arXiv:2306.17479 · Physical Review Research

Nuclear norm regularized loop optimization for tensor network

Kenji Homma, Tsuyoshi Okubo, Naoki Kawashima

Published Nov 5, 2024• 19 citations

Abstract

We propose a loop optimization algorithm based on nuclear norm regularization for the tensor network. The key ingredient of this scheme is to introduce a rank penalty term proposed in the context of data processing. Compared to the standard variational periodic matrix product states method, this algorithm can circumvent the local minima related to short-ranged correlation in a simpler fashion. We demonstrate its performance when used as a part of the tensor network renormalization algorithms [S. Yang, Z.-C. Gu, and X.-G. Wen, ] for the critical two-dimensional Ising model. The scale invariance of the renormalized tensors is attained with higher accuracy while the higher parts of the scaling dimension spectrum are obtained in a more stable fashion. Published by the American Physical Society 2024

DOI: 10.1103/physrevresearch.6.043102

Related papers

Loop Optimization for Tensor Network Renormalization

Shuo Yang, Zheng‐Cheng Gu, Xiao-Gang Wen

We introduce a tensor renormalization group scheme for coarse graining a two-dimensional tensor network that can be successfully applied to both classical and quantum systems on and off criticality. The key innovation in our scheme is to deform a 2D tensor network into small loops and then optimize the tensors on each loop. In this way, we remove short-range entanglement at each iteration step and significantly improve the accuracy and stability of the renormalization flow. We demonstrate our algorithm in the classical Ising model and a frustrated 2D quantum model.

arXiv:1512.04938Physical Review LettersMar 15, 2017167 citations

Variational Adiabatic Transport of Tensor Networks

Hyeongjin Kim, Matthew Fishman, Dries Sels

We discuss a tensor network method for constructing the adiabatic gauge potential—the generator of adiabatic transformations—as a matrix product operator, which allows us to adiabatically transport matrix product states. Adiabatic evolution of tensor networks offers a wide range of applications, of which two are explored in this paper: improving tensor network optimization and scanning phase diagrams. By efficiently transporting eigenstates to quantum criticality and performing intermediary density-matrix renormalization group (DMRG) optimizations along the way, we demonstrate that we can compute ground and low-lying excited states faster and more reliably than a standard DMRG method at or near quantum criticality. We demonstrate a simple automated step size adjustment and detection of the critical point based on the norm of the adiabatic gauge potential. Remarkably, we are able to reliably transport states through the critical point of the models we study. Published by the American Physical Society 2024

arXiv:2311.00748PRX QuantumJun 14, 20248 citations

Global optimization of tensor renormalization group using the corner transfer matrix

Satoshi Morita, Naoki Kawashima

A tensor network renormalization algorithm with global optimization based on the corner transfer matrix is proposed. Since the environment is updated by the corner transfer matrix renormalization group method, the forward-backward iteration is unnecessary, which is a time-consuming part of other methods with global optimization. In addition, a further approximation reducing the order of the computational cost of contraction for the calculation of the coarse-grained tensor is proposed. The computational time of our algorithm in two dimensions scales as the sixth power of the bond dimension, while the higher-order tensor renormalization group and the higher-order second renormalization group methods have the seventh power. We perform benchmark calculations in the Ising model on the square lattice and show that the time-to-solution of the proposed algorithm is faster than that of other methods.

arXiv:2009.01997Physical review. B./Physical review. BJan 22, 202112 citations

Tensor Network Renormalization

Glen Evenbly, Guifré Vidal

We introduce a coarse-graining transformation for tensor networks that can be applied to study both the partition function of a classical statistical system and the Euclidean path integral of a quantum many-body system. The scheme is based upon the insertion of optimized unitary and isometric tensors (disentanglers and isometries) into the tensor network and has, as its key feature, the ability to remove short-range entanglement or correlations at each coarse-graining step. Removal of short-range entanglement results in scale invariance being explicitly recovered at criticality. In this way we obtain a proper renormalization group flow (in the space of tensors), one that in particular (i) is computationally sustainable, even for critical systems, and (ii) has the correct structure of fixed points, both at criticality and away from it. We demonstrate the proposed approach in the context of the 2D classical Ising model.

arXiv:1412.0732Physical Review LettersOct 29, 2015362 citations

Variational boundary based tensor network renormalization group

Feng-Feng Song, Naoki Kawashima

We propose a real-space renormalization group algorithm for accurately coarse-graining two-dimensional tensor networks. The central innovation of our method lies in utilizing variational boundary tensors as a globally optimized environment for the entire system. Based on this optimized environment, we construct renormalization projectors that significantly enhance accuracy. By leveraging the canonical form of tensors, our algorithm maintains the same computational complexity as the original tensor renormalization group (TRG) method, yet achieves higher accuracy than existing approaches that do not incorporate entanglement filtering. Our work offers a practical pathway for extending TRG methods to higher dimensions while keeping computational costs manageable.

arXiv:2508.10418Physical review. B./Physical review. BDec 10, 20251 citation

Automatic structural search of tensor network states including entanglement renormalization

Ryo Watanabe, Hiroshi Ueda

Tensor network (TN) states, including entanglement renormalization (ER), can encompass a wider variety of entangled states. When the entanglement structure of the quantum state of interest is nonuniform in real space, accurately representing the state with a limited number of degrees of freedom hinges on appropriately configuring the TN to align with the entanglement pattern. However, a proposal has yet to show a structural search of ER due to its high computational cost and the lack of flexibility in its algorithm. In this study, we conducted an optimal structural search of TN, including ER, based on the reconstruction of their local structures with respect to variational energy. First, we demonstrated that our algorithm for the spin- 1 / 2 tetramer singlets model could calculate exact ground energy using the multiscale entanglement renormalization ansatz (MERA) structure as an initial TN structure. Subsequently, we applied our algorithm to the random XY models with the two initial structures: MERA and the suitable structure underlying the strong disordered renormalization group. We found that, in both cases, our algorithm achieves improvements in variational energy, fidelity, and entanglement entropy. The degree of improvement in these quantities is superior in the latter case compared to the former, suggesting that utilizing an existing TN design method as a preprocessing step is important for maximizing our algorithm's performance. Published by the American Physical Society 2024

arXiv:2405.06534Physical Review ResearchSep 6, 20243 citations