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Register a new userAutomatic Differentiation for Second Renormalization of Tensor Networks
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Tensor renormalization group (TRG) constitutes an important methodology for accurate simulations of strongly correlated lattice models. Facilitated by the automatic differentiation technique widely used in deep learning, we propose a uniform framework of differentiable TRG ($partial$TRG) that can be applied to improve various TRG methods, in an automatic fashion. Essentially, $partial$TRG systematically extends the concept of second renormalization [PRL 103, 160601 (2009)] where the tensor environment is computed recursively in the backward iteration, in the sense that given the forward process of TRG, $partial$TRG automatically finds the gradient through backpropagation, with which one can deeply train the tensor networks. We benchmark $partial$TRG in solving the square-lattice Ising model, and demonstrate its power by simulating one- and two-dimensional quantum systems at finite temperature. The deep optimization as well as GPU acceleration renders $partial$TRG manybody simulations with high efficiency and accuracy.
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We propose a second renormalization group method to handle the tensor-network states or models. This method reduces dramatically the truncation error of the tensor renormalization group. It allows physical quantities of classical tensor-network models or tensor-network ground states of quantum systems to be accurately and efficiently determined.
We present a new strategy for contracting tensor networks in arbitrary geometries. This method is designed to follow as strictly as possible the renormalization group philosophy, by first contracting tensors in an exact way and, then, performing a controlled truncation of the resulting tensor. We benchmark this approximation procedure in two dimensions against an exact contraction. We then apply the same idea to a three dimensional system. The underlying rational for emphasizing the exact coarse graining renormalization group step prior to truncation is related to monogamy of entanglement.
We discuss in detail algorithms for implementing tensor network renormalization (TNR) for the study of classical statistical and quantum many-body systems. Firstly, we recall established techniques for how the partition function of a 2D classical many-body system or the Euclidean path integral of a 1D quantum system can be represented as a network of tensors, before describing how TNR can be implemented to efficiently contract the network via a sequence of coarse-graining transformations. The efficacy of the TNR approach is then benchmarked for the 2D classical statistical and 1D quantum Ising models; in particular the ability of TNR to maintain a high level of accuracy over sustained coarse-graining transformations, even at a critical point, is demonstrated.
Automatic Differentiation (AD) allows to determine exactly the Taylor series of any function truncated at any order. Here we propose to use AD techniques for Monte Carlo data analysis. We discuss how to estimate errors of a general function of measured observables in different Monte Carlo simulations. Our proposal combines the $Gamma$-method with Automatic differentiation, allowing exact error propagation in arbitrary observables, even those defined via iterative algorithms. The case of special interest where we estimate the error in fit parameters is discussed in detail. We also present a freely available fortran reference implementation of the ideas discussed in this work.
As an intrinsically-unbiased method, quantum Monte Carlo (QMC) is of unique importance in simulating interacting quantum systems. Unfortunately, QMC often suffers from the notorious sign problem. Although generically curing sign problem is shown to be hard (NP-hard), sign problem of a given quantum model may be mitigated (sometimes even cured) by finding better choices of simulation scheme. A universal framework in identifying optimal QMC schemes has been desired. Here, we propose a general framework using automatic differentiation (AD) to automatically search for the best continuously-parameterized QMC scheme, which we call automatic differentiable sign mitigation (ADSM). We further apply the ADSM framework to the honeycomb lattice Hubbard model with Rashba spin-orbit coupling and demonstrate ADSMs effectiveness in mitigating its sign problem. For the model under study, ADSM leads a significant power-law acceleration in computation time (the computation time is reduced from $M$ to the order of $M^{ u}$ with $ uapprox 2/3$).
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