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arXiv:2609.21290

Efficient MPO Construction for Long-Range Hamiltonians with Periodic Boundary Conditions: Application to Many-Body Dynamics

Xin Wang, Bo Xiong

Published Sep 18, 2026• 0 citations
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Abstract

Matrix product operator (MPO) serves as a fundamental component in tensor network simulations of quantum many-body dynamics. We employ an MPO construction that introduces additional propagation channels to embed both periodic boundary conditions and finite-range couplings directly into an open boundary MPO. We apply this construction within the time-dependent variational principle (TDVP) framework to simulate quench dynamics in a spin-1/2 chain with finite-range interactions, and benchmark the results numerically against the fourth-order Runge-Kutta method, finding excellent agreement for both single-body and two-body observables. The approach offers a practical route for tensor network simulations of many-body dynamics in periodic finite-range systems.

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Matrix product operator (MPO) serves as a fundamental component in tensor network simulations of quantum many-body dynamics. We employ an MPO construction that introduces additional propagation channels to embed both periodic boundary conditions and finite-range couplings directly into an open boundary MPO. We apply this construction within the time-dependent variational principle (TDVP) framework to simulate quench dynamics in a spin-1/2 chain with finite-range interactions, and benchmark the results numerically against the fourth-order Runge-Kutta method, finding excellent agreement for both single-body and two-body observables. The approach offers a practical route for tensor network simulations of many-body dynamics in periodic finite-range systems.
Abstract This article concerns the matrix product operators for one-dimensional spin Hamiltonians under open and periodic boundary conditions. A novel approach called ‘the dynamics of blocks’ is proposed to obtain the matrix product operators in the case of Hamiltonians with periodic boundary conditions. This approach works universally for any Hamiltonian with two-particle as well as single-particle interactions. The main significance of the proposed approach is that, the dimensions of the matrix product operators obtained are the same for both open and periodic boundary conditions. This method gives us a generic way of constructing MPOs for any Hamiltonian with arbitrary neighboring interactions, immaterial of how the interaction strength falls. A general numerical implementation of the ‘dynamics of blocks’ method is also carried out. The DMRG algorithm was implemented, and the ground state eigenvalue was obtained using these MPOs to numerically validate the method.
Combining the time-dependent variational principle (TDVP) algorithm with the parallelization scheme introduced by Stoudenmire and White for the density matrix renormalization group (DMRG), we present the first parallel matrix product state (MPS) algorithm capable of time evolving one-dimensional (1D) quantum lattice systems with long-range interactions. We benchmark the accuracy and performance of the algorithm by simulating quenches in the long-range Ising and XY models. We show that our code scales well up to 32 processes, with parallel efficiencies as high as 8686%86. Finally, we calculate the dynamical correlation function of a 201-site Heisenberg XXX spin chain with 1/r21/{r}^{2}1/r2 interactions, which is challenging to compute sequentially. These results pave the way for the application of tensor networks to increasingly complex many-body systems.
We introduce a matrix product operator (MPO) encoding of the Magnus expansion and the Dyson series for one-dimensional quantum lattice models with time-dependent Hamiltonians. The MPO construction can be made accurate up to arbitrary order in the time step, it can be applied to both finite and infinite systems, and it can handle long-range interactions. The resulting MPO can be combined with state-of-the-art time evolution algorithms based on matrix product states, allowing for drastic improvements in simulating evolution under time-dependent Hamiltonians. Our MPO construction can also be used for the optimization of quantum circuits in the context of quantum simulation of time-dependent Hamiltonians.
Matrix product states (MPSs) and matrix product operators (MPOs) are fundamental tools in the study of quantum many-body systems, particularly in the context of tensor network methods such as time-evolving block decimation (TEBD). However, constructing compact MPO representations for Hamiltonians with interactions beyond nearest neighbors, such as those arising in atomic, molecular, and optical (AMO) systems or in systems with ring geometry, remains a challenge. In this paper, we propose an approach for the direct construction of compact MPOs tailored specifically for the exponential of spin Hamiltonians. This approach allows for a more efficient time evolution, using TEBD, of spin systems with interactions beyond nearest neighbors, such as long-range spin chains, periodic systems, and more complex cluster models, with interactions involving more than two spins.
We present a general method for simulating lattice gauge theories in low dimensions using infinite matrix product states (iMPS). A central challenge in Hamiltonian formulations of gauge theories is the unbounded local Hilbert space associated with gauge degrees of freedom. In one spatial dimension, Gauss's law permits these gauge fields to be integrated out, yielding an effective Hamiltonian with long-range interactions among matter fields. We construct efficient matrix product operator (MPO) representations of these Hamiltonians directly in the thermodynamic limit. Our formulation naturally includes background fields and θθθ-terms, requiring no modifications to the standard iDMRG algorithm. This provides a broadly applicable framework for 1+1D gauge theories and can be extended to quasi-two-dimensional geometries such as infinite cylinders, where tensor-network methods remain tractable. As a benchmark, we apply our construction to the Schwinger model, reproducing expected features including confinement, string breaking, and the critical behavior at finite mass. Because the method alters only the MPO structure, it can be incorporated with little effort into a wide range of iMPS and infinite-boundary-condition algorithms, opening the way to efficient studies of both equilibrium and non-equilibrium gauge dynamics.
Matrix product operators (MPOs) are at the heart of the second-generation density matrix renormalization group (DMRG) algorithm formulated in matrix product state language. We first summarize the widely known facts on MPO arithmetic and representations of single-site operators. Second, we introduce three compression methods (rescaled SVD, deparallelization, and delinearization) for MPOs and show that it is possible to construct efficient representations of arbitrary operators using MPO arithmetic and compression. As examples, we construct powers of a short-ranged spin-chain Hamiltonian, a complicated Hamiltonian of a two-dimensional system and, as proof of principle, the long-range four-body Hamiltonian from quantum chemistry.