Differentiable Programming of Isometric Tensor Networks

Abstract

Differentiable programming is a new programming paradigm which enables large scale optimization through automatic calculation of gradients also known as auto-differentiation. This concept emerges from deep learning, and has also been generalized to tensor network optimizations. Here, we extend the differentiable programming to tensor networks with isometric constraints with applications to multiscale entanglement renormalization ansatz (MERA) and tensor network renormalization (TNR). By introducing several gradient-based optimization methods for the isometric tensor network and comparing with Evenbly-Vidal method, we show that auto-differentiation has a better performance for both stability and accuracy. We numerically tested our methods on 1D critical quantum Ising spin chain and 2D classical Ising model. We calculate the ground state energy for the 1D quantum model and internal energy for the classical model, and scaling dimensions of scaling operators and find they all agree with the theory well.

17 pages, 22 figures

References in corpus (18)

• On the difficulty of training Recurrent Neural Networks
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• A class of quantum many-body states that can be efficiently simulated
• Tensor renormalization group approach to 2D classical lattice models
• Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions
• Entanglement renormalization, scale invariance, and quantum criticality
• Quantum simulation with hybrid tensor networks
• From Path Integrals to Tensor Networks for AdS/CFT
• Riemannian optimization of isometric tensor networks
• TensorNetwork for Machine Learning
• Supervised Learning with Projected Entangled Pair States
• Control landscapes for two-level open quantum systems
• Optimization search effort over the control landscapes for open quantum systems with Kraus-map evolution
• Efficient Riemannian Optimization on the Stiefel Manifold via the Cayley Transform
• RG-Flow: A hierarchical and explainable flow model based on renormalization group and sparse prior
• Entanglement and Tensor Networks for Supervised Image Classification
• Adaptive Learning of Tensor Network Structures
• Learning Non-linear Wavelet Transformation via Normalizing Flow

Cited by in corpus (4)

• The ITensor Software Library for Tensor Network Calculations
• Classical Shadow Tomography with Locally Scrambled Quantum Dynamics
• Scalable and Flexible Classical Shadow Tomography with Tensor Networks
• Riemannian quantum circuit optimization for Hamiltonian simulation