Completely Positive and Trace Preserving Schemes with Tensor Train Compression for the Lindblad Equation
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| Format: | Preprint |
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2026
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| _version_ | 1866917454567964672 |
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| author | DelMastro, Peter Appelö, Daniel Cheng, Yingda |
| author_facet | DelMastro, Peter Appelö, Daniel Cheng, Yingda |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_01494 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Completely Positive and Trace Preserving Schemes with Tensor Train Compression for the Lindblad Equation DelMastro, Peter Appelö, Daniel Cheng, Yingda Numerical Analysis Quantum Physics 65L99, 15A69, 81Q99 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. |
| title | Completely Positive and Trace Preserving Schemes with Tensor Train Compression for the Lindblad Equation |
| topic | Numerical Analysis Quantum Physics 65L99, 15A69, 81Q99 |
| url | https://arxiv.org/abs/2605.01494 |