Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2411.12731 · Physical review. B./Physical review. B

Finite-size scaling on the torus with periodic projected entangled-pair states

Gleb Fedorovich, Lukas Devos, Jutho Haegeman, Laurens Vanderstraeten, Frank Verstraete, Atsushi Ueda

Published Apr 14, 2025• 3 citations
arXivHTMLPDF

Abstract

Projected entangled pair states (PEPS) provide a powerful variational class for strongly correlated ground states in two dimensions, but how to apply them to clusters with periodic boundary conditions was an open question. To solve this problem, a novel algorithm is introduced to contract PEPS on a torus and variational ground state optimization is performed using gradient methods in combination with state-of-the-art automatic differentiation. The authors observe finite-size scaling of critical (2+1)d quantum models and use this to successfully extract scaling dimensions.

DOI
10.1103/physrevb.111.165124

Related papers

Gradient optimization of finite projected entangled pair states

Wen-Yuan Liu, Shaojun Dong, Yong‐Jian Han +2 more

Projected entangled pair states (PEPS) methods have been proven to be powerful tools to solve strongly correlated quantum many-body problems in two dimensions. However, due to the high computational scaling with the virtual bond dimension $D$, in a practical application, PEPS are often limited to rather small bond dimensions, which may not be large enough for some highly entangled systems, for instance, frustrated systems. Optimization of the ground state using the imaginary time evolution method with a simple update scheme may go to a larger bond dimension. However, the accuracy of the rough approximation to the environment of the local tensors is questionable. Here, we demonstrate that by combining the imaginary time evolution method with a simple update, Monte Carlo sampling techniques and gradient optimization will offer an efficient method to calculate the PEPS ground state. By taking advantage of massive parallel computing, we can study quantum systems with larger bond dimensions up to $D=10$ without resorting to any symmetry. Benchmark tests of the method on the ${J}_{1}\ensuremath{-}{J}_{2}$ model give impressive accuracy compared with exact results.

arXiv:1611.09467•Physical review. B./Physical review. B•May 24, 2017•59 citations

Gradient methods for variational optimization of projected entangled-pair states

Laurens Vanderstraeten, Jutho Haegeman, Philippe Corboz +1 more

The authors present a new method for variationally optimizing projected entangled-pair states (PEPS) in the thermodynamic limit based on the global energy functional and novel tensor contraction strategies. It is successfully benchmarked using Ising and Heisenberg Hamiltonians. The method is completely independent from previous PEPS implementations and improves on the variational energies and order parameters as compared to other state-of-the-art PEPS simulations. The algorithm presented here should prove valuable in simulating ground-state properties of strongly correlated quantum spin systems, can be used with other variational ansatzes, and can be applied to tensor network states in general.

arXiv:1606.09170•Physical review. B./Physical review. B•Oct 14, 2016•206 citations

Efficient Projected Entangled Pair States Methods for Periodic Quantum Systems

Shaojun Dong, Chao Wang, Hao Zhang +2 more

Projected entangled pair states (PEPS) are recognized as a potent tool for exploring two-dimensional quantum many-body systems. However, a significant challenge emerges when applying conventional PEPS methodologies to systems with periodic boundary conditions (PBCs), attributed to the prohibitive computational scaling with the bond dimension. This has notably restricted the study of systems with complex boundary conditions. To address this challenge, we have developed a strategy that involves the superposition of PEPS with open boundary conditions (OBCs) to treat systems with PBCs. This approach significantly reduces the computational complexity of such systems while maintaining their translational invariance and the PBCs. We benchmark this method against the Heisenberg model and the J_{1}-J_{2} model, demonstrating its capability to yield highly accurate results at low computational costs, even for large system sizes. We further apply the method to study the Chern numbers of the hard-core Harper-Hofstadter model at different filling factors using twisted boundary conditions. This advancement significantly broadens the applicability of the PEPS approach, opening new avenues for studying a wide range of quantum many-body phenomena.

Physical Review Letters•Jun 16, 2025•3 citations

Variational methods for contracting projected entangled-pair states

Laurens Vanderstraeten, Lander Burgelman, Boris Ponsioen +5 more

Projected entangled-pair states (PEPS) provide excellent variational wave functions for ground states of strongly correlated quantum systems in two dimensions. Determining the expectation value of such a PEPS wave function requires the contraction of a two-dimensional tensor network. Here, the authors show that this contraction can itself be formulated as a variational problem, allowing one to compare the accuracies of two commonly used contraction schemes. The authors find good agreement between the different schemes, demonstrating the robustness of PEPS simulations.

arXiv:2110.12726•Physical review. B./Physical review. B•May 25, 2022•26 citations

Beyond Boundaries: efficient Projected Entangled Pair States methods for periodic quantum systems

Shaojun Dong, Chao Wang, Zhang, Hao +2 more

Projected Entangled Pair States (PEPS) are recognized as a potent tool for exploring two-dimensional quantum many-body systems. However, a significant challenge emerges when applying conventional PEPS methodologies to systems with periodic boundary conditions (PBC), attributed to the prohibitive computational scaling with the bond dimension. This has notably restricted the study of systems with complex boundary conditions. To address this challenge, we have developed a strategy that involves the superposition of PEPS with open boundary conditions (OBC) to treat systems with PBC. This approach significantly reduces the computational complexity of such systems while maintaining their translational invariance and the PBC. We benchmark this method against the Heisenberg model and the $J_1$-$J_2$ model, demonstrating its capability to yield highly accurate results at low computational costs, even for large system sizes. The techniques are adaptable to other boundary conditions, including cylindrical and twisted boundary conditions, and therefore significantly expands the application scope of the PEPS approach, shining new light on numerous applications.

arXiv:2407.15333•Jul 22, 2024•0 citations

Implicit differentiation of tensor network algorithms

Lander Burgelman, Anna Francuz, Paul Brehmer +4 more

The current leading approach to the variational optimization of projected entangled-pair states (PEPS) is based on automatic differentiation, which allows for a convenient evaluation of the energy gradient with respect to the local variational degrees of freedom. However, evaluating the energy gradient not only remains a major computational bottleneck of the optimization procedure, but also suffers from frequent numerical instabilities. In this work, we adopt recent advances in implicit differentiation techniques to address these challenges in PEPS optimization. By reformulating the core step of the gradient computation in terms of a single characteristic equation for the contraction environment, we reduce the cost of the gradient computation and improve its scaling with the problem size. By choosing a suitable parametrization of this characteristic equation based on the intrinsic symmetries of the contraction environment, we can directly remove instabilities from the global gradient computation that would otherwise arise from the derivatives of subroutines of the contraction algorithm. Finally, we demonstrate how this approach drastically simplifies the practical implementation of stable gradient-based PEPS optimization.

quant-pharXiv:2607.15030•Jul 17, 2026•0 citations
Projected entangled pair states (PEPS) provide a powerful variational class for strongly correlated ground states in two dimensions, but how to apply them to clusters with periodic boundary conditions was an open question. To solve this problem, a novel algorithm is introduced to contract PEPS on a torus and variational ground state optimization is performed using gradient methods in combination with state-of-the-art automatic differentiation. The authors observe finite-size scaling of critical (2+1)d quantum models and use this to successfully extract scaling dimensions.
Projected entangled pair states (PEPS) methods have been proven to be powerful tools to solve strongly correlated quantum many-body problems in two dimensions. However, due to the high computational scaling with the virtual bond dimension DDD, in a practical application, PEPS are often limited to rather small bond dimensions, which may not be large enough for some highly entangled systems, for instance, frustrated systems. Optimization of the ground state using the imaginary time evolution method with a simple update scheme may go to a larger bond dimension. However, the accuracy of the rough approximation to the environment of the local tensors is questionable. Here, we demonstrate that by combining the imaginary time evolution method with a simple update, Monte Carlo sampling techniques and gradient optimization will offer an efficient method to calculate the PEPS ground state. By taking advantage of massive parallel computing, we can study quantum systems with larger bond dimensions up to D=10D=10D=10 without resorting to any symmetry. Benchmark tests of the method on the ${J}_{1}\ensuremath{-}{J}_{2}$ model give impressive accuracy compared with exact results.
The authors present a new method for variationally optimizing projected entangled-pair states (PEPS) in the thermodynamic limit based on the global energy functional and novel tensor contraction strategies. It is successfully benchmarked using Ising and Heisenberg Hamiltonians. The method is completely independent from previous PEPS implementations and improves on the variational energies and order parameters as compared to other state-of-the-art PEPS simulations. The algorithm presented here should prove valuable in simulating ground-state properties of strongly correlated quantum spin systems, can be used with other variational ansatzes, and can be applied to tensor network states in general.
Projected entangled pair states (PEPS) are recognized as a potent tool for exploring two-dimensional quantum many-body systems. However, a significant challenge emerges when applying conventional PEPS methodologies to systems with periodic boundary conditions (PBCs), attributed to the prohibitive computational scaling with the bond dimension. This has notably restricted the study of systems with complex boundary conditions. To address this challenge, we have developed a strategy that involves the superposition of PEPS with open boundary conditions (OBCs) to treat systems with PBCs. This approach significantly reduces the computational complexity of such systems while maintaining their translational invariance and the PBCs. We benchmark this method against the Heisenberg model and the J_{1}-J_{2} model, demonstrating its capability to yield highly accurate results at low computational costs, even for large system sizes. We further apply the method to study the Chern numbers of the hard-core Harper-Hofstadter model at different filling factors using twisted boundary conditions. This advancement significantly broadens the applicability of the PEPS approach, opening new avenues for studying a wide range of quantum many-body phenomena.
Projected entangled-pair states (PEPS) provide excellent variational wave functions for ground states of strongly correlated quantum systems in two dimensions. Determining the expectation value of such a PEPS wave function requires the contraction of a two-dimensional tensor network. Here, the authors show that this contraction can itself be formulated as a variational problem, allowing one to compare the accuracies of two commonly used contraction schemes. The authors find good agreement between the different schemes, demonstrating the robustness of PEPS simulations.
Projected Entangled Pair States (PEPS) are recognized as a potent tool for exploring two-dimensional quantum many-body systems. However, a significant challenge emerges when applying conventional PEPS methodologies to systems with periodic boundary conditions (PBC), attributed to the prohibitive computational scaling with the bond dimension. This has notably restricted the study of systems with complex boundary conditions. To address this challenge, we have developed a strategy that involves the superposition of PEPS with open boundary conditions (OBC) to treat systems with PBC. This approach significantly reduces the computational complexity of such systems while maintaining their translational invariance and the PBC. We benchmark this method against the Heisenberg model and the J1J_1J1​-J2J_2J2​ model, demonstrating its capability to yield highly accurate results at low computational costs, even for large system sizes. The techniques are adaptable to other boundary conditions, including cylindrical and twisted boundary conditions, and therefore significantly expands the application scope of the PEPS approach, shining new light on numerous applications.
The current leading approach to the variational optimization of projected entangled-pair states (PEPS) is based on automatic differentiation, which allows for a convenient evaluation of the energy gradient with respect to the local variational degrees of freedom. However, evaluating the energy gradient not only remains a major computational bottleneck of the optimization procedure, but also suffers from frequent numerical instabilities. In this work, we adopt recent advances in implicit differentiation techniques to address these challenges in PEPS optimization. By reformulating the core step of the gradient computation in terms of a single characteristic equation for the contraction environment, we reduce the cost of the gradient computation and improve its scaling with the problem size. By choosing a suitable parametrization of this characteristic equation based on the intrinsic symmetries of the contraction environment, we can directly remove instabilities from the global gradient computation that would otherwise arise from the derivatives of subroutines of the contraction algorithm. Finally, we demonstrate how this approach drastically simplifies the practical implementation of stable gradient-based PEPS optimization.