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Neural-Network Quantum States for Spin-1 Systems: Spin-Basis and Parameterization Effects on Compactness of
Michael Y Pei1, Stephen R Clark1
1H.H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, UK.
Entropy (Basel, Switzerland)
|August 6, 2021
Summary
We introduce a new spin-1 Restricted Boltzmann Machine (RBM) for neural network quantum states (NQS). This method achieves high accuracy for spin-1 systems with fewer parameters than previous approaches.
Area of Science:
- Quantum Many-Body Physics
- Computational Quantum Physics
- Machine Learning in Physics
Background:
- Neural network quantum states (NQS) are effective for spin-1/2 systems.
- Extending NQS to larger on-site dimensions (e.g., spin-1) is less explored, often relying on spin-1/2 Restricted Boltzmann Machines (RBMs) with specific encodings.
- Existing methods for spin-1 systems face limitations in parameter efficiency and direct generalization.
Purpose of the Study:
- To propose a direct generalization of RBMs for spin-1 systems.
- To retain key properties of spin-1/2 RBMs, including trivial product states and labeling freedom.
- To investigate the efficiency and representation capabilities of the new RBM for complex quantum states.
Main Methods:
- Development of a generalized spin-1 RBM.
- Variational Monte Carlo (VMC) calculations for the spin-1 anti-ferromagnetic Heisenberg (AFH) model.
- Benchmarking against one-hot/unary encoded RBMs.
- Analysis of hidden unit complexity based on local spin basis.
- Construction of analytic NQS representations using the tensor network version of the RBM.
Main Results:
- The proposed spin-1 RBM achieves accuracy comparable to existing methods but with significantly fewer variational parameters.
- The hidden unit complexity of NQS is shown to depend on the chosen local single-spin basis.
- An analytic NQS representation of the Affleck-Kennedy-Lieb-Tasaki (AKLT) state in the xyz spin-1 basis is constructed with M=2N hidden units.
- Strong evidence suggests an exact, compact NQS representation of the AKLT state exists in the xyz basis with only M=N hidden units.
Conclusions:
- The generalized spin-1 RBM offers a more efficient approach for studying larger quantum systems.
- Basis choice significantly impacts the complexity of NQS representations.
- The findings provide crucial insights for adapting NQS frameworks to more complex quantum systems, particularly in condensed matter physics and quantum information.
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