Neural Quantum States in Non-Stabilizer Regimes: Benchmarks with Atomic Nuclei

Abstract

As neural networks are known to efficiently represent classes of tensor-network states as well as volume-law-entangled states, identifying which properties determine the representational capabilities of neural quantum states (NQS) remains an open question. We construct NQS representations of ground states of medium-mass atomic nuclei, which typically exhibit significant entanglement and non-stabilizerness, to study their performance in relation to the quantum complexity of the target state. Leveraging a second-quantized formulation of NQS tailored for nuclear-physics applications, we perform calculations in active orbital spaces using a restricted Boltzmann machine (RBM), a prototypical NQS ansatz. For a fixed number of configurations, we find that states with larger non-stabilizerness are systematically harder to learn, as evidenced by reduced accuracy. This finding suggests that non-stabilizerness is a primary factor governing the compression and representational efficiency of RBMs in entangled regimes, and motivates extending these studies to more sophisticated network architectures.

Figures (5)

• Figure 1: Mapping the ISM to the RBM ansatz: active proton and neutron orbitals are mapped onto the $N = N_\nu + N_\pi$ visible nodes which carry the occupation numbers $n_i$ of these orbitals, and are correlated via their connections to the hidden nodes.
• Figure 2: Infidelity $\mathcal{I}_{\boldsymbol{\theta}} = 1- \mathcal{F}_{\boldsymbol{\theta}}$ of the nuclear NQS obtained from fidelity maximization, as a function of non-stabilizerness $\mathcal{M}_2(\ket{\Psi_{ex}})$ in the exact ISM state, for different values of $\alpha$. The color code denotes the number of configurations $N_\mathrm{conf}$ in the chosen symmetry sector. The number of parameters $N_{\boldsymbol{\theta}}$ in the NQS is also indicated for each $\alpha$.
• Figure 3: Relative energy error $\varepsilon_{\boldsymbol{\theta}}$ and infidelity $\mathcal{I}_{\boldsymbol{\theta}}$ of the nuclear RBMs obtained via energy minimization with VMC.
• Figure 4: Fidelity of the nuclear NQS versus multi-partite 8-orbital entanglement in the exact state, in the mixed proton-neutron (left), pure neutron (middle) and pure proton (right) sectors, for $\alpha=1$.
• Figure 5: Non-stabilizerness and bi-partite proton-neutron entanglement of the nuclear NQS versus exact values, for $\alpha=1,2,8$.