A Riemannian Approach to the Lindbladian Dynamics of a Locally Purified Tensor Network
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916897062125568 |
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| author | Godinez-Ramirez, Emiliano Milbradt, Richard Mendl, Christian B. |
| author_facet | Godinez-Ramirez, Emiliano Milbradt, Richard Mendl, Christian B. |
| contents | Tensor networks offer a valuable framework for implementing Lindbladian dynamics in many-body open quantum systems with nearest-neighbor couplings. In particular, a tensor network ansatz known as the Locally Purified Density Operator employs the local purification of the density matrix to guarantee the positivity of the state at all times. Within this framework, the dissipative evolution utilizes the Trotter-Suzuki splitting, yielding a second-order approximation error. However, due to the Lindbladian dynamics' nature, employing higher-order schemes results in non-physical quantum channels. In this work, we leverage the gauge freedom inherent in the Kraus representation of quantum channels to improve the splitting error. To this end, we formulate an optimization problem on the Riemannian manifold of isometries and find a solution via the second-order trust-region algorithm. We validate our approach using two nearest-neighbor noise models and achieve an improvement of orders of magnitude compared to other positivity-preserving schemes. In addition, we demonstrate the usefulness of our method as a compression scheme, helping to control the exponential growth of computational resources, which thus far has limited the use of the locally purified ansatz. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_08127 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Riemannian Approach to the Lindbladian Dynamics of a Locally Purified Tensor Network Godinez-Ramirez, Emiliano Milbradt, Richard Mendl, Christian B. Quantum Physics Strongly Correlated Electrons Tensor networks offer a valuable framework for implementing Lindbladian dynamics in many-body open quantum systems with nearest-neighbor couplings. In particular, a tensor network ansatz known as the Locally Purified Density Operator employs the local purification of the density matrix to guarantee the positivity of the state at all times. Within this framework, the dissipative evolution utilizes the Trotter-Suzuki splitting, yielding a second-order approximation error. However, due to the Lindbladian dynamics' nature, employing higher-order schemes results in non-physical quantum channels. In this work, we leverage the gauge freedom inherent in the Kraus representation of quantum channels to improve the splitting error. To this end, we formulate an optimization problem on the Riemannian manifold of isometries and find a solution via the second-order trust-region algorithm. We validate our approach using two nearest-neighbor noise models and achieve an improvement of orders of magnitude compared to other positivity-preserving schemes. In addition, we demonstrate the usefulness of our method as a compression scheme, helping to control the exponential growth of computational resources, which thus far has limited the use of the locally purified ansatz. |
| title | A Riemannian Approach to the Lindbladian Dynamics of a Locally Purified Tensor Network |
| topic | Quantum Physics Strongly Correlated Electrons |
| url | https://arxiv.org/abs/2409.08127 |