Approximation and composition of functions in quantized tensor trains via orthogonal polynomial expansions
quant-ph, cs.NA, math.NA
Submitted: 2024-07-12
Updated: 2026-09-16
Comments: 19 pages, 16 figures. Published in Linear Algebra and its Applications
Journal ref: Linear Algebra and its Applications 752 (2027) 278-307
DOI: 10.1016/j.laa.2026.08.022
License: http://creativecommons.org/licenses/by/4.0/
The gist: This work explores the representation of univariate and multivariate functions as quantized tensor trains (QTT).
Terminology
Abstract
This work explores the representation of univariate and multivariate functions as quantized tensor trains (QTT). It develops a constructive algorithm that employs expansions in orthogonal polynomial bases and Clenshaw evaluations to represent analytic and highly differentiable functions as QTT approximants. This approach provides a general framework for function composition in QTT form that generalizes efficiently to multidimensional settings. Without loss of generality, the presentation focuses on the Chebyshev basis, where the method demonstrates rapid convergence for highly differentiable functions, in agreement with classical theoretical predictions, and remains numerically stable at high polynomial orders. The performance of the algorithm is compared with that of tensor cross-interpolation (TCI) and multiscale interpolative constructions, demonstrating competitive performance and favorable scaling behavior in certain scenarios.
Sources
- Quantum-inspired algorithms for multivariate analysis: from interpolation to partial differential equations
- Quantum Fourier analysis for multivariate functions and applications to a class of Schr\"odinger-type partial differential equations
- A Quantum Inspired Approach to Exploit Turbulence Structures
- Parallel cross interpolation for high-precision calculation of high-dimensional integrals
- Tensorization of neural networks for improved privacy and interpretability
- Multiscale interpolative construction of quantized tensor trains
- Direct interpolative construction of the discrete Fourier transform as a matrix product operator
- Functional Tensor-Train Chebyshev Method for Multidimensional Quantum Dynamics Simulations
- Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness
- Efficient MPS representations and quantum circuits from the Fourier modes of classical image data
- Approximation Theory of Tree Tensor Networks: Tensorized Multivariate Functions
- Approximation by tree tensor networks in high dimensions: Sobolev and compositional functions
- Quasioptimality of maximum-volume cross interpolation of tensors
- Comparative study of matrix product state/quantized tensor-train algorithms for solving time-independent partial differential equations
- SeeMPS: A Python-based Matrix Product State and Tensor Train Library
- Rank Bounds for Approximating Gaussian Densities in the Tensor-Train Format
- Multigrid Renormalization
- Black box approximation in the tensor train format initialized by ANOVA decomposition
Related papers
- Reconquering Bell sampling on qudits: stabilizer learning and testing, quantum pseudorandomness bounds, and more
- Encrypted clones can leak: Classification of informative subsets in Quantum Encrypted Cloning
- Polynomial-time classical and quantum simulation of quantum impurity models
- A quasiparticle-protected superconducting qubit
- Theory of quantum-enhanced interferometry with general Markovian light sources
- A convergent hierarchy of spectral gap certificates for qubit Hamiltonians