A Riemannian Approach to the Lindbladian Dynamics of a Locally Purified Tensor Network

TL;DR

This paper introduces a Riemannian optimization method to improve the accuracy of Lindbladian dynamics simulations in tensor networks, enhancing positivity preservation and computational efficiency.

Contribution

It develops a Riemannian manifold optimization approach to reduce splitting errors in locally purified tensor networks for open quantum systems.

Findings

Significant error reduction compared to existing schemes

Enhanced positivity preservation in Lindbladian dynamics

Improved tensor network compression for quantum simulations

Abstract

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.