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:2607.15030

Implicit differentiation of tensor network algorithms

Lander Burgelman, Anna Francuz, Paul Brehmer, Lukas Devos, Jutho Haegeman, Frank Verstraete, Bram Vanhecke

quant-phcond-mat.str-elphysics.comp-phSubmitted Jul 16, 2026• Announced Jul 17, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

Related papers

Efficient 2D Tensor Network Simulation of Quantum Systems

Yuchen Pang, Tianyi Hao, Annika Dugad +2 more

Simulation of quantum systems is challenging due to the exponential size of the state space. Tensor networks provide a systematically improvable approximation for quantum states. 2D tensor networks such as Projected Entangled Pair States (PEPS) are well-suited for key classes of physical systems and quantum circuits. However, direct contraction of PEPS networks has exponential cost, while approximate algorithms require computations with large tensors. We propose new scalable algorithms and software abstractions for PEPS-based methods, accelerating the bottleneck operation of contraction and refactorization of a tensor subnetwork. We employ randomized SVD with an implicit matrix to reduce cost and memory footprint asymptotically. Further, we develop a distributed-memory PEPS library and study accuracy and efficiency of alternative algorithms for PEPS contraction and evolution on the Stampede2 supercomputer. We also simulate a popular near-term quantum algorithm, the Variational Quantum Eigensolver (VQE), and benchmark Imaginary Time Evolution (ITE), which compute ground states of Hamiltonians.

arXiv:2006.15234•International Conference for High Performance Computing, Networking, Storage and Analysis (SC)•Nov 1, 2020•17 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

Gradient methods for variational optimisation of projected entangled-pair states

Laurens Vanderstraeten, Jutho Haegeman, Philippe Corboz +1 more

We present a conjugate-gradient method for the ground-state optimisation of projected entangled-pair states (PEPS) in the thermodynamic limit, as a direct implementation of the variational principle within the PEPS manifold. Our optimisation is based on an efficient and accurate evaluation of the gradient of the global energy functional by using effective corner environments, and is robust with respect to the initial starting points. It has the additional advantage that physical and virtual symmetries can be straightforwardly implemented. We provide the tools to compute static structure factors directly in momentum space, as well as the variance of the Hamiltonian. We benchmark our method on Ising and Heisenberg models, and show a significant improvement on the energies and order parameters as compared to algorithms based on imaginary-time evolution.

•Jun 29, 2016•0 citations

Accelerating two-dimensional tensor network optimization by preconditioning

Xing-Yu Zhang, Qi Yang, Philippe Corboz +2 more

We revisit gradient-based optimization for infinite projected entangled pair states (iPEPS), a tensor network ansatz for simulating many-body quantum systems. This approach is hindered by two major challenges: the high computational cost of evaluating energies and gradients, and an ill-conditioned optimization landscape that slows convergence. To reduce the number of optimization steps, we introduce an efficient preconditioner derived from the leading term of the metric tensor. We benchmark our method against standard optimization techniques on the Heisenberg and Kitaev models, demonstrating substantial improvements in overall computational efficiency. Our approach is broadly applicable across various contraction schemes, unit cell sizes, and Hamiltonians, highlighting the potential of preconditioned optimization to advance tensor network algorithms for strongly correlated systems.

arXiv:2511.09546•Physical review. B./Physical review. B•Feb 13, 2026•3 citations

Differentiable Programming Tensor Networks

Haijun Liao, Jin-Guo Liu, Lei Wang +1 more

Differentiable programming is a fresh programming paradigm which composes parameterized algorithmic components and optimizes them using gradient search.The concept emerges from deep learning but is not limited to training neural networks.We present the theory and practice of programming tensor network algorithms in a fully differentiable way.By formulating the tensor network algorithm as a computation graph, one can compute higher-order derivatives of the program accurately and efficiently using automatic differentiation.We present essential techniques to differentiate through the tensor networks contraction algorithms, including numerical stable differentiation for tensor decompositions and efficient backpropagation through fixed-point iterations.As a demonstration, we compute the specific heat of the Ising model directly by taking the second-order derivative of the free energy obtained in the tensor renormalization group calculation.Next, we perform gradient-based variational optimization of infinite projected entangled pair states for the quantum antiferromagnetic Heisenberg model and obtain state-ofthe-art variational energy and magnetization with moderate efforts.Differentiable programming removes laborious human efforts in deriving and implementing analytical gradients for tensor network programs, which opens the door to more innovations in tensor network algorithms and applications.

arXiv:1903.09650•Physical Review X•Sep 5, 2019•251 citations

On the origin of finite entanglement scaling

Luke Hodgkiss, Laurens Lootens, Atsushi Ueda +2 more

The concept of finite entanglement scaling forms one of the pillars on which the tensor network ecosystem is built. In this paper, we resolve the open problem of determining the actual perturbations induced by matrix product state approximations of critical systems, and we demonstrate that these can be quite different than the ones predicted by conformal field theory. To that aim, we develop a sparse linear solver to calculate the forward and backward derivatives of 2-dimensional tensor networks with respect to their defining parameters in an implicit way. This algorithm is of independent interest as it provides a primitive for the variational optimization of projected entangled pair states that circumvents the instabilities plaguing traditional automatic differentiation methods.

quant-pharXiv:2607.15124•Jul 17, 2026•0 citations
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.
Simulation of quantum systems is challenging due to the exponential size of the state space. Tensor networks provide a systematically improvable approximation for quantum states. 2D tensor networks such as Projected Entangled Pair States (PEPS) are well-suited for key classes of physical systems and quantum circuits. However, direct contraction of PEPS networks has exponential cost, while approximate algorithms require computations with large tensors. We propose new scalable algorithms and software abstractions for PEPS-based methods, accelerating the bottleneck operation of contraction and refactorization of a tensor subnetwork. We employ randomized SVD with an implicit matrix to reduce cost and memory footprint asymptotically. Further, we develop a distributed-memory PEPS library and study accuracy and efficiency of alternative algorithms for PEPS contraction and evolution on the Stampede2 supercomputer. We also simulate a popular near-term quantum algorithm, the Variational Quantum Eigensolver (VQE), and benchmark Imaginary Time Evolution (ITE), which compute ground states of Hamiltonians.
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.
We present a conjugate-gradient method for the ground-state optimisation of projected entangled-pair states (PEPS) in the thermodynamic limit, as a direct implementation of the variational principle within the PEPS manifold. Our optimisation is based on an efficient and accurate evaluation of the gradient of the global energy functional by using effective corner environments, and is robust with respect to the initial starting points. It has the additional advantage that physical and virtual symmetries can be straightforwardly implemented. We provide the tools to compute static structure factors directly in momentum space, as well as the variance of the Hamiltonian. We benchmark our method on Ising and Heisenberg models, and show a significant improvement on the energies and order parameters as compared to algorithms based on imaginary-time evolution.
We revisit gradient-based optimization for infinite projected entangled pair states (iPEPS), a tensor network ansatz for simulating many-body quantum systems. This approach is hindered by two major challenges: the high computational cost of evaluating energies and gradients, and an ill-conditioned optimization landscape that slows convergence. To reduce the number of optimization steps, we introduce an efficient preconditioner derived from the leading term of the metric tensor. We benchmark our method against standard optimization techniques on the Heisenberg and Kitaev models, demonstrating substantial improvements in overall computational efficiency. Our approach is broadly applicable across various contraction schemes, unit cell sizes, and Hamiltonians, highlighting the potential of preconditioned optimization to advance tensor network algorithms for strongly correlated systems.
Differentiable programming is a fresh programming paradigm which composes parameterized algorithmic components and optimizes them using gradient search.The concept emerges from deep learning but is not limited to training neural networks.We present the theory and practice of programming tensor network algorithms in a fully differentiable way.By formulating the tensor network algorithm as a computation graph, one can compute higher-order derivatives of the program accurately and efficiently using automatic differentiation.We present essential techniques to differentiate through the tensor networks contraction algorithms, including numerical stable differentiation for tensor decompositions and efficient backpropagation through fixed-point iterations.As a demonstration, we compute the specific heat of the Ising model directly by taking the second-order derivative of the free energy obtained in the tensor renormalization group calculation.Next, we perform gradient-based variational optimization of infinite projected entangled pair states for the quantum antiferromagnetic Heisenberg model and obtain state-ofthe-art variational energy and magnetization with moderate efforts.Differentiable programming removes laborious human efforts in deriving and implementing analytical gradients for tensor network programs, which opens the door to more innovations in tensor network algorithms and applications.
The concept of finite entanglement scaling forms one of the pillars on which the tensor network ecosystem is built. In this paper, we resolve the open problem of determining the actual perturbations induced by matrix product state approximations of critical systems, and we demonstrate that these can be quite different than the ones predicted by conformal field theory. To that aim, we develop a sparse linear solver to calculate the forward and backward derivatives of 2-dimensional tensor networks with respect to their defining parameters in an implicit way. This algorithm is of independent interest as it provides a primitive for the variational optimization of projected entangled pair states that circumvents the instabilities plaguing traditional automatic differentiation methods.