Extension of TensorFlow optimizers on Riemannian manifolds that often arise in quantum mechanics (the Complex Stiefel manifold, the manifold of Choi matrices, etc).

Documentation

The documentation is available here https://qgopt.readthedocs.io

Basic Example

Here we create an example of the Stiefel manifold with canonical metric and Cayley retraction.

import QGOpt as qgo
import tensorflow as tf

stiefel_manifold = qgo.manifolds.StiefelManifold(retraction='cayley', metric='canonical')

You can create a Riemannian optimizer using the Stiefel manifold above. This optimizer works almost like TF optimizer.

learning_rate = 0.1
opt = qgo.optimizers.RAdam(stiefel_manifold, learning_rate)  # Riemannian Adam

One can create tf.Variable describing point on the Stiefel manifold. tf.Variable must have float dtype, and shape (..., n, m, 2), where (...) enumerates a manifold, n>=m for the Stiefel manifold, last index enumerates the real part [0] and the imag part [1]. You can also use real_to_complex and complex_to_real functions to switch between the float representation of a point from a manifold and the complex representation of a point from a manifold.

U = tf.random.normal((5, 100, 30, 2))  # random initial matrices
U = qgo.manifolds.real_to_complex(U)  # turns matrices to the complex repr. (shape (5, 100, 30, 2) -> (5, 100, 30))
U, _ = tf.linalg.qr(U)  # makes matrices isometric (Stiefel manifold)
U = qgo.manifolds.complex_to_real(U)  # turns matrices back to the float repr. (shape (5, 100, 30) -> (5, 100, 30, 2))
U = tf.Variable(U)  # tf.Variable to be optimized

Now you can perform an optimization step of your target function.

with tf.GradientTape() as tape:
    U_complex = qgo.manifolds.real_to_complex(U)  # turns U to the complex representation
    target = target_function(U_complex)
grad = tape.gradient(target, [U])  # gradient
opt.apply_gradients(zip([grad], [U]))  # optimization step

For more examples, see ipython notebooks and documentation.

Types of manifolds

The current version of the package includes six types of manifolds: the complex Stiefel manifold, the manifold of density matrices, the manifold of Choi matrices, the manifold of Hermitian matrices, the manifold of POVMs and the manifold of positive-definite matrices (positive-definite cone).

stiefel = qgo.manifolds.StiefelManifold()
density_matrix = qgo.manifolds.DensityMatrix()
choi_matrix = qgo.manifolds.ChoiMatrix()
hermitian_matrix = qgo.manifolds.HermitianMatrix()
positive_cone = qgo.manifolds.PositiveCone()
povm = qgo.manifolds.POVM()

For some manifolds, one can also choose a type of reaction and metric.

Types of optimizers

The current version of the package includes Riemannian versions of popular first-order optimizers that are used in Deep Learning (for more information please read arXiv:1810.00760, arXiv:2002.01113).

lr = 0.01  # learning rate
m = qgo.manifolds.StiefelManifold()  # example of a manifold
momentum = 0.9

gd_optimizer = qgo.optimizers.RSGD(m, lr)
gd_with_momentum_optimizer = qgo.optimizers.RSGD(m, lr, momentum)
nesterov_gd_with_momentum_optimizer = qgo.optimizers.RSGD(m, lr, momentum, use_nesterov=True)
adam_optimizer = qgo.optimizers.RAdam(m, lr)
amsgrad_optimizer = qgo.optimizers.RAdam(m, lr, ams=True)

Installation

Make sure you have TensorFlow >= 2.0. One can install the package from GitHub (is recommended)

pip install git+https://github.com/LuchnikovI/QGOpt

or from pypi (might be different in comparison with the current state of master)

pip install QGOpt

Papers

We have a tutorial paper and a paper with overview of possible application. If you use QGOpt we kindly ask you to cite these papers:

@Article{10.21468/SciPostPhys.10.3.079,
	title={{QGOpt: Riemannian optimization for quantum technologies}},
	author={I. A. Luchnikov and A. Ryzhov and S. N. Filippov and H. Ouerdane},
	journal={SciPost Phys.},
	volume={10},
	issue={3},
	pages={79},
	year={2021},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.10.3.079},
	url={https://scipost.org/10.21468/SciPostPhys.10.3.079},
}

@article{luchnikov2021riemannian,
  title={Riemannian geometry and automatic differentiation for optimization problems of quantum physics and quantum technologies},
  author={Luchnikov, Ilia A and Krechetov, Mikhail E and Filippov, Sergey N},
  journal={New Journal of Physics},
  volume={23},
  number={7},
  pages={073006},
  year={2021},
  publisher={IOP Publishing}
}

Riemannian gradient optimization over Stiefel manifolds for controlling open quantum systems for tasks of quantum technologies was developed in the following work (of which we were not aware before) which we also ask to cite:

@article{OzaJPA2009.42.205305,
doi = {10.1088/1751-8113/42/20/205305},
url = {https://doi.org/10.1088/1751-8113/42/20/205305},
year = 2009,
month = {may},
publisher = {{IOP} Publishing},
volume = {42},
number = {20},
pages = {205305},
author = {Anand Oza and Alexander Pechen and Jason Dominy and Vincent Beltrani and Katharine Moore and Herschel Rabitz},
title = {Optimization search effort over the control landscapes for open quantum systems with Kraus-map evolution},
journal = {Journal of Physics A: Mathematical and Theoretical},
}