Deep tensor networks with matrix product operators

Tao Xiang

This chapter introduces the tensor network representation of physical operators, especially the matrix product representation of model Hamiltonians, called the matrix product operators (MPO), and the quantum transfer matrix representation of partition functions with different boundary conditions or with an impurity. The leading eigenvalue and eigenvectors of the quantum transfer matrix determine all thermodynamic quantities. It allows us to investigate thermodynamics without solving the full energy spectra of the Hamiltonian.

Timo Felser

Matthew Fishman, Steven R. White, Edwin Miles Stoudenmire

ITensor is a system for programming tensor network calculations with an interface modeled on tensor diagrams, allowing users to focus on the connectivity of a tensor network without manually bookkeeping tensor indices. The ITensor interface rules out common programming errors and enables rapid prototyping of algorithms. After discussing the philosophy behind the ITensor approach, we show examples of each part of the interface including Index objects, the ITensor product operator, tensor factorizations, tensor storage types, algorithms for matrix product state (MPS) and matrix product operator (MPO) tensor networks, quantum number conserving block sparse tensors, and the NDTensors library. We also review publications that have used ITensor for quantum many-body physics and for other areas where tensor networks are increasingly applied. To conclude we discuss promising features and optimizations to be added in the future.

Samuel Desrosiers, Glen Evenbly, Thomas E. Baker

Eric M. Howard

Matthew J. O’Rourke, Garnet Kin‐Lic Chan

Matrix product states and matrix product operators (MPOs) are one-dimensional tensor networks that underlie the modern density matrix renormalization group (DMRG) algorithm. The use of MPOs accounts for the high level of generality and wide range of applicability of DMRG. However, current algorithms for two-dimensional (2D) tensor network states, known as projected entangled-pair states, rarely employ the associated 2D tensor network operators, projected entangled-pair operators (PEPOs), due to their computational cost and conceptual complexity. To lower these two barriers, we describe how to reformulate a PEPO into a set of tensor network operators that resemble MPOs by considering the different sets of local operators that are generated from sequential bipartitions of the 2D system. The expectation value of a PEPO can then be evaluated on the fly using only the action of MPOs and generalized MPOs at each step of the approximate contraction of the 2D tensor network. This technique allows for the simpler construction and more efficient energy evaluation of 2D Hamiltonians that contain finite-range interactions, and provides an improved strategy to encode long-range interactions that is orders of magnitude more accurate and efficient than existing schemes.