Non-stabilizerness of Neural Quantum States

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 Rényi 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.

💡 Research Summary

The paper introduces a practical framework for quantifying non‑stabilizerness—also called “magic”—in many‑body quantum states represented by Neural Quantum States (NQS). Magic is a crucial resource for quantum computational advantage beyond what can be simulated efficiently with Clifford circuits, and its quantitative assessment has been hampered by the exponential cost of existing methods. The authors adopt the Stabilizer Rényi Entropy (SRE) as the figure of merit, focusing on the α = 2 case (M₂) because it is monotonic for pure qubit states and thus a valid resource‑theoretic measure.

Two Monte‑Carlo estimators are developed to evaluate M₂ from a variational wave function without requiring an explicit tensor‑network representation. The first, the “replicated estimator,” constructs a four‑fold replicated state |Φ⟩ = |Ψ*,Ψ*,Ψ*,Ψ⟩ and measures the expectation value of a non‑local operator U that couples the replicas. By sampling configurations from the Born distribution of |Φ⟩, the exponential of –M₂ is obtained as ⟨Φ|U|Φ⟩. This estimator is universally applicable to any NQS but suffers from large statistical fluctuations because U is non‑local, leading to sign‑problem‑like outliers. The authors mitigate this by employing U‑statistics and an annealed importance‑sampling scheme.

The second estimator, the “Bell‑basis estimator,” exploits the identity that the expectation value of a Pauli string can be expressed as an amplitude ratio on a Bell‑state of two copies. It requires the ground state of a transformed Hamiltonian H̃ = C†(H⊗I + I⊗H*)C, where C is a Clifford circuit that entangles the two copies. Sampling the Born distribution of the resulting state |Γ⟩ yields exp(–M₂) directly. This method is statistically more stable because the sampling distribution is proportional to the estimator itself, but it is limited to ground‑state calculations and demands knowledge of the transformed ground state, which can be difficult for complex models. Consequently, the authors employ the replicated estimator for all the main results.

A detailed sample‑complexity analysis shows that if SRE grows at most logarithmically with system size N, the number of Monte‑Carlo samples required scales polynomially, making the estimators efficient. For generic volume‑law states where SRE scales linearly with N, the required samples grow exponentially, yet still far more favorable than the exact 2^{2N} cost.

The methodology is first benchmarked on an ensemble of random Restricted‑Boltzmann‑Machine (RBM) NQS with zero visible/hidden biases, complex weights drawn uniformly from a prescribed range, and a hidden‑to‑visible ratio α = 1. For system sizes up to N = 60, the averaged M₂ exhibits a clear linear dependence on N, yielding a magic density m₂ ≈ 0.241 ± 0.005 in the thermodynamic limit. This demonstrates that NQS can simultaneously encode extensive entanglement and a finite amount of magic, confirming their expressive power beyond that of conventional tensor‑network states.

The authors then apply the framework to the J₁‑J₂ Heisenberg model. In one dimension, the SRE vanishes at the Majumdar‑Ghosh point (J₂ = J₁/2), consistent with the known stabilizer nature of the exact dimerized ground state. In two dimensions, a pronounced dip in M₂ appears around J₂/J₁ ≈ 0.6, the region of maximal frustration. The authors interpret this dip as a signature of a Valence‑Bond‑Solid (VBS) phase separating two antiferromagnetic orders, aligning with previous theoretical expectations.

Overall, the paper establishes that Neural Quantum States, combined with Monte‑Carlo estimators of Stabilizer Rényi Entropy, provide a scalable and versatile tool for probing magic in strongly correlated quantum systems of arbitrary dimension. This opens new avenues for exploring the interplay between entanglement, magic, and phase transitions, and for assessing quantum computational resources in many‑body physics.