Hyperbolic Restricted Boltzmann Machine Neural Quantum State

We construct the first type of non-Euclidean non-autoregressive neural quantum state (NQS) in the form of the hyperbolic Restricted Boltzmann Machine (HRBM), which is studied in the variational Monte-Carlo (VMC) setting of the Quantum Sherrington-Kirkpatrick (QSK) model whose ground state exhibits volume-law entanglement. Across a 512-fold increase in the Hilbert space dimension corresponding to a system size increase from $N=14$ to $N=24$, HRBM NQS robustly outperforms its Euclidean version, the RBM NQS, in terms of better ground state energy optimization as well as lower Renyi-2 $S_2$ and von Neumann $S_{vN}$ absolute entanglement entropy reconstruction errors. More importantly, for all QSK system sizes, HRBM NQS demonstrates a superior expressivity in faithfully reproducing the entire entanglement spectrum of the QSK model from the top eigenvalues down to the tail end across 15 orders of magnitude, while RBM NQS consistently overestimates the sub-dominant modes. This work furnishes a proof-of-concept demonstrating that hyperbolic non-autoregressive NQS ansatzë, thanks to the exponential volume of the hyperbolic geometry underlying their constructions, might be more natural at representing volume-law quantum systems than conventional Euclidean NQS. Furthermore, an interesting byproduct of this work is the polynomial scaling result of RBM-type NQS ansatzë in the QSK volume-law system as the Hilbert space increases exponentially.