# Differentiable Programming Tensor Networks

## TL;DR

This paper introduces a differentiable programming framework for tensor network algorithms, enabling automatic differentiation and efficient gradient computations, which simplifies development and enhances optimization in tensor network applications.

## Contribution

It presents a novel approach to differentiable tensor network programming, including techniques for stable differentiation and backpropagation through complex tensor operations.

## Key findings

- Computed the specific heat of the Ising model via second derivatives.
- Achieved state-of-the-art variational energy and magnetization for quantum antiferromagnetic Heisenberg model.
- Demonstrated efficient gradient-based optimization in tensor network algorithms.

## Abstract

Differentiable programming is a fresh programming paradigm which composes parameterized algorithmic components and trains them using automatic differentiation (AD). The concept emerges from deep learning but is not only limited to training neural networks. We present theory and practice of programming tensor network algorithms in a fully differentiable way. By formulating the tensor network algorithm as a computation graph, one can compute higher order derivatives of the program accurately and efficiently using AD. We present essential techniques to differentiate through the tensor networks contractions, including stable AD for tensor decomposition and efficient backpropagation through fixed point iterations. As a demonstration, we compute the specific heat of the Ising model directly by taking the second order derivative of the free energy obtained in the tensor renormalization group calculation. Next, we perform gradient based variational optimization of infinite projected entangled pair states for quantum antiferromagnetic Heisenberg model and obtain start-of-the-art variational energy and magnetization with moderate efforts. Differentiable programming removes laborious human efforts in deriving and implementing analytical gradients for tensor network programs, which opens the door to more innovations in tensor network algorithms and applications.