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# Simulation of open quantum systems by automated compression of arbitrary environments

## Abstract

Studies of the dynamics of open quantum systems are limited by the large Hilbert space of typical environments, which is too large to be treated exactly. In some cases, approximate descriptions of the system are possible, for example, when the environment has a short memory time or only interacts weakly with the system. Accurate numerical methods exist, but these are typically restricted to baths with Gaussian correlations, such as non-interacting bosons. Here we present a method for simulating open quantum systems with arbitrary environments that consist of a set of independent degrees of freedom. Our approach automatically reduces the large number of environmental degrees of freedom to those which are most relevant. Specifically, we show how the process tensor describing the effect of the environment can be iteratively constructed and compressed using matrix product state techniques. We demonstrate the power of this method by applying it to a range of open quantum systems, including bosonic, fermionic and spin environments. The versatility and efficiency of our automated compression of environments method provides a practical general-purpose tool for open quantum systems.

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## Data availability

The data presented in the figures including the parameter files to generate them are available online in the ‘examples’ subdirectory of the Zenodo repository at https://doi.org/10.5281/zenodo.5214128.

## Code availability

The C++ computer code including documentation is available online at https://doi.org/10.5281/zenodo.5214128.

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## Acknowledgements

M. Cosacchi and V.M.A. are grateful for funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project no. 419036043. A.V. acknowledges support from the Russian Science Foundation under Project 18-12-00429 and from the Basic Research Program at the HSE University. M. Cygorek and E.M.G. acknowledge funding from EPSRC grant no. EP/T01377X/1. B.W.L. and J.K. were supported by EPSRC grant no. EP/T014032/1.

## Author information

Authors

### Contributions

M. Cygorek, M. Cosacchi, A.V., and V.M.A. developed the concept of explicitly constructing the PT to simulate open quantum systems with arbitrary system–environment couplings. M. Cygorek, B.W.L., J.K. and E.M.G. contributed the idea of using MPO representations for efficient storage and evaluation of the PT. M. Cygorek is responsible for the details of the algorithm, implementation in the form of C++ code and generation of data. All the authors analysed and discussed the results and contributed to writing the Article.

### Corresponding authors

Correspondence to Moritz Cygorek or Erik M. Gauger.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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## Supplementary information

### Supplementary Information

Supplementary Sections 1–6.

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Cygorek, M., Cosacchi, M., Vagov, A. et al. Simulation of open quantum systems by automated compression of arbitrary environments. Nat. Phys. 18, 662–668 (2022). https://doi.org/10.1038/s41567-022-01544-9

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• DOI: https://doi.org/10.1038/s41567-022-01544-9