Non-stabilizerness of Neural Quantum States

TL;DR

The paper addresses the challenge of quantifying non-stabilizerness (magic) in highly entangled many-body quantum states by introducing two Monte Carlo schemes to estimate Stabilizer Rényi Entropy M2 for Neural Quantum States (NQS). The replicated estimator and Bell basis estimator enable magic assessment in arbitrary variational states, leveraging NQS to overcome tensor-network limitations in higher dimensions. Across an ensemble of random NQS and the J1-J2 Heisenberg model in 1D and 2D, the authors find that random NQS maintain finite magic with volume-law entanglement, while in 1D the SRE vanishes at the Majumdar-Ghosh point, consistent with a stabilizer ground state, and in 2D a dip near maximum frustration suggests a Valence Bond Solid phase. Overall, the work demonstrates that NQS can capture both large entanglement and finite non-stabilizerness, broadening the toolkit for exploring magic in complex quantum systems and beyond Tensor Network methods.

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 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.

Figures (6)

• Figure 1: Sketch of the two approaches for estimating the SRE $M_2$ of a many-qubit quantum state $\ket{\Psi}$ through Monte Carlo sampling. The replicated estimator consists of considering four unentangled copies of $\ket{\Psi}$ and sampling the expectation value of an operator $\hat{U}$ which creates interaction between the replicas. The Bell basis estimator involves two entangled copies of $\ket{\Psi}$ and computes the non-stabilizerness by averaging amplitude ratios. Each copy $\ket{\Psi}$ in the quadrupled state $\ket{\Phi}$ and the entire doubled entangled state $\ket{\Gamma}$ are encoded as NQS. The stochastic estimates of the $M_2$ are performed by Monte Carlo sampling from the Born distributions of $\ket{\Phi}$ and $\ket{\Gamma}$.
• Figure 2: Averaged SRE $M_2$ in the ensemble of random RBM for various system sizes $N$. Each point is the mean over $10^3$ independent realizations of the ensemble. The values of the $M_2$ are estimated using $N_s = 2^{22} \approx 4 \cdot 10^6$ samples. The dashed line corresponds to a linear fit on the last $7$ data points. The inset displays the SRE density $m_2=M_2/N$ without the offset coming from the intercept of the fit. The dashed line in the inset indicates the asymptotic value $m^*_2$ of the SRE density for $N \rightarrow \infty$ which coincides with the slope of the fit.
• Figure 3: Pictorial diagram illustrating the expressive power of some Amplitude Ratio quantum states havlicek2023amplitude in terms of entanglement and non-stabilizerness. Here TN indicates all the planar Tensor Network ansätze, including MPS as well as Projected Entangled Pair States (PEPS) verstraete2004renormalization.
• Figure 4: SRE density $m_2$ in the ground state of the 1D $J_1$-$J_2$ Heisenberg model for different values of $J_2/J_1$. Results are shown for chains with $N=16, 32, 64$ spins. The values of the SRE density are estimated using $N_s \approx 10^{9}$ samples. The inset shows the size scaling of the $m_2$ at the Majumdar-Ghosh point $J_2 = J_1/2$, denoted as $m_2^{\text{MG}}$. The ground state is approximated using a complex RBM with $\alpha=4$ for $N=16$ and the ViT for $N=32, 64$. The dashed lines are guides to the eye.
• Figure 5: SRE density $m_2$ in the ground state of the 2D $J_1$-$J_2$ Heisenberg model for different values of $J_2/J_1$. Results are shown for the $6 \times 6$ and the $8 \times 8$ lattices. The values of the SRE density are estimated using $N_s \approx 4 \cdot 10^{6}$ samples. The ground state is approximated by employing the ViT architecture. The dashed lines are guides to the eye.