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.
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 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.
This work introduces an extension to the Tensor Network Quantum Virtual Machine (TNQVM) tool, enhancing the existing stack of ExaScale Tensor Network (ExaTN), ExaScale Accelerator (XACC), and TNQVM. It features a new plugin that enables efficient simulation of a Projected Entangled Pair State (PEPS), a 2D tensor network. To improve simulation efficiency for PEPS, we have implemented the snake boundary contraction algorithm. By integrating this capability into the existing stack, we enhance the overall functionality and versatility of the framework. We tested this new PEPS topology for a simple GHZ bell-pair generation quantum circuit and saw that its runtime is very close to that of the MPS topology. We estimate that the real potential of the PEPS topology becomes discernible when quantum circuits with multidimensional entanglement are simulated using tensor networks. In such cases, 1D tensor networks fail to represent or contract them efficiently.
Tensor network algorithms have proven to be very powerful tools for studying one- and two-dimensional quantum many-body systems. However, their application to three-dimensional (3D) quantum systems has so far been limited, mostly because the efficient contraction of a 3D tensor network is very challenging. In this paper, we develop and benchmark two contraction approaches for infinite projected entangled-pair states (iPEPS) in 3D. The first approach is based on a contraction of a finite cluster of tensors including an effective environment to approximate the full 3D network. The second approach performs a full contraction of the network by first iteratively contracting layers of the network with a boundary iPEPS, followed by a contraction of the resulting quasi-2D network using the corner transfer matrix renormalization group. Benchmark data for the Heisenberg and Bose-Hubbard models on the cubic lattice show that the algorithms provide competitive results compared to other approaches, making iPEPS a promising tool to study challenging open problems in 3D.
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) 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.
These notes provide an introduction to tensor-network methods in quantum many-body physics, with an emphasis on matrix-product states (MPS). They develop the basic tensor-network language, including graphical notation, virtual indices, bond dimensions, gauge freedom, canonical forms, QR and singular-value decompositions, and the role of entanglement in controlling the efficiency of the representation. The main MPS algorithms are then introduced, including contractions, correlation functions, matrix-product operators, DMRG, and time-evolution methods. The notes also briefly discuss projected entangled-pair states (PEPS) as a higher-dimensional generalization of MPS, together with the basic ideas behind approximate PEPS contraction. Finally, tensor-network representations of mixed states, quantum channels, and Lindblad dynamics are presented, with applications to thermal states and open quantum systems. The presentation is accompanied by short Julia code examples based on ITensor, ITensorMPS, and TensorMixedStates. These notes were written for the 9th Les Houches Summer School on Computational Physics: Open Quantum Systems, held in June 2026.