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All-mode approach for general tensor renormalization group method

This paper introduces a stochastic all-mode approach for general tensor renormalization group methods that eliminates systematic errors in coarse-graining by utilizing a "squeezer" construction, demonstrating its effectiveness through higher-order TRG applications to two- and three-dimensional Ising models.

Original authors: Katsumasa Nakayama, Shinji Takeda

Published 2026-10-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Katsumasa Nakayama, Shinji Takeda

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern physics, researchers often face a paradox: the most interesting phenomena, like the sudden shift of a material from a magnet to a non-magnet, happen when countless tiny particles interact in complex, chaotic ways. To understand these shifts, scientists use powerful mathematical maps called tensor networks. Think of these networks as a way to compress a massive, tangled web of information into a manageable shape, allowing computers to calculate the behavior of matter without getting lost in the details. For decades, a specific method known as the tensor renormalization group has been a leading tool for this job, successfully predicting how materials behave in two dimensions. However, as scientists tried to apply this method to more complex, three-dimensional systems, they hit a wall. The calculations became so heavy that they required cutting corners, which introduced hidden, systematic errors that were difficult to measure or remove. This left a gap in our ability to precisely predict the properties of matter in the real, three-dimensional world.

A team of physicists from RIKEN and Kanazawa University has now developed a new approach to bridge this gap, offering a way to perform these complex calculations without those stubborn, hidden errors. Their work, published under the identifier KANAZAWA-26-07, introduces a technique that blends the precision of tensor networks with the randomness of statistical sampling. Instead of forcing the computer to keep every single piece of information or blindly discarding the rest, the researchers devised a method that uses random noise to represent the discarded information. By running the calculation thousands of times with different patterns of this noise and then averaging the results, they can recover the exact answer, with any remaining uncertainty coming only from the randomness of the noise itself, which can be made as small as desired by running more samples. This "all-mode" technique allows the team to apply these powerful calculations to three-dimensional systems for the first time with a level of reliability that was previously out of reach.

The core of this new method relies on a clever mathematical tool they call a "squeezer." In the standard approach, when a computer tries to simplify a complex network of data, it must decide which parts of the information are important enough to keep and which can be thrown away. This decision is usually made by looking at the strength of the connections, but throwing anything away inevitably introduces a small error that accumulates as the calculation progresses. The researchers found that they could construct these squeezers in a way that captures all the information, including the parts that would normally be discarded, by embedding them within a structure that includes random noise. Imagine trying to weigh a heavy object on a scale that can only hold a fraction of the weight; instead of removing the excess, you could use a series of random, tiny jolts to distribute the weight in a way that the scale can read, provided you average the results over many trials. In their simulation, the researchers used a specific type of random noise, where values are simply positive or negative, to create these jolts. This allowed them to keep the full complexity of the data in their mathematical model without the computational cost exploding, effectively keeping every "mode" or piece of information alive in the calculation.

To test if this idea worked in practice, the team applied their new method to the Ising model, a classic theoretical framework used to describe how magnets work. They first ran the simulation on a two-dimensional grid, a scenario where the exact answer is already known, to verify that their new technique could reproduce the correct results. They used a lattice size of 29 by 29 and a bond dimension of 6, running 100 independent samples of the random noise. The results matched the known exact solution perfectly, with the statistical errors being so small they were invisible on their graphs. More importantly, they observed that the error decreased exactly as predicted by statistical theory as they increased the number of samples, confirming that the method was working as intended. They then pushed the method into three dimensions, a much harder challenge where previous methods struggled with accuracy. By extending their "all-mode" approach to a cubic lattice, they successfully calculated the free energy of the system, a key measure of its stability. The results showed that the method scales efficiently, meaning the time required to get a precise answer grows in a manageable way even as the system gets larger.

The significance of this work lies in its ability to eliminate a specific type of error that has plagued high-precision physics calculations for years. In previous methods, researchers had to guess how much error was introduced by simplifying the math, often by running the calculation multiple times with different settings and hoping the results converged. This new approach removes that guesswork entirely. The only error remaining is statistical, arising from the random noise, and this is a type of error that scientists know how to control and reduce simply by doing more calculations. The team demonstrated that for the three-dimensional Ising model, their method could achieve high accuracy without the systematic drift that usually plagues these simulations. While the computational cost is still significant, scaling with the seventh power of the bond dimension for two-dimensional systems and the eleventh power for three-dimensional ones, the trade-off is a result that is free from the hidden biases of truncation. This opens the door for more reliable studies of critical phenomena in three dimensions, where the behavior of materials changes dramatically, providing a clearer window into the fundamental laws that govern the physical world.

Technical Summary: All-mode approach for general tensor renormalization group method

Problem Statement
Tensor Renormalization Group (TRG) methods have become powerful tools for computing physical quantities in lattice systems, particularly for critical phenomena where Monte Carlo methods suffer from the sign problem. However, a persistent challenge in TRG-based approaches is the assessment and control of systematic errors arising from the truncation of singular values during the coarse-graining process. While truncation errors can be estimated locally via discarded singular values, the propagation and accumulation of these errors through iterative coarse-graining steps are not fully understood, making reliable error estimation for physical observables difficult. Existing stochastic approaches, such as the all-mode method originally proposed for lattice QCD and applied to 2D TRG, offer a way to eliminate systematic truncation errors by introducing stochastic sampling. However, generalizing this technique to higher-dimensional formulations (e.g., Higher-Order TRG or HOTRG) and other tensor network algorithms has remained an open task.

Methodology
The authors propose a generalization of the all-mode technique applicable to a wide class of tensor network algorithms, specifically targeting Higher-Order TRG (HOTRG) in two and three dimensions. The core of the methodology relies on the construction of "squeezers," a concept introduced in boundary TRG contexts.

Squeezer Construction: In standard HOTRG, information compression is achieved via isometries derived from Singular Value Decomposition (SVD). The authors utilize a decomposition where a tensor network $N$ is approximated as $N \approx AP_1P_2B$ . The matrices $P_1$ and $P_2$ (squeezers) are constructed to minimize the Frobenius norm of the difference between the original and compressed networks. Crucially, these squeezers can be derived from the eigenvalue decompositions of $A^\dagger A$ and $BB^\dagger$ , avoiding the full SVD of large intermediate tensors.
Stochastic Noise Integration: To eliminate the systematic error introduced by the truncation of singular values in the standard squeezer construction, the authors introduce stochastic noise. They define a noise vector $\eta$ with $Z_2$ components ( $\pm 1$ ) and embed it into a matrix $\xi$ . By averaging over an ensemble of independent noise realizations ( $N_{stat}$ ), the stochastic squeezers satisfy an approximate completeness relation.
Generalized Algorithm:
- 2D HOTRG: The method inserts stochastic squeezers into a $2 \times 2$ fundamental lattice region. The tensors are contracted to form new even ( $E$ ) and odd ( $O$ ) tensors. A second set of stochastic squeezers is applied to the remaining bonds to perform the full coarse-graining step. This process is repeated iteratively along the $x$ and $y$ directions.
- 3D HOTRG: The approach is extended to cubic lattices using a $2 \times 1 \times 2$ fundamental region. Stochastic squeezers are inserted along the $x$ , $y$ , and $z$ directions in a nested construction. The computational cost for the dominant contraction step in 3D scales as $O(D^{11})$ , where $D$ is the bond dimension.
Error Profile: The resulting method is entirely free from systematic truncation errors inherent to the coarse-graining step. The only remaining errors are statistical, arising from the finite number of noise samples ( $N_{stat}$ ), which scale as $1/\sqrt{N_{stat}}$ .

Key Contributions

Generalization of All-Mode Technique: The paper successfully extends the all-mode stochastic technique, previously limited to 2D TRG, to higher-dimensional systems (HOTRG) and general tensor network schemes describable by squeezers.
Systematic Error Elimination: The method provides a framework where the systematic error due to bond dimension truncation is removed in the limit of infinite statistics, leaving only controllable statistical noise.
Implementation in 2D and 3D: The authors provide explicit formulations and algorithmic steps for applying this technique to both 2D and 3D Ising models using HOTRG.

Numerical Results
The authors demonstrate the method using the Ising model on square (2D) and cubic (3D) lattices.

2D Ising Model: For a lattice volume $V = 2^9 \times 2^9$ and bond dimension $D=6$ , the free energy density calculated using the all-mode HOTRG agrees with the exact thermodynamic limit solution within statistical errors. The statistical error was found to increase near the critical point, similar to the behavior of systematic errors in conventional TRG.
Convergence and Scaling:
- The relative deviation of the free energy decreases as $N_{stat}$ increases, becoming consistent with zero within statistical errors for sufficient sample sizes.
- Statistical errors scale as $1/\sqrt{N_{stat}}$ , confirming the expected stochastic behavior.
- The accuracy improves exponentially with increasing bond dimension $D$ in the large $D$ regime.
3D Ising Model: The method was applied to 3D systems, demonstrating feasibility despite the higher computational cost ( $O(D^{11})$ ).
Limitations: The authors note that for small bond dimensions and insufficient statistics, the distribution of the partition function estimator can become broad, potentially yielding negative samples. This can render the real-valued free energy density ill-defined in specific leave-one-out jackknife estimates, a known issue in stochastic estimators of partition functions.

Significance and Claims
The paper claims that the proposed approach enables the application of the all-mode technique to systems beyond two dimensions, specifically targeting higher-dimensional formulations like HOTRG. By utilizing squeezers, the method becomes applicable to any tensor network algorithm where information compression can be described in this manner, including anisotropic TRG and triad TRG. The primary significance lies in offering a pathway to high-precision calculations where systematic errors are not merely estimated but eliminated through stochastic sampling, provided sufficient computational resources are allocated to reduce statistical noise. The authors position this as a step toward establishing a reliable and systematic error estimation scheme essential for future high-precision tensor network calculations.

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