Neural-Network Quantum States for Spin-1 Systems: Spin-Basis and Parameterization Effects on Compactness of Representations

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