Neural-network Quantum States for Spin-1 systems: spin-basis and parameterization effects on compactness of representations

TL;DR

This paper extends neural network quantum states to spin-1 systems, demonstrating more efficient representations and basis-dependent compactness, with practical benchmarks on the AKLT state and the Heisenberg model.

Contribution

It introduces a direct generalization of RBMs for spin-1, showing improved efficiency and basis-dependent compactness in NQS representations.

Findings

Achieves similar accuracy with fewer parameters than unary-encoded RBMs.

Constructs an exact compact NQS for the AKLT state in the $xyz$ basis.

Provides evidence of basis-dependent NQS compactness for complex states.

Abstract

Neural network quantum states (NQS) have been widely applied to spin-1/2 systems where they have proven to be highly effective. The application to systems with larger on-site dimension, such as spin-1 or bosonic systems, has been explored less and predominantly using spin-1/2 Restricted Boltzmann Machines (RBMs) with a one-hot/unary encoding. Here we propose a more direct generalisation of RBMs for spin-1 that retains the key properties of the standard spin-1/2 RBM, specifically trivial product states representations, labelling freedom for the visible variables and gauge equivalence to the tensor network formulation. To test this new approach we present variational Monte Carlo (VMC) calculations for the spin-1 antiferromagnetic Heisenberg (AFH) model and benchmark it against the one-hot/unary encoded RBM demonstrating that it achieves the same accuracy with substantially fewer variational parameters. Further to this we investigate how the hidden unit complexity of NQS depend on the local single-spin basis used. Exploiting the tensor network version of our RBM we construct an analytic NQS representation of the Affleck-Kennedy-Lieb-Tasaki (AKLT) state in the $xyz$ spin-1 basis using only $M = 2N$ hidden units, compared to $M \sim O(N^2)$ required in the $S^z$ basis. Additional VMC calculations provide strong evidence that the AKLT state in fact possesses an exact compact NQS representation in the $xyz$ basis with only $M=N$ hidden units. These insights help to further unravel how to most effectively adapt the NQS framework for more complex quantum systems.