Emiliano Godinez-Ramirez, Richard Milbradt, Christian B. Mendl
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A 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 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.