Completely Positive and Trace Preserving Schemes with Tensor Train Compression for the Lindblad Equation
This work addresses the computational bottleneck of simulating large open quantum systems by providing an efficient numerical method.
The authors propose a low-rank, completely positive and trace preserving scheme for the Lindblad equation using tensor train compression, enabling simulation of systems with up to 10^19 degrees of freedom with modest compute resources.
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.