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arXiv:2402.05198 · Physical review. B./Physical review. B

Numerically efficient unitary evolution for Hamiltonians beyond nearest neighbors

Alberto Giuseppe Catalano

Published Nov 13, 2025• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1103/jpt3-xjjf

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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 show how to construct relevant families of matrix product operators (MPOs) in one and higher dimensions. These form the building blocks for the numerical simulation methods based on matrix product states and projected entangled pair states. In particular, we construct translationally invariant MPOs suitable for time evolution, and show how such descriptions are possible for Hamiltonians with long-range interactions. We show how these tools can be exploited for constructing new algorithms for simulating quantum spin systems.
Abstract Many-body quantum systems are subjected to the Curse of Dimensionality: The dimension of the Hilbert space H, where these systems live in, grows exponentially with number of their components. However, with some systems, it is possible to escape the curse by using a low-rank tensor approximation known as “matrix-product state/operator (MPS/O) representation” in the quantum community and “tensor-train decomposition” among applied mathematicians. Motivated by recent advances in computational quantum physics, we consider chains of N spins coupled by nearest-neighbor interactions. The spins are subjected to an action coming from the environment. Spatially disordered interaction and environment-induced decoherence drive systems into non-trivial asymptotic states. The dissipative evolution is modeled with a Markovian master equation in the Lindblad form. By implementing the MPO technique and propagating system states with the time-evolving block decimation scheme, which allows keeping the length of the state descriptions fixed, it is in principle possible to reach the asymptotic states. We propose and realize a cluster implementation of this idea. The implementation on four nodes allowed us to resolve the asymptotic states of the model systems with N = 128 spins (total dimension of the Hilbert space dimH = 2128 ≈ 1039).
Simulating many-body open quantum systems is an extremely challenging problem, with methods often restricted to either models with nearest-neighbor interactions or semi-classical approximations. In particular, modeling two-dimensional systems with realistic long-range interactions, in addition to dissipation, is of vital importance to the development of modern quantum computing and simulation platforms. In this paper, we present a construction of the time-evolution operator, as a projected entangled pair operator (denoted tePEPO), that can be used to evolve a tensor network ansatz through time. Interactions beyond nearest-neighbor, including interactions between sites not collinear in the lattice, can be represented efficiently as a tePEPO. Furthermore, we obtain approximations to realistic radial long-range interactions decaying with a power-law, that give accurate results with small tePEPO bond dimension. Finally, we consider a physical example of a Rydberg atom Hamiltonian with long-range dipolar interactions, and show evidence of a dipole-dipole blockading effect in presence of dissipation. This work demonstrates the applicability of tensor networks to two-dimensional systems widely studied in experiments, but previously inaccessible to non-semi-classical methods.
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