[Paper Review] Quantics Tensor Cross Interpolation for High-Resolution, Parsimonious Representations of Multivariate Functions in Physics and Beyond

Marc K. Ritter, Yuriel Núñez Fernández+4 more|arXiv (Cornell University)|Mar 21, 2023

Computational Physics and Python Applications4 citations

TL;DR

This paper introduces Quantics Tensor Cross Interpolation (QTCI), a method that combines the quantics representation of multivariate functions with tensor cross interpolation to achieve high-resolution, low-memory representations. It enables efficient computation of complex physical quantities—such as the Chern number in the Haldane model—using only 4×10⁵ samples and 20 seconds on a single core, achieving a relative error tolerance of 10⁻¹⁰ with maximum bond dimension D_max = 50.

ABSTRACT

Multivariate functions of continuous variables arise in countless branches of science. Numerical computations with such functions typically involve a compromise between two contrary desiderata: accurate resolution of the functional dependence, versus parsimonious memory usage. Recently, two promising strategies have emerged for satisfying both requirements: (i) The quantics representation, which expresses functions as multi-index tensors, with each index representing one bit of a binary encoding of one of the variables; and (ii) tensor cross interpolation (TCI), which, if applicable, yields parsimonious interpolations for multi-index tensors. Here, we present a strategy, quantics TCI (QTCI), which combines the advantages of both schemes. We illustrate its potential with an application from condensed matter physics: the computation of Brillouin zone integrals.

Motivation & Objective

• To address the challenge of balancing high-resolution representation and memory efficiency in numerical computations of multivariate functions.
• To develop a practical, scalable method for constructing low-rank tensor train (TT) representations of functions with scale-separated structures.
• To enable efficient computation of integrals, Fourier transforms, and topological invariants in high-dimensional function spaces.
• To demonstrate the method's effectiveness on a physically relevant problem: computing the Chern number in the Haldane model.
• To establish a framework applicable beyond physics, particularly to functions with hierarchical or multiscale structure.

Proposed method

• The quantics representation encodes each variable using R binary digits, transforming continuous variables into multi-index tensors with L = nR indices, enabling exponential resolution via binary bit decomposition.
• Tensor cross interpolation (TCI) is applied to the quantics tensor to construct a low-rank tensor train (QTT) approximation with computational cost linear in L = nR, avoiding expensive SVDs.
• The method leverages the hierarchical structure of the quantics grid to sample function values adaptively, focusing on regions of high variation such as peaks and singularities.
• Once the QTCI is constructed, it enables ultrafast algorithms for integration, convolution, and Fourier transforms at O(L) cost, far below standard FFTs.
• The approach uses a fused quantics representation to maintain resolution across multiple scales, ensuring fine sampling near critical features like band gaps.
• The algorithm is implemented using ITensors.jl in Julia, with adaptive sampling guided by error tolerance and convergence criteria.

Experimental results

Research questions

• RQ1 Can the quantics representation combined with tensor cross interpolation (QTCI) efficiently capture multiscale structures in multivariate functions with minimal memory usage?
• RQ2 To what extent can QTCI outperform standard sampling or SVD-based methods in resolving sharp features such as peaks and discontinuities in high-dimensional functions?
• RQ3 How accurately can QTCI compute topological invariants like the Chern number in condensed matter systems with fine-scale momentum-space structures?
• RQ4 What is the minimal number of samples required by QTCI to achieve a given accuracy, and how does this scale with resolution R?
• RQ5 Under what conditions does a multivariate function admit a low-rank QTT representation, and how can this be detected efficiently?

Key findings

• QTCI computed the Chern number of the Haldane model with a relative error of less than 10⁻⁶ from the expected value of -1, using only 4×10⁵ samples and 20 seconds on a single-core Apple M1 processor.
• The method achieved a relative tolerance of ε = 10⁻¹⁰ with a maximum bond dimension D_max = 50 in the QTT approximation, demonstrating high accuracy and compression.
• For R = 20, the QTCI resolved the peak in the Berry curvature with a width of order δₘ = 10⁻⁵, confirming that the mesh spacing 2⁻²⁰ was sufficiently fine.
• With smaller R (e.g., R = 10–14), the mesh became too coarse, leading to incorrect plateau-like behavior at -1/2 instead of a sharp transition at δₘ = 0, highlighting the importance of resolution.
• Random sampling failed to capture the sharp structures, resulting in lower out-of-sample error than in-sample error, whereas QTCI successfully identified and resolved them.
• The QTCI method reduced the required number of function evaluations from 2²⁰ ≈ 10¹² (for direct summation) to 4×10⁵, demonstrating exponential efficiency gain.