Stable and efficient differentiation of tensor network algorithms

Abstract

Gradient based optimization methods are the established state-of-the-art paradigm to study strongly entangled quantum systems in two dimensions with Projected Entangled Pair States. However, the key ingredient, the gradient itself, has proven challenging to calculate accurately and reliably in the case of a corner transfer matrix (CTM)-based approach. Automatic differentiation (AD), which is the best known tool for calculating the gradient, still suffers some crucial shortcomings. Some of these are known, like the problem of excessive memory usage and the divergences which may arise when differentiating a singular value decomposition (SVD). Importantly, we also find that there is a fundamental inaccuracy in the currently used backpropagation of SVD that had not been noted before. In this paper, we describe all these problems and provide them with compact and easy to implement solutions. We analyse the impact of these changes and find that the last problem -- the use of the correct gradient -- is by far the dominant one and thus should be considered a crucial patch to any AD application that makes use of an SVD for truncation.

References in corpus (6)

• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• Tensor renormalization group approach to 2D classical lattice models
• Accurate determination of tensor network state of quantum lattice models in two dimensions
• Systematic construction of spin liquids on the square lattice from tensor networks with SU(2) symmetry
• Time evolution of an infinite projected entangled pair state: a neighborhood tensor update
• Determining topological order from infinite projected entangled pair states

Cited by in corpus (9)

• Critical behavior of Fredenhagen-Marcu string order parameters at topological phase transitions with emergent higher-form symmetries
• Variationally optimizing infinite projected entangled-pair states at large bond dimensions: A split corner transfer matrix renormalization group approach
• Quantum spin liquid phase in the Shastry-Sutherland model revealed by high-precision infinite projected entangled-pair states
• Accelerating two-dimensional tensor network optimization by preconditioning
• The Cytnx Library for Tensor Networks
• Forward-mode automatic differentiation for the tensor renormalization group and its relation to the impurity method
• Variational optimization of projected entangled-pair states on the triangular lattice
• Accelerating two-dimensional tensor network contractions using QR decompositions
• Emergence of Hermitian topology from non-Hermitian knots