Completely Positive and Trace Preserving Schemes with Tensor Train Compression for the Lindblad Equation

TL;DR

This paper introduces a low-rank, tensor train compressed scheme for simulating the Lindblad equation in quantum systems, enabling efficient computations for extremely large systems.

Contribution

It develops a novel two-level low-rank tensor train approach integrated with a Kraus scheme for efficient Lindblad equation simulation.

Findings

Efficient tensor train operations for density matrices.

Convergence demonstrated through extensive numerical experiments.

Able to simulate systems with up to 10^{19} degrees of freedom.

Abstract

We propose a family of low-rank, completely positive and trace preserving schemes for the Lindblad equation, a common model for open quantum systems. Low-rank representation is employed at two levels: the density matrix is factorized into the product of tall-skinny matrices, and the columns of these matrices are further represented using the tensor train (TT) format, also know as matrix product states (MPS). This two-level low-rank format fits naturally into our existing Kraus is King scheme (arXiv:2409.08898v2 [math.NA]) for the Lindblad equation, whose underlying operations are arithmetic on the columns of the tall-skinny matrices. We show how these operations can be performed efficiently in the TT/MPS format, with particular emphasis on density matrix rank-truncation. We conclude with extensive numerical experiments demonstrating the convergence of this scheme and its efficiency in simulating systems with up to $10^{19}$ degrees of freedom using only modest compute resources.