# Automatic Differentiation for Second Renormalization of Tensor Networks

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Abstract:

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 metho...More

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Introduction

- In the course of TRG process, environment of local tensors should be considered for conducting a precise truncation through isometric renormalization transformations in tensor bases.
- Tensor renormalization group (TRG) constitutes an important methodology for accurate simulations of strongly correlated lattice models.
- ∂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, ∂TRG automatically finds the gradient through backpropagation, with which one can deeply “train” the tensor networks.

Highlights

- Facilitated by the automatic differentiation technique widely used in deep learning, we propose a uniform framework of differentiable Tensor renormalization group (∂Tensor renormalization group) that can be applied to improve various Tensor renormalization group methods, in an automatic fashion
- ∂Tensor renormalization group 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 Tensor renormalization group, ∂Tensor renormalization group automatically finds the gradient through backpropagation, with which one can deeply “train” the tensor networks
- In the course of Tensor renormalization group process, environment of local tensors should be considered for conducting a precise truncation through isometric renormalization transformations in tensor bases
- Differentiable tensor renormalization group. — Being aware of the intimate relation between the backpropagation and second renormalization group, we extend the latter to a more flexible framework, ∂Tensor renormalization group, with the help of well-developed automatic differentiation packages [39], e.g., autograd [51] and PyTorch [35, 52]
- Conclusion and outlook.— Inspired by the essential correspondence between the backpropagation algorithm and second renormalization group of tensor networks, we propose the framework of ∂Tensor renormalization group

Results

- The authors benchmark ∂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 ∂TRG manybody simulations with high efficiency and accuracy.
- In SRG, the environment of local tensors is computed recursively between different scales of a hierarchical network, with which a global optimization is feasible.
- In ∂TRG, the forward TRG process is made fully differentiable, and the renormalization transformations are optimized globally and automatically through the backpropagation.
- The authors apply ∂TRG to simulate thermal equilibrium states at finite temperature, and achieve significantly improved accuracy over previous methods [13, 19].
- The efficiency is demonstrated by implementing ∂TRG with PyTorch [35, 52], which facilitates the GPU computing and shows a high performance of about 40 times acceleration over a single CPU core.
- The authors consider two different ∂TRG schemes following the HOTRG [7] and exponential TRG (XTRG) [19], as shown in Figs.
- In Fig. 2, the authors show the accuracies of ∂TRG implementations, together with the HOTRG and HOSRG data for comparisons.
- The results are shown in Fig. 3(a), where the relative error |δ f / f | curves rise up from very small values at high temperature and increase monotonically as T decreases.
- On the other hand, the enhancement of accuracy is marginal due to the limited expressibility of the tensor network with a given bond dimension D = 32.
- [42], suggest that GPU acceleration constitutes a very promising technique to be fully explored in quantum manybody computations, in tensor network simulations.

Conclusion

- One can observe a high accuracy with an optimization depth nd = 3, which continuously improves upon increasing the bond dimension D.
- Large-scale simulations and finite-temperature phase m transition.— the authors conduct ∂TRG calculations of quantum Ising model on wide cylinders with various widths W and lengths L.
- Conclusion and outlook.— Inspired by the essential correspondence between the backpropagation algorithm and SRG of tensor networks, the authors propose the framework of ∂TRG.

Summary

- In the course of TRG process, environment of local tensors should be considered for conducting a precise truncation through isometric renormalization transformations in tensor bases.
- Tensor renormalization group (TRG) constitutes an important methodology for accurate simulations of strongly correlated lattice models.
- ∂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, ∂TRG automatically finds the gradient through backpropagation, with which one can deeply “train” the tensor networks.
- The authors benchmark ∂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 ∂TRG manybody simulations with high efficiency and accuracy.
- In SRG, the environment of local tensors is computed recursively between different scales of a hierarchical network, with which a global optimization is feasible.
- In ∂TRG, the forward TRG process is made fully differentiable, and the renormalization transformations are optimized globally and automatically through the backpropagation.
- The authors apply ∂TRG to simulate thermal equilibrium states at finite temperature, and achieve significantly improved accuracy over previous methods [13, 19].
- The efficiency is demonstrated by implementing ∂TRG with PyTorch [35, 52], which facilitates the GPU computing and shows a high performance of about 40 times acceleration over a single CPU core.
- The authors consider two different ∂TRG schemes following the HOTRG [7] and exponential TRG (XTRG) [19], as shown in Figs.
- In Fig. 2, the authors show the accuracies of ∂TRG implementations, together with the HOTRG and HOSRG data for comparisons.
- The results are shown in Fig. 3(a), where the relative error |δ f / f | curves rise up from very small values at high temperature and increase monotonically as T decreases.
- On the other hand, the enhancement of accuracy is marginal due to the limited expressibility of the tensor network with a given bond dimension D = 32.
- [42], suggest that GPU acceleration constitutes a very promising technique to be fully explored in quantum manybody computations, in tensor network simulations.
- One can observe a high accuracy with an optimization depth nd = 3, which continuously improves upon increasing the bond dimension D.
- Large-scale simulations and finite-temperature phase m transition.— the authors conduct ∂TRG calculations of quantum Ising model on wide cylinders with various widths W and lengths L.
- Conclusion and outlook.— Inspired by the essential correspondence between the backpropagation algorithm and SRG of tensor networks, the authors propose the framework of ∂TRG.

Funding

- This work was supported by the National Natural Science Foundation of China (Grant Nos. 11774420, 11834014, 11974036, and 11774398), the National R&D Program of China (Grants Nos. 2016YFA0300503, 2017YFA0302900) and German Research Foundation (DFG WE4819/3-1) under Germany’s Excellence Strategy - EXC2111 - 390814868

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