Schedule

Projected entangled-pair states (PEPS) have become a cornerstone of tensor network methods for simulating strongly correlated quantum systems in two dimensions. The combination of infinite PEPS with the corner transfer matrix renormalization group (CTMRG) provides a powerful variational framework for studying ground states directly in the thermodynamic limit. In this talk, I will introduce the key ideas behind infinite PEPS and the CTMRG algorithm on the conventional square lattice, highlighting how efficient environment renormalization enable accurate computation of observables. I will then discuss a recent generalization of the CTMRG tailored to the triangular lattice geometry. This native triangular-lattice PEPS formulation captures the entanglement structure of the system more faithfully, at the expense of higher algorithmic complexity and contraction effort. These developments open the door to more natural and accurate simulations of frustrated quantum systems on non-square geometries.

In this introductory talk, we will examine in some detail the geometric structure of the manifold of Projected Entangled-Pair States (PEPS) and its tangent space. Next, we will discuss the challenges in computing expectation values, and consider boundary techniques, in particular through the Variational Uniform Matrix Product State (VUMPS) algorithm, as one possible strategy. In this context, we will also focus on the role of the role of gauge freedom and gauge choices in the PEPS parameterisation.

Then we will discuss the implications of these structures and challenges when variationally optimizing a PEPS using gradient optimization techniques. In particular, I will give a brief introduction to Riemannian optimization, and discuss the lessons we can learn from this. These insights can for example be used to design preconditioners that can speed up the convergence of optimization schemes, or to design schemes that preserve certain gauge choices on the PEPS parameterisation. 

We propose a simple and generic construction of the variational tensor network operators to study the quantum spin systems by the synergy of ideas from the imaginary-time evolution and variational optimization of trial wave functions. By applying these operators to simple initial states, accurate variational ground state wave functions with extremely few parameters can be obtained. Furthermore, the framework can be applied to study spontaneously symmetry breaking, symmetry protected topological, and intrinsic topologically ordered phases, and we show that symmetries of the local tensors associated with these phases can emerge directly after the optimization without any gauge fixing. This provides a universal way to identify quantum phase transitions without prior knowledge of the system.

One of the key challenges in iPEPS calculations is the accurate extrapolation of the ground-state energy to the infinite bond-dimension limit. In this talk I present an approach to accurately compute the energy variance of an iPEPS, enabling systematic extrapolations of the energy. The method is based on contracting a large cell of tensors using the corner transfer matrix renormalization group method to evaluate the correlators between pairs of local Hamiltonian terms. I will show that the accuracy of this approach is substantially higher than that of previous methods and demonstrate the usefulness of the variance extrapolation for the Heisenberg model, for a free fermionic model, and for the Shastry-Sutherland model. The approach can also be used to compute <L\dagger L> for an open quantum system described by the Liouvillian L, in order to assess the quality of the steady-state solution and to locate first-order phase transitions. Finally, I will discuss progress in evaluating the iPEPS excitation ansatz, which is technically closely related.

TensorKit is the backbone of a group of Julia tensor network simulation libraries developed at the University of Ghent. Over the past six months we have been working to add GPU acceleration and support for Julia's next generation of automatic differentiation libraries. This talk will cover the improvements so far, challenges remaining, and priorities for future development. In particular, important tools needed for large scale simulations which are not easily supported with the existing GPU and AD frameworks will be discussed.

A core component of the iPEPS calculation is the approximate contraction of the iPEPS, where the computational bottleneck typically lies in the singular value or eigenvalue decompositions involved in the renormalization step.  We propose a contraction scheme for C4v-symmetric tensor networks based on combining the corner transfer matrix renormalization group (CTMRG) with QR-decompositions and discuss the extension to generic setting. 

This tutorial introduces the CTMRG algorithm with an emphasis on how to implement it in practice. Using TensorKit.jl and interactive Pluto notebooks, this hands-on session will build the algorithm up step by step, from a primer on using TensorKit.jl in Julia to converged environment tensors. We will demonstrate how to leverage this to compute expectation values and correlation functions, and conclude by integrating automatic differentiation to enable gradient-based variational optimization of PEPS ground states. Finally, if time permits we will showcase some of the more advanced features of higher-level libraries like PEPSKit.jl that enable more advanced PEPS workflows for a wider variety of applications. 

In order to follow the session, an installation of Julia v1.10 or v1.11 along with a modern browser (Chrome or Firefox) is required. For more information, visit : https://github.com/lkdvos/PEPSTutorial

We benchmark time evolution methods for Projected Entangled Pair States (PEPS) in two scenarios: real-time evolution of the Ising model on a finite lattice and imaginary-time evolution for purification of the t-J model on an infinite lattice. PEPS time evolution employs a Trotter decomposition, where each application of a local gate is followed by truncation of the enlarged virtual bond dimension. We discuss how the accuracy of the results depends on the truncation procedure—particularly the choice of environment used during truncation—and highlight additional simulation choices that can have unexpected impacts on the quality of the results.

We investigate the class of isometric tensor network states (isoTNSs), which generalize the isometry condition of one-dimensional matrix-product states to tensor networks in higher dimensions. Notably, the isometry condition allows for both efficient classical simulation and a simple sequential preparation on quantum computers. First, we benchmark the variational power of isoTNS for finding ground states of local Hamiltonians and performing time evolution. Second, we identify model systems that have exact isoTNS representable ground states. 

Projected Entangled Pair States (PEPS) provide a powerful framework for simulating strongly correlated quantum many-body systems in two and higher dimensions. However, their application to finite systems has long been hampered by the prohibitive computational cost of standard contraction methods. Thus, developing efficient and accurate algorithms for finite PEPS remains a central challenge in computational quantum physics. In this talk, I will present significant advances in finite PEPS simulations achieved within a Variational Monte Carlo (VMC) framework. This approach proves to be a powerful and versatile tool, enabling accurate studies of frustrated spin systems, fermionic models, lattice gauge theories, and real-time dynamics, while also bridging connections with neural network representations.

While there are controlled methods for computations with PEPS, they can be expensive in practice. Belief propagation, a method dating back to the 1930's, gives an alternative either as a cheap, uncontrolled approach or can be made controlled with loop corrections and related strategies. The belief propagation framework allows affordable dynamics simulations even in 3D and serves as an important conceptual framework for understanding other PEPS algorithms. I will demonstrate belief propagation for dynamics with applications to quantum simulators.

The disorder parameter, defined as the expectation value of the symmetry transformation acting on a subsystem, can be used to characterize symmetric phases as an analogy to detecting spontaneous symmetry-breaking phases using local order parameters. In a dual picture, disorder parameters actually detect spontaneous symmetry breaking of higher-form symmetries. In this talk, I will show that the nonlocal disorder parameters can be conveniently and efficiently evaluated using infinite projected entangled pair states. Moreover, we propose a finite correlation length scaling theory of the disorder parameter within the quantum critical region and validate the scaling theory with variationally optimized infinite projected entangled pair states. We find from the finite correlation length scaling that the disorder parameter satisfies perimeter law at a critical point, i.e., it decays exponentially with the boundary size of the subsystem, indicating spontaneous higher-form symmetry breaking at the critical point of the dual model.

Hamiltonian learning aims to reconstruct unknown Hamiltonians from measurements of quantum states or processes, offering a powerful route to uncover the underlying physics of quantum systems. Here we show that Hamiltonian learning of quantum states—identifying local Hamiltonians for which a given state is exactly or approximately an eigenstate—can be achieved from static structure factors. Focusing on quantum states represented by infinite projected entangled pair states (iPEPS), we evaluate structure factors efficiently using the generating function method. Our approach recovers both frustration-free and non-frustration-free parent Hamiltonians and can construct approximate parent Hamiltonians whose local term support is small even when the exact ones require a large support. Moreover, it can identify Hamiltonians such that the iPEPS are excited quantum many-body scars eigenstates. These results demonstrate that Hamiltonian learning of iPEPS is feasible despite its contraction complexity and extends beyond the conventional parent-Hamiltonian framework.

This talk presents a differentiable programming framework for tensor network simulations. I will explain how to formulate tensor network contractions as differentiable computational graphs, which enables gradient-based optimization and higher-order derivative calculations via automatic differentiation. Key technical challenges—including the stable differentiation of tensor decompositions and memory-efficient backpropagation—will be addressed. The framework is demonstrated through applications in statistical mechanics and the variational optimization of quantum states, offering a systematic approach to automating and innovating tensor network algorithm design. 

In this session we will discuss the Python library "variPEPS" for variational PEPS ground state simulations. In the first part, we present an overview of the current functionality and features of the library, as well as a discussion of the obstacles encountered during implementation. In the second part, you will have the opportunity to play around with the library yourself. To join this hands-on session, it would be helpful to have a current Python installation with the pip package manager ready.

Tensor networks offer a natural language for computational physicists talking about the renormalization group and scaling properties. I review some developments in the tensor-network-based real space renormalization group (TNRG) in which I am involved. It includes: improvement of quality of the environment, entanglement filtering by nuclear norm regularization, TNRG procedure respecting the lattice reflection symmetries, linearized TNRG map for obtaining the scaling dimensions without resorting to dimension-specific CFT formula, etc.

Recent progress in generalized symmetry and topological holography has shown that, in conformal field theory (CFT), topological data from one dimensional higher can play a key role in determining local dynamics. Based on this insight, a fixed-point (FP) tensor complex (TC) for CFT has recently been constructed. In this work, we develop a TC renormalization (TCR) algorithm adapted to this CFT-based structure, forming a renormalization-group (RG) framework with generalized symmetry. We show that the full FP tensor can emerge from the RG flow starting with only the three-point function of the primary fields. Remarkably, even when starting solely from topological data, the RG process can still reconstruct the full FP tensor--a method we call as topological bootstrap. This approach deepens the connection between the topological and dynamical aspects of CFT and suggests pathways toward a fully algebraic description of gapless quantum states, with potential extensions to higher dimensions.

TBA

I will discuss a few examples of classical statistical mechanics problems which can be written as (relatively simple) 2D tensor networks, where the contraction can fail. Ad-hoc solutions involve increasing significantly the numerical precision, finding a specific non-local transformation expressed as an MPO, or writing the partition function as a different tensor network. I will emphasize the absence of systematic solution to this problem and how it seems related to non-local constraints. 

Projected entangled-pair states (PEPS) constitute a powerful variational ansatz for capturing ground state physics of two-dimensional quantum systems. However, accurately computing and minimizing the energy expectation value remains challenging, in part because the impact of the gauge degrees of freedom that are present in the tensor network representation is poorly understood. We analyze the role of gauge transformations for the case of a U(1)-symmetric PEPS with point group symmetry, thereby reducing the gauge degrees of freedom to a single class. We show how gradient-based optimization strategies exploit the gauge freedom, causing the tensor network contraction to become increasingly inaccurate and to produce artificially low variational energies. Furthermore, we develop a gauge-fixed optimization strategy that largely suppresses this effect, resulting in a more robust optimization. Our study underscores the need for gauge-aware optimization strategies to guarantee reliability of variational PEPS in general settings.

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.

Reliably simulating two-dimensional many-body quantum dynamics with projected entangled pair states (PEPS) has long been a difficult challenge. In this work, we overcome this barrier for low-energy quantum dynamics by developing a stable and efficient time-dependent variational Monte Carlo (tVMC) framework for PEPS. By analytically removing all gauge redundancies of the PEPS manifold and exploiting tensor locality, we obtain a numerically well-conditioned stochastic reconfiguration (SR) equation amenable to robust solution using the efficient Cholesky decomposition, enabling long-time evolution in previously inaccessible regimes. We demonstrate the power and generality of the method through four representative real-time problems in two dimensions: (I) chiral edge propagation in a free-fermion Chern insulator; (II) fractionalized charge transport in a fractional Chern insulator; (III) vison confinement dynamics in the Higgs phase of a Z2 lattice gauge theory; and (IV) superfluidity and critical velocity in interacting bosons. All simulations are performed on 12x12 or 13x13 lattices with evolution times T = 10 to 12 using modest computational resources (1 to 5 days on a single GPU card). Where exact benchmarks exist (case I), PEPS-tVMC matches free-fermion dynamics with high accuracy up to T = 12. These results establish PEPS-tVMC as a practical and versatile tool for real-time quantum dynamics in two dimensions. The method extends the reach of classical tensor-network simulations for studying elementary excitations in quantum many-body systems and provides a valuable computational counterpart to emerging quantum simulators. (arXiv:2512.06768)

Cuprate superconductors host intertwined electronic orders—d-wave superconductivity, pair-density-wave (PDW), charge-density-wave (CDW), and spin order. The t-J model is a candidate minimal description, yet obtaining controlled results at low doping on two-dimensional clusters remains challenging. Here we combine state-of-the-art density matrix renormalization group (DMRG) with finite-size projected entangled pair states (PEPS) equipped with Monte Carlo sampling to study the lightly doped ($\delta\lesssim 0.10$) t-J model on open-boundary stripe geometries and C_4-symmetric clusters. We scan the next-nearest-neighbor hopping t_2 across electron-like (t_2>0), hole-like (t_2<0), and t_2=0 regimes. On small clusters, DMRG and PEPS agree quantitatively on both energies and local observables; on larger systems our PEPS calculations reach up to 16x16 sites. In this doping range we find signatures of a 4a_0x4a_0 CDW coexisting with antiferromagnetic backgrounds. Superconductivity is predominantly d-wave and is enhanced for t_2>0 but suppressed for t_2<0 relative to t_2=0, which is consistent with many recent relevant numerical studies. Strikingly, on top of the uniform d-wave component we resolve a plaquette PDW with period two lattice spacings; this PDW is markedly strengthened for t_2<0 and strongly suppressed for t_2>0, in contrast to the trend of the uniform d-wave component. Taken together, our results highlight a plaquette PDW with period two as a feature of the lightly doped t-J model—strengthened for t_2<0 and suppressed for t_2>0—and provide a concrete benchmark for future theory and experiment.

A panel discussion on the past and future of PEPS methods for the simulations of strongly-correlated quantum systems : What are the most pressing challenges in the near future? How do PEPS methods compare with other computational methods? What are the advantages and disadvantages? 

The panel members will first be asked to express their views and opinions in 5 minutes, afterwards there will be an open discussion. Questions from the audience are very much appreciated.