) and $0.1$(right) and different lattice sizes $\ell$. The SRE is estimated using $2000$ samples per state. Error-bars are smaller than symbol size.