局部纯化张量网络Lindbladian动力学的黎曼方法
A Riemannian Approach to the Lindbladian Dynamics of a Locally Purified Tensor Network
AI总结：
针对局部纯化张量网络Lindbladian动力学的分裂误差问题，提出基于Kraus表示规范自由度的黎曼流形优化方法，有效降低误差并控制计算资源指数增长。
AI中文摘要：
张量网络为在具有最近邻耦合的多体开放量子系统中实现Lindbladian动力学提供了有价值的框架。特别是，一种称为局部纯化密度算符的张量网络拟设，通过密度矩阵的局部纯化来保证状态在所有时刻的正定性。在此框架内，耗散演化利用Trotter-Suzuki分裂，产生二阶近似误差。然而，由于Lindbladian动力学的本质，采用高阶方案会导致非物理量子信道。在本工作中，我们利用量子信道Kraus表示中固有的规范自由度来改善分裂误差。为此，我们在等距映射的黎曼流形上构建优化问题，并通过二阶信赖域算法求解。我们使用两种最近邻噪声模型验证了该方法，与其他保持正定性的方案相比，实现了数量级的改进。此外，我们证明了该方法作为压缩方案的有效性，有助于控制计算资源的指数增长，迄今为止这种增长限制了局部纯化拟设的使用。
英文摘要：
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.
