Researchers tested how well a simple neural quantum state can represent the ground states of medium-mass atomic nuclei. They found that when a nuclear state contains more “non-stabilizerness” (also called quantum magic), the neural network is systematically worse at learning it. In plain terms, certain kinds of quantum complexity make the nuclei harder to compress into a compact neural description.

The team adapted a second-quantized neural quantum state method to nuclear physics. They used a Restricted Boltzmann Machine (RBM) ansatz defined on occupation-number configurations. In this view each orbital becomes a visible node that records whether a proton or neutron occupies it. Hidden nodes capture correlations between occupations. The calculations were carried out in the sd-shell active space with a well-known nuclear Hamiltonian (the USDB interaction). Exact diagonalization inside that active space, called the interacting shell model, provided the reference ground states.

To measure learning quality the authors maximized fidelity, which tracks how close the RBM state is to the exact ground state. They also optimized energies using variational Monte Carlo. To quantify the kind of quantum complexity present in the nuclei they used a stabilizer Rényi entropy M2: a higher M2 means more non-stabilizerness or quantum magic. The RBM’s size was controlled by the hidden-unit density α, and the number of free parameters grows roughly like α times the square of the number of visible nodes.

The main numerical finding is clear. For a fixed budget of configurations, states with larger M2 were harder to represent accurately. Increasing the network size (larger α) improved accuracy overall, but the correlation between higher non-stabilizerness and lower fidelity remained. For example, 24Mg in the chosen symmetry sector had about 28,503 many-body configurations, showed the largest non-stabilizerness in the set, and was represented with the lowest fidelity. The authors also note that simple single-orbital entanglement measures did not explain the pattern of errors; the multi-partite, proton–neutron features and non-stabilizerness correlated with representational difficulty instead.