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

TL;DR

This work introduces Tree Tensor Network Operators (TTNOs) as compact representations for Hamiltonians with long-range pairwise interactions within tree tensor networks. By connecting the hierarchical low-rank structure of the interaction matrix $\boldsymbol{\beta}$ to TTNOs, the authors show that TTNOs can be constructed from Hierarchical Semi-Separable (HSS) decompositions, achieving TTNO ranks bounded by $r_\tau \le 2 + k_\tau$ when $\boldsymbol{\beta}$ has HSS rank $k_\tau$. They derive error bounds for TTNOs obtained via HSS compression, including spectral and Frobenius-norm guarantees, and demonstrate through numerical experiments on closed and open quantum spin systems that TTNOs offer advantages over MPOs in handling long-range interactions. The results highlight the practical impact of using balanced TTN formats and HSS-based TTNOs to enable scalable simulations of high-dimensional quantum and scientific computing problems with long-range couplings.

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 semi-separable matrix. Numerical experiments for different quantum spin systems validate our results and highlight the potential advantages of TTNOs over matrix product operators.

Figures (11)

• Figure 1: Graphical representation of the tree $\bar{\tau}=(\tau_1,\tau_2)$ with $\tau_1 = (1,2,3)$ and $\tau_2 = (4,5,6)$.
• Figure 1: \newlabelfig:lemmapart0 Different recursive block-partitions of an $8 \times 8$ interaction matrix $\boldsymbol{\beta}$. Left: Recursive block-partition corresponding to a balanced binary tree. Right: Recursive block-partition corresponding to a degenerate tree.
• Figure 1: Long-range unitary Hamiltonian with given parameters: $\Omega=3$, $\Delta=-2$, $\nu = 2$, $\alpha = 1$ and HSS tolerance $10^{-12}$. Left: Relative error of the TTNO vs the number of particles. Right: Representation rank of the TTNO (solid line) and the excepted ranks (dashed line) versus the number of particles.
• Figure 2: Long-range unitary Hamiltonian with given parameters: $\Omega=3$, $\Delta=-2$, $\nu = 2$, $\alpha = 1$ and $d=256$. Left: Relative error of the TTNO vs the HSS tolerance. Right: Representation rank of the TTNO versus the HSS tolerance.
• Figure 3: Long-range unitary Hamiltonian with given parameters: $\Omega=3$, $\Delta=-2$, $\nu = 2$ and $d=256$. Left: Representation rank of the TTNO versus different values of $\alpha$ computed with HSS tolerance $10^{-4}$. Right: Representation rank of the TTNO versus different values of $\alpha$ computed with HSS tolerance $10^{-12}$.

Theorems & Definitions (17)

• Definition 2.1: dimension tree
• Definition 2.2: Tree tensor network
• Theorem 2.3
• Definition 2.4: TTNO
• Lemma 3.1
• Proof 1
• Lemma 3.2
• Proof 2
• Theorem 3.3
• Proof 3