Cost scaling of MPS and TTNS simulations for 2D and 3D systems with area-law entanglement

TL;DR

This work analyzes the cost scaling of matrix product states (MPS) and tree tensor network states (TTNS) for simulating quantum many-body systems in dimensions $D\ge 2$ under area-law and log-area entanglement. By linking bond dimensions to Rényi entropies and invoking area-law scaling, the authors derive that bond dimensions scale as $M_i \sim q^{|\partial \,\mathcal{A}_i|}$, and compare contraction costs: MPS costs scale as $\mathcal{O}(M^3)$ while TTNS scale as $\mathcal{O}(M^{z+1})$ with $z=3$ for binary TTNS. Across 2D and 3D geometries and various boundary conditions, the results show that, asymptotically, MPS (with snake or infinite-MPS mappings) incur far smaller costs than TTNS, with an exponential separation in system size $L^{D-1}$ (e.g., $\mathcal{O}(q^{3L^{D-1}})$ for MPS vs $\mathcal{O}(q^{8L^{D-1}})$ for TTNS). The findings imply that, for large systems, MPS are typically more efficient than TTNS for area-law states, though polynomial factors and specific boundary setups can affect crossover scales; the work also situates TTNS within the broader context of tensor-network approaches such as PEPS and MERA and notes potential relevance to non-quantum domains.

Abstract

Tensor network states are an indispensable tool for the simulation of strongly correlated quantum many-body systems. In recent years, tree tensor network states (TTNS) have been successfully used for two-dimensional systems and to benchmark quantum simulation approaches for condensed matter, nuclear, and particle physics. In comparison to the more traditional approach based on matrix product states (MPS), the graph distance of physical degrees of freedom can be drastically reduced in TTNS. Surprisingly, it turns out that, for large systems in $D>1$ spatial dimensions, MPS simulations of low-energy states are nevertheless more efficient than TTNS simulations. With a focus on $D=2$ and 3, the scaling of computational costs for different boundary conditions is determined under the assumption that the system obeys an entanglement (log-)area law, implying that bond dimensions scale exponentially in the surface area of the associated subsystems.

Figures (5)

• Figure 1: (a) Part of a TTNS with vertex degree $z=3$, orthogonality center $x$ with edges $1,2,3$, and the corresponding center tensor $C_x$, which has bond dimensions $M_1,M_2,M_3$. Arrows indicate isometric properties of tensors according to the convention of Ref. Barthel2025_12. Vertical lines represent site-basis indices. The remaining panels show cost-dominant TTNS operations: (b) SVDs which yield Schmidt spectra $\lambda_i$, are used for truncations, and can move the orthogonality center, (c) contraction of Hamiltonian environment tensors $\mathcal{E}_i$ and $\tilde{\mathcal{E}}_i$, where the square is a tensor of the Hamiltonian TTN operator with bond dimensions $k_i$, (d) contraction of $C_x$ with its effective Hamiltonian $H^\text{eff}_x$, corresponding to the energy gradient $\partial\langle\hat{H}\rangle/\partial C_x^\dag$.
• Figure 2: 2D geometries considered in Sec. \ref{['sec:2D']}: (a) $L\times L$ lattice on a cylinder with PBC in the $y$ direction, (b) long $L_x\times L$ cylinder with $L_x=2^k L$, (c) $L\times L$ square with OBC in both directions, (d) $L\times L$ lattice on a torus, i.e., PBC in both directions.
• Figure 3: (a) In MPS simulations for 2D systems, we can arrange the MPS sites along a Hamiltonian path that covers the entire lattice, following trails that fully traverse the system in the $y$ direction before progressing in $x$ direction. (b) Cutting the MPS tensor network at bond $i$ corresponds to a spatial bipartition into subsystems $\mathcal{A}_i$ and $\mathcal{B}_i$. With Eq. \ref{['eq:Mbound']}, the bond dimensions $M_i$ needed to achieve a certain approximation accuracy can be bounded from above and below by associated Rényi entanglement entropies $S_{\alpha,i}$ of the target state. For area-law systems, $S_{\alpha,i}$ is proportional to the interface area $|\partial A_i|$.
• Figure 4: Simulation of 2D systems with binary TTNS: (a) For square geometries, layers split the system alternately in the $x$ and $y$ directions and only the lowest-layer tensors carry physical indices $\sigma_x$. (b) For long strips or cylinders, the first layers correspond to a 1D TTNS, splitting the system in $x$ directions until reaching $L\times L$ segments. (c) A sequence of spatial subsystems, corresponding to bond indices (renormalized sites) after the action of the tensors in layer $n=0,1,2,\dotsc$. We follow a specific sequence, where subsystem $\mathcal{A}_{n-1}$ is split by a tensor of layer $n$ into subsystems $\mathcal{A}_n$ and $\mathcal{A}_n'$ with surface areas $|\partial A_n|\geq |\partial A_n'|$. The surface areas can depend on the boundary conditions as specified in Table \ref{['tab:2D']}. The scheme works for any 2D lattice, where non-square lattices only require some inessential modifications in the lowest layers.
• Figure 5: Simulation of 3D systems with binary TTNS: For cubic geometries, layers split the system cyclically in the $x$, $y$, and $z$ directions. Here, we follow a specific sequence of subsystems, where $\mathcal{A}_{n-1}$ is split by a layer-$n$ tensor into subsystems $\mathcal{A}_n$ and $\mathcal{A}_n'$. The subsystem surface areas are given in Table \ref{['tab:3D']} and depend on the boundary conditions.