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arXiv · 2605.01494

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

Peter DelMastro·Daniel Appelö·Yingda Cheng
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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.

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Peter DelMastro, Daniel Appelö, Yingda Cheng. 2026-05-02. Completely Positive and Trace Preserving Schemes with Tensor Train Compression for the Lindblad Equation. https://arxiv.org/abs/2605.01494

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