On low-rank tensor train approximability for linear nearest neighbor systems

Low-rank tensor methods are an important tool in the numerical treatment of equations with a high-dimensional state space.

Nearest neighbor interaction systems like the Ising model or more general Markov jump processes, as well as 1D finite-state quantum systems are examples of such problems.

While low-rank tensor train/matrix product state models have been shown to be highly efficient for the simulation of such systems, providing theoretical justification for this remains a challenging task.

One approach for obtaining estimates on required ranks for certain accuracies is to investigate the rank increase in Krylov subspace methods for solving the problem at hand.

In the context of area laws for ground states of 1D spin systems, nontrivial results on rank-increasing properties of nearest neighbor operator polynomials have been obtained in work of Arad et al. [arXiv:1301.1162] by studying the partial commutativity of local operators.

In the present work, this technique is applied to polynomial methods for definite linear equations and dissipative linear ODEs with nearest neighbor structure.

This allows to derive corresponding low-rank approximability statements for solutions of such problems which are independent of the system size.

Numerical simulations of high-dimensional nearest neighbor systems illustrate the theoretical findings.