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 generalization of RBMs for spin-1 that retains the key properties of the standard spin-1/2 RBM, specifically trivial product states representations, labeling 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 anti-ferromagnetic 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. Furthermore, 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 spin-1 basis using only hidden units, compared to required in the basis. Additional VMC calculations provide strong evidence that the AKLT state in fact possesses an exact compact NQS representation in the basis with only hidden units. These insights help to further unravel how to most effectively adapt the NQS framework for more complex quantum systems.
Highlights
Rather than representing a physical degree of freedom directly with one visible unit this approach encodes the possible local physical states into a set of binary visible units. While this approach leverages the power of binary or spin- 12 Restricted Boltzmann Machines (RBMs), it multiplies the number of visible units by a factor d, significantly increasing the parameter count and complexity of the optimization
Complementary to this, here, we propose and study a direct generalization of the RBMs to spin-1 systems that retains key properties of spin- 12 RBM with a minimal increase in variational parameters
We demonstrate the effectiveness of the new formulation via variational Monte Carlo (VMC) calculations for the spin-1 anti-ferromagnetic Heisenberg (AFH) model, where it is seen to deliver the same accuracy as unary encoding but with substantially fewer variational parameters
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Rather than representing a physical degree of freedom directly with one visible unit this approach encodes the possible local physical states into a set of binary visible units While this approach leverages the power of binary or spin- 12 RBMs, it multiplies the number of visible units by a factor d, significantly increasing the parameter count and complexity of the optimization. The ability to describe arbitrary product states without hidden units, invariance of the parameterization to the values assigned to visible variables (labeling freedom), and equivalence to the tensor network formulation This leads to the introduction of new quadratic bias and interaction weights in the RBM effective energy function. We demonstrate the effectiveness of the new formulation via VMC calculations for the spin-1 AFH model, where it is seen to deliver the same accuracy as unary encoding but with substantially fewer variational parameters.
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