toshaoki/matrix-model

Variational Monte Carlo with generative flows for mini-BMN matrix models

Deep Quantum Geometry of Matrices

Here is the implementation of the variational quantum Monte Carlo for low-energy states in mini-BMN matrix models. The wavefunction ansatz consists of generative flows for bosons and superposition of free states for fermions. For more details please see arXiv:1906.08781.

Setup

I recommend Anaconda (https://www.anaconda.com/) for environment setup. Please install Anaconda from the website and create an environment as follows:

1. Run the command conda create -n tf tensorflow=1.13 tensorflow-probability=0.6.0 matplotlib to create an environment named tf (feel free to use other names);
2. Activate the environment by conda activate tf;
3. There are several bugs in implementations of gradients for complex matrices in TensorFlow 1.13. As a quick fix, run pip show tensorflow to see where the TensorFlow packages are located, and replace tensorflow/python/ops/linalg_grad.py in the TensorFlow directory by the copy in this folder;
4. I recommend running the tests python tests.py to make sure that the environment is setup properly. All tests should pass (in ~1h) and you will see OK after the command finishes.

Demo

Demonstration code is in demo.py; please check out the source code for its arguments and usage. As an example, the command python demo.py maf 2 1.0 -f 2 will search for the ground state of the N = 2 and nu = 1.0 mini-BMN with -f 2 fermions, using Masked Autoregressive Flows.

Structure of the repository

• algebra.py is utility code for Lie groups and algebras
• bent_identity.py includes an implementation of the bent-identity nonlinearity
• demo.py is for demonstration
• dist.py includes implementations of probability distributions
• ent.py is used to compute entanglement of the wavefunction
• linalg_grad.py should be replaced into the TensorFlow library
• obs.py includes observables and hamiltonians
• tests.py is for unit tests
• train.py includes procedures for training neural networks
• wavefunc.py implements wavefunction ansatz
• fig/ includes temporary files of plots for probability distributions from tests
• results/ is where trained parameters are stored in files
• data/ is for additional data files (spin matrices etc)