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:2102.03143 · Physical Review Letters

Scaling Hypothesis for Projected Entangled-Pair States

Bram Vanhecke, Juraj Hašík, Frank Verstraete, Laurens Vanderstraeten

Published Nov 8, 2022• 25 citations
arXivHTMLPDF

Abstract

We introduce a new paradigm for scaling simulations with projected entangled-pair states (PEPS) for critical strongly correlated systems, allowing for reliable extrapolations of PEPS data with relatively small bond dimensions D. The key ingredient consists of using the effective correlation length ξ for inducing a collapse of data points, f(D,χ)=f(ξ(D,χ)), for arbitrary values of D and the environment bond dimension χ. As such we circumvent the need for extrapolations in χ and can use many distinct data points for a fixed value of D. Here, we need that the PEPSs have been optimized using a fixed-χ gradient method, which can be achieved using a novel tensor-network algorithm for finding fixed points of 2D transfer matrices, or by using the formalism of backwards differentiation. We test our hypothesis on the critical 3D dimer model, the 3D classical Ising model, and the 2D quantum Heisenberg model.

DOI
10.1103/physrevlett.129.200601

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

Variational approach to projected entangled pair states at finite temperature

Piotr Czarnik, Jacek Dziarmaga

The projected entangled pair state (PEPS) ansatz can represent a thermal state in a strongly correlated system. We introduce a variational algorithm to optimize this tensor network whose essential ingredient is an auxiliary tree tensor network (TTN). Since the full tensor environment is taken into account, with increasing bond dimension the PEPS-TTN ansatz provides the exact Gibbs state. Our presentation opens with a 1D version for a matrix product state (MPS-TTN) and then generalizes to PEPS-TTN in 2D. Benchmark results in the quantum Ising model are presented.

arXiv:1503.01077•Physical Review B•Jul 29, 2015•88 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

Variationally optimizing infinite projected entangled-pair states at large bond dimensions: A split corner transfer matrix renormalization group approach

Jan Naumann, Erik Lennart Weerda, Jens Eisert +2 more

Projected entangled-pair states (PEPS) have become a powerful tool for studying quantum many-body systems in the condensed matter and quantum materials context, particularly with advances in variational energy optimization methods. A key challenge within this framework is the computational cost associated with the contraction of the two-dimensional lattice, crucial for calculating state vector norms and expectation values. The conventional approach, using the corner transfer matrix renormalization group (CTMRG), involves combining two tensor network layers, resulting in significant time and memory demands. In this work, we introduce an alternative split-CTMRG algorithm, which maintains separate PEPS layers and leverages modified environment tensors, reducing computational complexity while preserving accuracy. Benchmarks on quantum lattice models demonstrate substantial speedups for variational energy optimization, rendering this method valuable for large-scale PEPS simulations.

arXiv:2502.10298•Physical review. B./Physical review. B•Jun 6, 2025•7 citations

Projected entangled pair states with flexible geometry

Siddhartha Patra, Sukhbinder Singh, Román Orús

Projected entangled pair states (PEPS) are a class of quantum many-body states that generalize matrix product states for one-dimensional systems to higher dimensions. In recent years, PEPS have advanced understanding of strongly correlated systems, especially in two dimensions, e.g., quantum spin liquids. Typically described by tensor networks on regular lattices (e.g., square, cubic), PEPS have also been adapted for irregular graphs; however, the computational cost becomes prohibitive for dense graphs with large vertex degrees. In this paper, we present a PEPS algorithm to simulate low-energy states and dynamics defined on arbitrary, fluctuating, and densely connected graphs. We introduce a cut-off, κ ∈ N , to constrain the vertex degree of the PEPS to a set but tunable value, which is enforced in the optimization by applying a simple edge-deletion rule, allowing the geometry of the PEPS to change and adapt dynamically to the system's correlation structure. We benchmark our flexible PEPS algorithm with simulations of classical spin glasses and quantum annealing on densely connected graphs with hundreds of spins, and also study the impact of tuning κ when simulating a uniform quantum spin model on a regular (square) lattice. Our paper opens the way to apply tensor network algorithms to arbitrary, even fluctuating, background geometries.

arXiv:2407.21140•Physical Review Research•Jan 6, 2025•6 citations

Corner entanglement scaling with projected entangled pair states

Chloé Van Bastelaere, Rui-Zhen Huang, Laurens Vanderstraeten

Entanglement scaling provides a powerful probe of universal properties at quantum critical points. In two dimensions, contributions originating from a corner-shaped bipartition exhibit a universal scaling, which is determined by the underlying conformal field theory. We develop a method to extract this corner entanglement entropy from projected entangled pair states directly in the thermodynamic limit. When applied to models at a quantum critical point, we show that the corner contribution exhibits scaling with the effective correlation length, in agreement with the hypothesis of finite-entanglement scaling. Our results for the corner coefficients are consistent with other methods, demonstrating the efficiency of our method for diagnosing strongly-correlated quantum critical points in two dimensions.

cond-mat.str-elCross-listedarXiv:2609.20020•Sep 18, 2026•0 citations
We introduce a new paradigm for scaling simulations with projected entangled-pair states (PEPS) for critical strongly correlated systems, allowing for reliable extrapolations of PEPS data with relatively small bond dimensions D. The key ingredient consists of using the effective correlation length ξ for inducing a collapse of data points, f(D,χ)=f(ξ(D,χ)), for arbitrary values of D and the environment bond dimension χ. As such we circumvent the need for extrapolations in χ and can use many distinct data points for a fixed value of D. Here, we need that the PEPSs have been optimized using a fixed-χ gradient method, which can be achieved using a novel tensor-network algorithm for finding fixed points of 2D transfer matrices, or by using the formalism of backwards differentiation. We test our hypothesis on the critical 3D dimer model, the 3D classical Ising model, and the 2D quantum Heisenberg model.
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 projected entangled pair state (PEPS) ansatz can represent a thermal state in a strongly correlated system. We introduce a variational algorithm to optimize this tensor network whose essential ingredient is an auxiliary tree tensor network (TTN). Since the full tensor environment is taken into account, with increasing bond dimension the PEPS-TTN ansatz provides the exact Gibbs state. Our presentation opens with a 1D version for a matrix product state (MPS-TTN) and then generalizes to PEPS-TTN in 2D. Benchmark results in the quantum Ising model are presented.
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) have become a powerful tool for studying quantum many-body systems in the condensed matter and quantum materials context, particularly with advances in variational energy optimization methods. A key challenge within this framework is the computational cost associated with the contraction of the two-dimensional lattice, crucial for calculating state vector norms and expectation values. The conventional approach, using the corner transfer matrix renormalization group (CTMRG), involves combining two tensor network layers, resulting in significant time and memory demands. In this work, we introduce an alternative split-CTMRG algorithm, which maintains separate PEPS layers and leverages modified environment tensors, reducing computational complexity while preserving accuracy. Benchmarks on quantum lattice models demonstrate substantial speedups for variational energy optimization, rendering this method valuable for large-scale PEPS simulations.
Projected entangled pair states (PEPS) are a class of quantum many-body states that generalize matrix product states for one-dimensional systems to higher dimensions. In recent years, PEPS have advanced understanding of strongly correlated systems, especially in two dimensions, e.g., quantum spin liquids. Typically described by tensor networks on regular lattices (e.g., square, cubic), PEPS have also been adapted for irregular graphs; however, the computational cost becomes prohibitive for dense graphs with large vertex degrees. In this paper, we present a PEPS algorithm to simulate low-energy states and dynamics defined on arbitrary, fluctuating, and densely connected graphs. We introduce a cut-off, κ ∈ N , to constrain the vertex degree of the PEPS to a set but tunable value, which is enforced in the optimization by applying a simple edge-deletion rule, allowing the geometry of the PEPS to change and adapt dynamically to the system's correlation structure. We benchmark our flexible PEPS algorithm with simulations of classical spin glasses and quantum annealing on densely connected graphs with hundreds of spins, and also study the impact of tuning κ when simulating a uniform quantum spin model on a regular (square) lattice. Our paper opens the way to apply tensor network algorithms to arbitrary, even fluctuating, background geometries.
Entanglement scaling provides a powerful probe of universal properties at quantum critical points. In two dimensions, contributions originating from a corner-shaped bipartition exhibit a universal scaling, which is determined by the underlying conformal field theory. We develop a method to extract this corner entanglement entropy from projected entangled pair states directly in the thermodynamic limit. When applied to models at a quantum critical point, we show that the corner contribution exhibits scaling with the effective correlation length, in agreement with the hypothesis of finite-entanglement scaling. Our results for the corner coefficients are consistent with other methods, demonstrating the efficiency of our method for diagnosing strongly-correlated quantum critical points in two dimensions.