Locally Purified Density Operators: Simulation, Tomography, and Topology

Studying open quantum systems is essential for exploring novel quantum phenomena and evaluating noisy quantum computation. We systematically explore the tensor network representation for mixed states, namely locally purified density operator (LPDO), and its applications in both quantum computation and topological quantum phases. Firstly, we address the problem of whether mixed states generated from noisy quantum circuits can be efficiently represented by LPDO. With a unified scheme for managing virtual and Kraus bonds, we derive a critical scaling of circuit depth with respect to error rate, which is numerically validated by noisy random quantum circuits with depths up to d=40, using fidelity and entanglement entropy as accuracy measures. Secondly, we introduce a new tomography approach that uses LPDO to represent mixed states and employs a classical optimization algorithm requiring only local measurements. Our proposed methodology (Grad-LPDO method) exhibits remarkable efficiency, accuracy, and robustness for various 1D and 2D systems. Experiments on the IBM and Quafu Quantum platforms complement these numerical simulations. Finally, we use LPDO to perform the classification and construction of average symmetry-protected topological (ASPT) phases in both 1D and 2D systems. In particular, we construct examples of intrinsic ASPT states without pure-state counterparts using the LPDO formalism. In short, these works advance our understanding of open quantum systems from different perspectives and promote the development of quantum information processing techniques with the tensor network method.