Non-stabilizerness of Neural Quantum States

Abstract: We introduce a methodology to estimate non-stabilizerness or "magic", a key resource for quantum complexity, with Neural Quantum States (NQS). Our framework relies on two schemes based on Monte Carlo sampling to quantify non-stabilizerness via Stabilizer R\'enyi Entropy (SRE) in arbitrary variational wave functions. When combined with NQS, this approach is effective for systems with strong correlations and in dimensions larger than one, unlike Tensor Network methods. Firstly, we study the magic content in an ensemble of random NQS, demonstrating that neural network parametrizations of the wave function capture finite non-stabilizerness besides large entanglement. Secondly, we investigate the non-stabilizerness in the ground state of the $J_1$-$J_2$ Heisenberg model. In 1D, we find that the SRE vanishes at the Majumdar-Ghosh point $J_2 = J_1/2$, consistent with a stabilizer ground state. In 2D, a dip in the SRE is observed near maximum frustration around $J_2/J_1 \approx 0.6$, suggesting a Valence Bond Solid between the two antiferromagnetic phases.