# Automatic Differentiation for Second Renormalization of Tensor Networks

Chen Bin-Bin
Gao Yuan
Guo Yi-Bin
Liu Yuzhi
Liao Hai-Jun
Cited by: 0|Bibtex|Views120
Weibo:
Facilitated by the automatic differentiation technique widely used in deep learning, we propose a uniform framework of differentiable Tensor renormalization group that can be applied to improve various Tensor renormalization group methods, in an automatic fashion

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

Code:

Data:

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
Reference
• T. Liu, W. Li, A. Weichselbaum, J. von Delft, and G. Su, “Simplex valence-bond crystal in the spin-1 kagome heisenberg antiferromagnet,” Phys. Rev. B 91, 060403 (2015).
• H. J. Liao, Z. Y. Xie, J. Chen, Z. Y. Liu, H. D. Xie, R. Z. Huang, B. Normand, and T. Xiang, “Gapless spin-liquid ground state in the s = 1/2 kagome antiferromagnet,” Phys. Rev. Lett. 118, 137202 (2017).
• L. Chen, D.-W. Qu, H. Li, B.-B. Chen, S.-S. Gong, J. von Delft, A. Weichselbaum, and W. Li, “Two-temperature scales in the triangular lattice Heisenberg antiferromagnet,” Phys. Rev. B 99, 140404(R) (2019).
• [5] M. Levin and C. P. Nave, “Tensor renormalization group approach to two-dimensional classical lattice models,” Phys. Rev. Lett. 99, 120601 (2007).
• [6] Z.-C. Gu and X.-G. Wen, “Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order,” Phys. Rev. B 80, 155131 (2009).
• [7] Z. Y. Xie, J. Chen, M. P. Qin, J. W. Zhu, L. P. Yang, and T. Xiang, “Coarse-graining renormalization by higher-order singular value decomposition,” Phys. Rev. B 86, 045139 (2012).
• [8] G. Evenbly and G. Vidal, “Tensor network renormalization,” Phys. Rev. Lett. 115, 180405 (2015).
• [9] G. Evenbly and G. Vidal, “Tensor network renormalization yields the multiscale entanglement renormalization ansatz,” Phys. Rev. Lett. 115, 200401 (2015).
• [10] S. Yang, Z.-C. Gu, and X.-G. Wen, “Loop optimization for tensor network renormalization,” Phys. Rev. Lett. 118, 110504 (2017).
• [11] M. Bal, M. Mariën, J. Haegeman, and F. Verstraete, “Renormalization group flows of Hamiltonians using tensor networks,” Phys. Rev. Lett. 118, 250602 (2017).
• [12] H. C. Jiang, Z. Y. Weng, and T. Xiang, “Accurate determination of tensor network state of quantum lattice models in two dimensions,” Phys. Rev. Lett. 101, 090603 (2008).
• [13] W. Li, S.-J. Ran, S.-S. Gong, Y. Zhao, B. Xi, F. Ye, and G. Su, “Linearized tensor renormalization group algorithm for the calculation of thermodynamic properties of quantum lattice models,” Phys. Rev. Lett. 106, 127202 (2011).
• [14] P. Czarnik, L. Cincio, and J. Dziarmaga, “Projected entangled pair states at finite temperature: Imaginary time evolution with ancillas,” Phys. Rev. B 86, 245101 (2012).
• [15] Y.-L. Dong, L. Chen, Y.-J. Liu, and W. Li, “Bilayer linearized tensor renormalization group approach for thermal tensor networks,” Phys. Rev. B 95, 144428 (2017).
• [16] A. Kshetrimayum, M. Rizzi, J. Eisert, and R. Orús, “Tensor network annealing algorithm for two-dimensional thermal states,” Phys. Rev. Lett. 122, 070502 (2019).
• [17] P. Czarnik and J. Dziarmaga, “Variational approach to projected entangled pair states at finite temperature,” Phys. Rev. B 92, 035152 (2015).
• [18] P. Czarnik and P. Corboz, “Finite correlation length scaling with infinite projected entangled pair states at finite temperature,” arXiv:1904.02476 (2019).
• [19] B.-B. Chen, L. Chen, Z. Chen, W. Li, and A. Weichselbaum, “Exponential thermal tensor network approach for quantum lattice models,” Phys. Rev. X 8, 031082 (2018).
• [20] H. Li, B.-B. Chen, Z. Chen, J. von Delft, A. Weichselbaum, and W. Li, “Thermal tensor renormalization group simulations of square-lattice quantum spin models,” Phys. Rev. B 100, 045110 (2019).
• [21] S. R. White, “Density matrix formulation for quantum renormalization groups,” Phys. Rev. Lett. 69, 2863–2866 (1992).
• [22] Z. Y. Xie, H. C. Jiang, Q. N. Chen, Z. Y. Weng, and T. Xiang, “Second renormalization of tensor-network states,” Phys. Rev. Lett. 103, 160601 (2009).
• [23] H. H. Zhao, Z. Y. Xie, Q. N. Chen, Z. C. Wei, J. W. Cai, and T. Xiang, “Renormalization of tensor-network states,” Phys. Rev. B 81, 174411 (2010).
• [24] H.-H. Zhao, Z.-Y. Xie, T. Xiang, and M. Imada, “Tensor network algorithm by coarse-graining tensor renormalization on finite periodic lattices,” Phys. Rev. B 93, 125115 (2016).
• [25] G. Carleo and M. Troyer, “Solving the quantum many-body problem with artificial neural networks,” Science 355, 602–606 (2017).
• [26] J. Carrasquilla and R. G. Melko, “Machine learning phases of matter,” Nature Physics 13, 431–434 (2017).
• [28] E. Stoudenmire and D. J. Schwab, “Supervised learning with tensor networks,” in Advances in Neural Information Processing Systems 29, edited by D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett (Curran Associates, Inc., 2016) pp. 4799–4807.
• [29] S. Foreman, J. Giedt, Y. Meurice, and J. Unmuth-Yockey, “Examples of renormalization group transformations for image sets,” Phys. Rev. E 98, 052129 (2018).
• [30] Z.-Y. Han, J. Wang, H. Fan, L. Wang, and P. Zhang, “Unsupervised generative modeling using matrix product states,” Phys. Rev. X 8, 031012 (2018).
• [31] C. Guo, Z. M. Jie, W. Lu, and D. Poletti, “Matrix product operators for sequence-to-sequence learning,” Phys. Rev. E 98, 042114 (2018).
• [32] S.-H. Li and L. Wang, “Neural network renormalization group,” Phys. Rev. Lett. 121, 260601 (2018).
• [33] M. Koch-Janusz and Z. Ringel, “Mutual information, neural networks and the renormalization group,” Nature Physics 14, 578–582 (2018).
• [34] H.-J. Liao, J.-G. Liu, L. Wang, and T. Xiang, “Differentiable programming tensor networks,” Phys. Rev. X 9, 031041 (2019).
• [35] A. Paszke, G. Chanan, Z. Lin, S. Gross, E. Yang, L. Antiga, and Z. Devito, “Automatic differentiation in PyTorch,” in Conference on Neural Information Processing Systems (NIPS 2017) (Long beach, CA, USA).
• [36] D. E. Rumelhart, G. E. Hinton, and R. J. Williams, “Learning representations by back-propagating errors,” Nature (London) 323, 533–536 (1986).
• [37] D. B. Parker, “Learning logic report,” (1985), MIT, TR-47.
• [38] Y. LeCun, L. D. Jackel, B. Boser, J. S. Denker, H. P. Graf, I. Guyon, D. Henderson, R. E. Howard, and W. Hubbard, “Handwritten digit recognition: applications of neural network chips and automatic learning,” IEEE Communications Magazine 27, 41–46 (1989).
• [40] G. Vidal, “Entanglement renormalization,” Phys. Rev. Lett. 99, 220405 (2007).
• [41] G. Evenbly and G. Vidal, “Algorithms for entanglement renormalization,” Phys. Rev. B 79, 144108 (2009).
• [42] A. Milsted, M. Ganahl, S. Leichenauer, J. Hidary, and G. Vidal, “TensorNetwork on TensorFlow: A Spin Chain Application Usfing Tree Tensor Networks,” (2019), arXiv:1905.01331.
• [43] B. Bauer et al., “The ALPS project release 2.0: open source software for strongly correlated systems,” Journal of Statistical Mechanics: Theory and Experiment 2011, P05001 (2011).
• [46] E. Efrati, Z. Wang, A. Kolan, and L. P. Kadanoff, “Real-space renormalization in statistical mechanics,” Rev. Mod. Phys. 86, 647–667 (2014).
• [47] K. G. Wilson, “The renormalization group: Critical phenomena and the kondo problem,” Rev. Mod. Phys. 47, 773–840 (1975).
• [48] L. P. Kadanoff, “Variational principles and approximate renormalization group calculations,” Phys. Rev. Lett. 34, 1005–1008 (1975).
• [49] Y. Lecun, Y. Bengio, and G. Hinton, “Deep learning,” Nature (London) 521, 436–444 (2015).
Full Text