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arXiv:2405.09952 · SIAM Journal on Scientific Computing

Low-Rank Tree Tensor Network Operators for Long-Range Pairwise Interactions

Gianluca Ceruti, Daniel Kreßner, Dominik Sulz

Published Aug 1, 2025• 4 citations
arXivHTMLPDF

Abstract

Abstract. Compactly representing and efficently applying linear operators are fundamental ingredients in tensor network methods for simulating quantum many-body problems and solving high-dimensional problems in scientific computing. In this work, we study such representations for tree tensor networks, the so-called tree tensor network operators (TTNOs), paying particular attention to Hamiltonian operators that involve long-range pairwise interactions between particles. Generalizing the work by Lin, Tong, and others on matrix product operators, we establish a direct connection between the hierarchical low-rank structure of the interaction matrix and the TTNO property. This connection allows us to arrive at very compact TTNO representations by compressing the interaction matrix into a hierarchically semiseparable matrix. Numerical experiments for different quantum spin systems validate our results and highlight the potential advantages of TTNOs over matrix product operators.

DOI
10.1137/24m1661996

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Abstract. Compactly representing and efficently applying linear operators are fundamental ingredients in tensor network methods for simulating quantum many-body problems and solving high-dimensional problems in scientific computing. In this work, we study such representations for tree tensor networks, the so-called tree tensor network operators (TTNOs), paying particular attention to Hamiltonian operators that involve long-range pairwise interactions between particles. Generalizing the work by Lin, Tong, and others on matrix product operators, we establish a direct connection between the hierarchical low-rank structure of the interaction matrix and the TTNO property. This connection allows us to arrive at very compact TTNO representations by compressing the interaction matrix into a hierarchically semiseparable matrix. Numerical experiments for different quantum spin systems validate our results and highlight the potential advantages of TTNOs over matrix product operators.
Tensor network operators, such as the matrix product operator (MPO) and the projected entangled-pair operator (PEPO), can provide efficient representation of certain linear operators in high-dimensional spaces. This paper focuses on the efficient representation of tensor network operators with long-range pairwise interactions such as the Coulomb interaction. For MPOs, we find that all existing efficient methods exploit an upper-triangular low-rank (UTLR) property, i.e., the upper-triangular part of the matrix can be well approximated by a low-rank matrix, while the matrix itself can be full-rank. This allows us to convert the problem of finding the efficient MPO representation into a matrix completion problem. We develop a modified incremental singular value decomposition method (ISVD) to solve this ill-conditioned matrix completion problem. This algorithm yields equivalent MPO representation to that developed in [E. M. Stoudenmire and S. R. White, Phys. Rev. Lett., 119 (2017), 046401]. In order to efficiently treat more general tensor network operators, we develop another strategy for compressing tensor network operators based on hierarchical low-rank matrix formats, such as the hierarchical off-diagonal low-rank (HODLR) format and the H\mathcal{H}H-matrix format. Though the preconstant in the complexity is larger, the advantage of using the hierarchical low-rank matrix format is that it is applicable to both MPOs and PEPOs. For the Coulomb interaction, the operator can be represented by a linear combination of O(log⁡(N)log⁡(N/ϵ))\mathcal{O}(\log(N)\log(N/\epsilon))O(log(N)log(N/ϵ)) MPOs/PEPOs, each with a constant bond dimension, where NNN is the system size and ϵ\epsilonϵ is the accuracy of the low-rank truncation. Neither the modified ISVD nor the hierarchical low-rank algorithm assumes that the long-range interaction takes a translation-invariant form.
This work is concerned with tree tensor network operators (TTNOs) for representing quantum Hamiltonians. We first establish a mathematical framework connecting tree topologies with state diagrams. Based on these, we devise an algorithm for constructing a TTNO given a Hamiltonian. The algorithm exploits the tensor product structure of the Hamiltonian to add paths to a state diagram, while combining local operators if possible. We test the capabilities of our algorithm on random Hamiltonians for a given tree structure. Additionally, we construct explicit TTNOs for nearest neighbour interactions on a tree topology. Furthermore, we derive a bound on the bond dimension of tensor operators representing arbitrary interactions on trees. Finally, we consider an open quantum system in the form of a Heisenberg spin chain coupled to bosonic bath sites as a concrete example. We find that tree structures allow for lower bond dimensions of the Hamiltonian tensor network representation compared to a matrix product operator structure. This reduction is large enough to reduce the number of total tensor elements required as soon as the number of baths per spin reaches 3.
This research introduces an improved framework for constructing matrix product operators and tree tensor network operators (TTNOs), crucial tools in quantum simulations. A given (Hamiltonian) operator typically has a known symbolic “sum of operator strings” form that can be translated into a tensor network structure. Combining the existing bipartite-graph-based approach and a newly introduced symbolic Gaussian elimination preprocessing step, our proposed method improves upon earlier algorithms in cases when Hamiltonian terms share the same prefactors. We test the performance of our method against established ones for benchmarking purposes. Finally, we apply our methodology to the model of a cavity filled with molecules in a solvent. This open quantum system is cast into the hierarchical equation of motion setting to obtain an effective Hamiltonian. Construction of the corresponding TTNO demonstrates a sublinear increase of the maximum bond dimension.
Tree tensor network states (TTNS) decompose the system wavefunction to the product of low-rank tensors based on the tree topology, serving as the foundation of the multi-layer multi-configuration time-dependent Hartree method. In this work, we present an algorithm that automatically constructs the optimal and exact tree tensor network operators (TTNO) for any sum-of-product symbolic quantum operator. The construction is based on the minimum vertex cover of a bipartite graph. With the optimal TTNO, we simulate open quantum systems, such as spin relaxation dynamics in the spin-boson model and charge transport in molecular junctions. In these simulations, the environment is treated as discrete modes and its wavefunction is evolved on equal footing with the system. We employ the Cole-Davidson spectral density to model the glassy phonon environment and incorporate temperature effects via thermo-field dynamics. Our results show that the computational cost scales linearly with the number of discretized modes, demonstrating the efficiency of our approach.
We have developed TTNOpt, a software package that utilizes tree tensor networks (TTNs) for quantum spin systems and high-dimensional data analysis. TTNOpt provides efficient and powerful TTN computations by locally optimizing the network structure, guided by the entanglement pattern of the target tensors. For quantum spin systems, TTNOpt searches for the ground state of Hamiltonians with bilinear spin interactions and magnetic fields, and computes physical properties of these states, including the variational energy, bipartite entanglement entropy (EE), single-site expectation values, and two-site correlation functions. Additionally, TTNOpt can target the lowest-energy state within a specified subspace, provided that the Hamiltonian conserves total magnetization. For high-dimensional data analysis, TTNOpt factorizes complex tensors into TTN states that maximize fidelity to the original tensors by optimizing the tensors and the network. When a TTN is provided as input, TTNOpt reconstructs the network based on the EE without referencing the fidelity of the original state. We present three demonstrations of TTNOpt:(1) Ground-state search for the hierarchical chain model with a system size of 256256256. The entanglement patterns of the ground state manifest themselves in a tree structure, and TTNOpt successfully identifies the tree.(2) Factorization of a quantic tensor of the 2242^{24}224 dimensions representing a three-variable function where each variant has a weak bit-wise correlation. The optimized TTN shows that its structure isolates the variables from each other.(3) Reconstruction of the matrix product network representing a 161616-variable normal distribution characterized by a tree-like correlation structure. TTNOpt can reveal hidden correlation structures of the covariance matrix.
The hierarchical equations of motion (HEOM) method is a numerically exact open quantum system dynamics approach. The method is rooted in an exponential expansion of the bath correlation function, which in essence strategically reshapes a continuous environment into a set of effective bath modes that allow for more efficient cutoff at finite temperatures. Based on this understanding, one can map the HEOM method into a Schrödinger-like equation, with a non-Hermitian super-Hamiltonian for an extended wave function being the tensor product of the central system wave function and the Fock state of these effective bath modes. In this work, we explore the possibility of representing the extended wave function as a tree tensor network state (TTNS) and the super-Hamiltonian as a tree tensor network operator of the same structure as the TTNS, as well as the application of a time propagation algorithm using the time-dependent variational principle. Our benchmark calculations based on the spin-boson model with a slow-relaxing bath show that the proposed HEOM+TTNS approach yields consistent results with those of the conventional HEOM method, while the computation is considerably sped up. In addition, the simulation with a genuine TTNS is four times faster than a one-dimensional matrix product state decomposition scheme.