Learning Quantum Spin Liquids with Neural Quantum States

Authors

Lucas Brito (lbrito2), Sam Bear (sbear1), Adam Remels (aremels)

Final Deliverables

Presentation Slides (Also found in Additional Info)
GitHub Repo
Final Writeup/Reflection

Introduction

Quantum spin liquids are a theorized state of matter proposed as a platform for quantum computing. We investigate whether neural quantum states—a form of Boltzmann machine designed to model quantum states with reinforcement learning [arXiv:1606.02318]—are capable of learning quantum spin liquid phases in one dimension.

Related work

Neural quantum states (NQS) were originally proposed in arXiv:1606.02318. There, the authors define NQS, propose a gradient descent algorithm suitable for quantum problems, and demonstrate NQS perform comparably to state-of-the-art techniques. We expect a significant portion of the project will be dedicated to reproducing those results. Our work proposes to study quantum spin liquids in one dimension by approximating the ground state of the Haldane-Shastry model (PhysRevLett.60.635, PhysRevLett.60.639). A similar study was performed in PhysRevX.7.021021, where a modified version of the Haldane-Shastry model is considered in the context of entanglement entropy.

Data

Our study does not require any data. Reinforcement learning is performed by minimizing the energy of the model, which is given in mathematical form as the model’s Hamiltonian.

Methodology

The model is a variation on the restricted Boltzmann machine, an energy-based model consisting of a visible layer and a hidden layer. Symmetries of the energy function (the Hamiltonian) such as translation invariance simply the architecture of the Boltzmann machine. Its parameters are trained by minimizing a given Hamiltonian with respect to the hidden layer. The descent is performed with stochastic reconfiguration, a form of Monte Carlo sampling.

Metrics

The Haldane-Shastry model has a known exact ground state—the Gutzwiller-projected wavefunction. Thus we may compare the NQS results (specifically the energy and correlation functions) with that of the exact solution as an accuracy metric. Our base goal is to reproduce the results of arXiv:1606.02318. Our target goal is to approximate the Haldane-Shastry ground state to a satisfactory degree of accuracy. Our stretch goal is to extend our results to other models, such as the spin-1 Haldane chain.

Ethics

The number of parameters of NQS models is not sufficiently large to warrant concerns about resource usage, and the domain of our project is not social in nature and thus there are no immediate concerns about, say, biased data or malicious usage. Should quantum spin liquids be employed for quantum computing, the broader implications of this new paradigm—e.g., post-quantum cryptography—are important ethical issues for the physics community to address. This is, however, presently a distant application of our study.