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

TL;DR

This work addresses the challenge of loading and manipulating generic functions within matrix product states by marrying Chebyshev interpolation with a finite-precision nonlinear algebra for MPS. The proposed method yields exponential convergence for analytic targets and supports function composition directly in MPS/QTT, enabling non-linear processing in quantum-inspired numerical analysis. Through comprehensive univariate and multivariate benchmarks, the authors show competitive performance against tensor cross-interpolation and multiscale interpolative constructions in several multivariate scenarios, while outlining regimes where TCI or classical methods may outperform. The study provides a practical framework for function loading, composition, and higher-dimensional extensions in MPS/QTT, with implications for non-linear problems and many-body statistical physics, and it supplies open-source tooling for replication.

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.

Figures (18)

• Figure 1: Matrix product state composed of $n$ mutually contracted rank-3 tensors $A_i$, with virtual indices $\alpha_i$ and physical indices $s_i$.
• Figure 2: Illustration of the tensor product of two MPS $\ket{A}$ and $\ket{B}$ in serial (a) and interleaved (b) orders.
• Figure 3: Representation of the first five Chebyshev polynomials.
• Figure 4: Four qualitatively different functions proposed for the univariate performance evaluation of the different methods. The color code labeling each function is preserved along the subsequent studies. In blue, a Gaussian probability distribution $f_G$\ref{['eq:3_func_gaussian']}. In red, an oscillating function $f_O$\ref{['eq:3_func_osc']}. In orange, the absolute value function $f_A$\ref{['eq:3_func_abs']}. Lastly, in green, the Heaviside step function $f_S$\ref{['eq:3_func_step']}.
• Figure 5: Results for the MPS/QTT Chebyshev approximation of the four univariate functions $f_G$\ref{['eq:3_func_gaussian']}, $f_O$\ref{['eq:3_func_osc']}, $f_A$\ref{['eq:3_func_abs']} and $f_S$\ref{['eq:3_func_step']}. This study considers nine independent experiments given by the combination of the three parameter settings with the three figures of merit. All axes are mutually shared and shown in logarithmic scale. The second row contains a thin dotted curve representing the maximum bond dimension of a SVD decomposition. The results are asymptotically fitted with thick dashed curves and the scalings are shown in the legend. The main messages of the results can be understood by columns. The first column suggests that MPS polynomial approximations are of low entanglement, as their bond dimensions saturate to a value that is independent of the grid and only function-dependent. The second column illustrates that the method converges according to the theoretical rates for Chebyshev expansions, but with a complexity that scales algebraically below the bound for MPS polynomials. The third column demonstrates that the local truncation error of the MPS, given by the tolerance $\epsilon$, provides a good proxy for the error and complexity entailed in the approximations.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2