Riemannian approach to the Lindbladian dynamics of a locally purified tensor network

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 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 nonphysical 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 .