We introduce an efficient algorithm for reducing bond dimensions in an arbitrary tensor network without changing its geometry. The method is based on a novel, quantitative understanding of local correlations in a network. Together with a tensor network coarse-graining algorithm, it yields a proper renormalization group (RG) flow. Compared to existing methods, the advantages of our algorithm are its low computational cost, simplicity of implementation, and applicability to any network. We benchmark it by evaluating physical observables for the 2D classical Ising model and find accuracy comparable with the best existing tensor network methods. Because of its graph independence, our algorithm is an excellent candidate for implementation of real-space RG in higher dimensions. We discuss some of the details and the remaining challenges in 3D. Source code for our algorithm is available as an ancillary file and on GitHub.

The critical two-dimensional classical Ising model on the square lattice has two topological conformal defects: the Z2 symmetry defect D and the Kramers-Wannier duality defect Dσ. These two defects implement antiperiodic boundary conditions and a more exotic form of twisted boundary conditions, respectively. On the torus, the partition function ZD of the critical Ising model in the presence of a topological conformal defect D is expressed in terms of the scaling dimensions ∆α and conformal spins sα of a distinct set of primary fields (and their descendants, or conformal towers) of the Ising conformal field theory. This characteristic conformal data {∆α, sα}D can be extracted from the eigenvalue spectrum of a transfer matrix MD for the partition function ZD. In this paper, we investigate the use of tensor network techniques to both represent and coarse-grain the partition functions ZD and ZD σ of the critical Ising model with either a symmetry defect D or a duality defect Dσ. We also explain how to coarse-grain the corresponding transfer matrices MD and MD σ , from which we can extract accurate numerical estimates of {∆α, sα}D and {∆α, sα}D σ . Two key new ingredients of our approach are (i) coarse-graining of the defect D, which applies to any (i.e. not just topological) conformal defect and yields a set of associated scaling dimensions ∆α, and (ii) construction and coarse-graining of a generalized translation operator using a local unitary transformation that moves the defect, which only exist for topological conformal defects and yields the corresponding conformal spins sα. arXiv:1512.03846v3 [cond-mat.str-el]

Several tensor networks are built of isometric tensors, i.e. tensors satisfying W\dagger W = \mathbb{1}W†W=1. Prominent examples include matrix product states (MPS) in canonical form, the multiscale entanglement renormalization ansatz (MERA), and quantum circuits in general, such as those needed in state preparation and quantum variational eigensolvers. We show how gradient-based optimization methods on Riemannian manifolds can be used to optimize tensor networks of isometries to represent e.g. ground states of 1D quantum Hamiltonians. We discuss the geometry of Grassmann and Stiefel manifolds, the Riemannian manifolds of isometric tensors, and review how state-of-the-art optimization methods like nonlinear conjugate gradient and quasi-Newton algorithms can be implemented in this context. We apply these methods in the context of infinite MPS and MERA, and show benchmark results in which they outperform the best previously-known optimization methods, which are tailor-made for those specific variational classes. We also provide open-source implementations of our algorithms.