Entanglement across scales: Quantics tensor trains as a natural framework for renormalization

TL;DR

The paper reveals that quantics tensor trains (QTT) offer a natural, analytical Renormalization Group framework for length-scale entanglement in quantum systems. By recasting a cyclic-reduction real-space RG for the one-dimensional n-th-nearest-neighbor tight-binding model into QTT language, it shows a precise correspondence: the QTT bond dimension equals the number of renormalized couplings, capturing how information flows across length scales. The authors derive exact QTT representations for the Green's function, prove bond-dimension saturation, and establish a priori error bounds for approximate QTT constructions under decaying couplings. Numerical analyses validate the theory in 1D and extend to interacting/higher-dimensional cases, highlighting the potential for analytical QTT techniques and deeper links to entanglement renormalization concepts such as MERA. Overall, the work positions QTT as both a powerful numerical tool and a new analytic lens for real-space RG across scales, with implications for multi-dimensional and interacting systems.

Abstract

Understanding entanglement remains one of the most intriguing problems in physics. While particle and site entanglement have been studied extensively, the investigation of length or energy scale entanglement, quantifying the information exchange between different length scales, has received far less attention. Here, we identify the quantics tensor train (QTT) technique, a matrix product state-inspired approach for overcoming computational bottlenecks in resource-intensive numerical calculations, as a renormalization group method by analytically expressing an exact cyclic reduction-based real-space renormalization scheme in QTT language, which serves as a natural formalism for the method. In doing so, we precisely match the QTT bond dimension, a measure of length scale entanglement, to the number of rescaled couplings generated in each coarse-graining renormalization step. While QTTs have so far been applied almost exclusively to numerical problems in physics, our analytical calculations demonstrate that they are also powerful tools for mitigating computational costs in semi-analytical treatments. We present our results for the one-dimensional tight-binding model with n-th-nearest-neighbor hopping, where the 2n rescaled couplings generated in the renormalization procedure precisely match the QTT bond dimension of the one-particle Green's function.

Figures (15)

• Figure 1: QTT decomposition with separation of exponentially different length scales as a natural framework of real-space RG methods.
• Figure 2: "Running" of the renormalized couplings ($\omega^{(r)}$ (a),(c),(e),(g); $t^{(r)}$ (b),(d),(f),(h)) in every step of the renormalization procedure in the case of $R=15$ for different values of $\omega,t$.
• Figure 3: (a) Decomposition of tensor $\mathbf{T}$ in QTT form with the indices $\sigma_j := (i_j^{(1)},i_j^{(2)},...,i_j^{(d)})$ and the connection to the corresponding length scales $2^{R-j}$. (b) Contraction of MPO $\mathbf{A}$ with QTT $\mathbf{B}$ leading to QTT $\mathbf{C}$.
• Figure 4: "Rosetta stone" of QTT formalisms. (a) Rank product of two tensor cores $\mathcal{U}, \mathcal{V}$ defined with the help of the strong Kronecker product ($\lrtimes$), as a tensor network diagram and in index notation. (b) Contraction of two tensor cores $\mathcal{U}, \mathcal{V}$ representing a MPO-MPS contraction defined with the help of the mode product ($\underset{*}{\bullet}$), as a tensor network diagram and in index notation.
• Figure 5: "Rosetta stone" of MPO-MPS contractions in different QTT formalisms. Contraction of MPO $\mathbf{U} = \mathcal{U}_1 \lrtimes \mathcal{U}_2$ and the MPS $\mathbf{V} = \mathcal{V}_1 \lrtimes \mathcal{V}_2$ in QTT form over the shared mode indices $j=(j_1,j_2)_2$.