Rank-adaptive time integration of tree tensor networks

TL;DR

This paper introduces a rank-adaptive integrator for high-order tensor differential equations using tree tensor networks, which efficiently updates bases and truncates ranks to control error, preserving key properties in quantum and gradient systems.

Contribution

The paper presents a novel rank-adaptive integrator for tree tensor networks that efficiently manages tensor ranks and preserves physical properties, with demonstrated numerical effectiveness.

Findings

Memory usage is linear in tensor order and mode dimension.

The integrator preserves norm and energy in Schrödinger equations.

Numerical experiments validate the robustness and efficiency of the method.

Abstract

A rank-adaptive integrator for the approximate solution of high-order tensor differential equations by tree tensor networks is proposed and analyzed. In a recursion from the leaves to the root, the integrator updates bases and then evolves connection tensors by a Galerkin method in the augmented subspace spanned by the new and old bases. This is followed by rank truncation within a specified error tolerance. The memory requirements are linear in the order of the tensor and linear in the maximal mode dimension. The integrator is robust to small singular values of matricizations of the connection tensors. Up to the rank truncation error, which is controlled by the given error tolerance, the integrator preserves norm and energy for Schrodinger equations, and it dissipates the energy in gradient systems. Numerical experiments with a basic quantum spin system illustrate the behavior of the proposed algorithm.