Chebyshev approximation and composition of functions in matrix product states for quantum-inspired numerical analysis

TL;DR

This paper introduces a Chebyshev-based algorithm for representing and approximating functions as matrix product states, demonstrating rapid convergence and efficient generalization to multidimensional and compositional tasks in quantum-inspired numerical analysis.

Contribution

It presents a novel iterative Chebyshev expansion method for MPS function approximation, extending the framework to function composition and multidimensional problems, with competitive performance against existing techniques.

Findings

Rapid convergence for highly-differentiable functions

Efficient generalization to multidimensional scenarios

Competitive performance in multivariate function approximation

Abstract

This work explores the representation of univariate and multivariate functions as matrix product states (MPS), also known as quantized tensor-trains (QTT). It proposes an algorithm that employs iterative Chebyshev expansions and Clenshaw evaluations to represent analytic and highly differentiable functions as MPS Chebyshev interpolants. It demonstrates rapid convergence for highly-differentiable functions, aligning with theoretical predictions, and generalizes efficiently to multidimensional scenarios. The performance of the algorithm is compared with that of tensor cross-interpolation (TCI) and multiscale interpolative constructions through a comprehensive comparative study. When function evaluation is inexpensive or when the function is not analytical, TCI is generally more efficient for function loading. However, the proposed method shows competitive performance, outperforming TCI in certain multivariate scenarios. Moreover, it shows advantageous scaling rates and generalizes to a wider range of tasks by providing a framework for function composition in MPS, which is useful for non-linear problems and many-body statistical physics.