Final project for Phys 449: Machine Learning in Physics taken during Fall 2021 with Pooya Ronagh.

Reproducing the results of a recently published paper https://journals.aps.org/pra/abstract/10.1103/PhysRevA.104.012401, we built a recurrent neural network that can reconstruct the spin-1/2 XY model's ground state wavefunction after training on sampled data using the density matrix renormalization group (DMRG) technique.

The spin-1/2 XY model has the following Hamiltonian:

$$ H = -J\sum_{\braket{ij}}(S_i^xS_i^x + S_i^yS_j^y) $$

where $\braket{ij}$ denotes nearest neighbor pairs over a one dimensional lattice of spins. The main innovation of the paper is the incorporation of the Hamiltonian's $U(1)$ symmetry within the network's architecture, which greatly improves its convergence.

To begin training, enter at the command line:

Python main.py -json <param.json file> -system_size <N> -results_path <res_directory> -symmetric <Symmetric>

Command line parameters:

• <param.json file> : List of hyperarameters
• <N> : Size of system, Number of spins
• <res_directory> : Where to save figures
• <Symmetric> : 1 for symmetry imposed, 0 for no symmetry imposed

The JSON file contains hyperparameters that can be found and changed:

• Learning rate
• Number of epochs
• Display Epochs (When verbose = True, will show progress for every integer * display epochs, I.e if Display Epochs = 10, will display metrics every 10 epochs)
• Model parameters
• Number of hidden units (Integer)
• Verbosity level (1 or 0 [True or false])

utilities.py contains utility functions, i.e:

• Support functions for visualizing results
• Percentage of samples/data that obeys U(1)-symmetry
• Computing fidelity of RNN’s wave function
• Loading needed data/constants
• Loading observables data such as ground state wave function, and energy difference(for testing fidelity/energy diff)
• Loading data generated by DMRG(for training/sampling)

visualizations.py plots observables/metrics over epochs for different system sizes both for normal RNN and U(1)-symmetry:

• Loss plot
• Energy difference plot
• Number of generated configurations with non-zero magnetizations plot
• Quantum state fidelity plot

training.py provides functions for training the recurrent neural network. Uses stochastic gradient descent to backpropagate through the hidden layers to minimize negative loss likelihood

main.py is the main entrypoint to the code:

• Loads model hyper parameters and data
• Creates directory to store results
• Initializes RNN
• Begins training
• Displays execution time

RNN.py contains the recurrent neural network architecture, which is as the class RNN

• data contains data from DMRG
• final_results contains final results presented after training
• notebooks contains notebooks that were used to plot results
• params contains the json file with the values used to analyze the data

• Jefferson Pule Mendez
• Luc Andre Ouellet
• Uzair Lakhani
• Sam Yu