Tensor-Network Algorithms

• Tensor-network based algorithms are frameworks that use tensor decompositions and renormalization methods to represent and optimize high-dimensional systems with local entanglement.
• They employ techniques such as TRG, HOTRG, and SRG sweeping to manage computational cost while preserving key system geometries and long-range correlations.
• These methods have broad applications in quantum many-body physics, classical statistical mechanics, and machine learning for accurate simulation of complex phenomena.

A tensor-network based algorithm is an algorithmic framework that leverages tensor network representations—such as Matrix Product States (MPS), Projected Entangled Pair States (PEPS), or more general higher-order tensor constructions—to efficiently simulate, contract, or optimize high-dimensional systems with local structure or entanglement constraints. These algorithms have become foundational in quantum many-body physics, classical statistical mechanics, machine learning, and combinatorial optimization, due to their ability to exploit the area-law scaling of entanglement and compress exponential state spaces into polynomially-parameterized objects.

1. Fundamental Principles and Coarse-Graining Strategies

Tensor-network based algorithms construct representations of large-scale objects (e.g., the partition function, wavefunction, or probabilistic model) as networks of lower-rank tensors connected via contracted indices. For two-dimensional lattice models, the core task is to efficiently contract the network, which otherwise exhibits exponential scaling in system size.

Two primary coarse-graining strategies have been developed to tackle this challenge on finite periodic lattices (Zhao et al., 2015):

• Tensor Renormalization Group (TRG): This method performs a local coarse-graining by decomposing pairs of neighboring site or bond tensors via singular value decomposition (SVD), truncating low-weight singular values to control the growth of bond dimensions. The TRG rewires the network (e.g., from a honeycomb to a triangle–honeycomb lattice), consolidates tensors, and iteratively reduces the network size.
• Higher-Order Tensor Renormalization Group (HOTRG): HOTRG extends the SVD approach by employing higher-order SVD (HOSVD) to treat all indices simultaneously. This enables the simultaneous contraction and truncation of multiple bonds, thereby capturing more global correlations and preserving the original lattice geometry, a critical factor for 3D extensions.

Strategy	Core Decomposition	Geometry Preservation	Scaling (Honeycomb/Square)
TRG	SVD (bond-based)	Low (network is rewired)	O(χ⁶) on honeycomb
HOTRG	HOSVD (multi-leg)	High (lattice geometry kept)	O(χ⁷) or higher in some cases

While TRG is efficient and conceptually simple, HOTRG is more accurate in capturing long-range correlations, especially in systems with significant geometric sensitivity or higher dimensions.

2. Sweeping Scheme and Second Renormalization Group (SRG)

A key development for achieving high-precision tensor contractions is the introduction of a sweeping scheme within the Second Renormalization Group (SRG) framework (Zhao et al., 2015). Analogous to finite-system DMRG sweeps, this involves

• Contracting the environment tensors (surroundings) of a localized region using backward iterations.
• Dressing local truncation procedures by incorporating environmental effects, which is performed by modifying the local matrices prior to SVD using the environment tensor, thus “globalizing” the optimization.
• Iteratively sweeping: After updating the tensors (system), the environment is recomputed, and another pass is performed with updated corrections. This feedback loop is repeated until convergence.

The SRG and its higher-order variant (HOSRG) thus minimize not only local truncation errors but also the error in the full partition function, systematically reducing global approximation errors.

3. Application to Benchmark Models

Tensor-network based algorithms have been explicitly validated on both classical and quantum lattice models (Zhao et al., 2015):

• Classical Ising Model (Square Lattice): The partition function is mapped to a tensor network, often by decomposing Boltzmann weights and reshaping the network (e.g., from square to honeycomb geometry). Finite-size SRG algorithms with periodic boundary conditions enable direct calculation of observables on finite lattices and support reliable extrapolation to thermodynamic limits via scaling forms such as $$f_\infty - f(L) = a/L^2 + b/L^4 + \mathcal{O}(L^{-6})$$.
• Kitaev Model (Honeycomb Lattice): The ground state is expressed as a PEPS, optimized using imaginary-time projection and environment-dressed truncations reached through finite SRG with sweeping. For challenging parameters (e.g., $D = 6,8$), the finite-size approach yields rapid convergence to the exact energetic value, surpassing “infinite-size” algorithms.

In both cases, finite-size tensor network algorithms combined with sweeping reduce truncation errors and produce more accurate physical results than their infinite-dimensional counterparts.

4. Mathematical Structure and Optimization

The mathematical underpinnings of tensor-network based algorithms center on repeated decompositions and contractions:

• Partition Function Representation:

$$Z = \operatorname{Tr}\prod_{i}A_{i;u_i,d_i,l_i,r_i}$$

• Local SVD Truncation (TRG):

$$M_{jn,km} \approx \sum_{z=1}^{\chi} U_{jn,z} \Lambda_z V_{km,z}$$

With truncation error measured as $$\epsilon = 1 - \left(\sum_{z=1}^{\chi} \Lambda_z^2\right) / \left(\sum \Lambda_z^2\right)$$.

• SRG Environment Incorporation:

Compute SVD of the environment tensor $E = X\Omega Y$, “dress” the local matrix:

$$\tilde{A}_{z_1,z_2} = \sum_{m,n,j,k} \Omega_{z_1}^{1/2} Y_{jn,z_1} M_{jn,km} X_{km,z_2} \Omega_{z_2}^{1/2}$$

and perform SVD on $\tilde{A}$ to update local tensors.

Iterative sweeps leverage these constructs, updating both local tensors and their environments, and thus optimizing the network globally rather than in isolation.

5. Accuracy, Scaling, and Performance

Finite-size algorithms with sweeping demonstrate several empirical properties (Zhao et al., 2015):

• Precision: For both Ising and Kitaev models, finite-size SRG with sweeping outperforms infinite-size SRG, especially at large system sizes (e.g., 216 sites), with clear numerical evidence that truncation and finite-size effects are systematically reduced.
• Computational Cost: The dominant costs scale as $O(\chi^6)$ (TRG on honeycomb) to $O(\chi^7)$ (HOTRG on square/higher-dimensional lattices). The ability to optimize tensor neighborhoods globally (via sweeping and environment-dressing) justifies the extra cost through significant gains in accuracy.
• Error Accumulation: Sweeping efficiently controls accumulation of truncation errors over successive coarse-graining steps—a key factor for reliable extrapolation and for controlling systematic errors in physical observables.

6. Broader Implications and Future Directions

The integration of SRG/HOSRG with sweeping for finite periodic systems establishes new benchmarks for accuracy in tensor network contractions.

• Extension Possibilities: The methods are applicable to disordered systems, systems without translational invariance, and higher-dimensional models, where each site may require a distinct environment tensor.
• Quantum Many-Body Systems: With further optimization, these coarse-graining and sweeping schemes could address frustrated magnets, interacting fermion systems, or quantum field theories, which are presently beyond the reach of many conventional techniques.
• Algorithmic Enhancements: Combining these strategies with “full-update” ideas from PEPS optimization or employing more efficient environment contractions may further improve performance.

7. Summary and Outlook

Tensor-network based algorithms equipped with advanced coarse-graining (TRG, HOTRG), environment-aware optimization (SRG, HOSRG), and global sweeping schemes offer a high-precision, controllable approach for contracting networks on finite periodic lattices, as established in leading benchmarks for both classical and quantum critical systems. Their systematic treatment of both local and global correlations paves the way for broader applicability to complex many-body phenomena, particularly where traditional methods are limited by sign problems or scaling bottlenecks (Zhao et al., 2015). The path forward includes extending these methods to systems with less symmetry, higher connectivity, and more challenging entanglement structure.