Hyperbolic restricted Boltzmann machines improve quantum state modeling
Hyperbolic Restricted Boltzmann Machine Neural Quantum State
Machine Learning
Summary
Modeling complex quantum states is difficult, especially when they have a lot of entanglement. The authors show that using a new type of model based on hyperbolic geometry (HRBM) works better than traditional Euclidean ones (RBM) for representing these states. Their model better captures energy and entanglement properties and more accurately reflects the full range of the system's quantum behaviors. This suggests that hyperbolic geometry can be a more natural way to handle complex quantum systems.
What this means in practice
- •For quantum computing engineers: Improve variational quantum algorithms by using HRBM models to better approximate states with high entanglement in quantum hardware simulations.
- •For computational physicists: Simulate complex quantum many-body systems more accurately and efficiently using hyperbolic neural quantum states as improved trial wavefunctions.
Authors
H. L. Dao
Abstract
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.