Dynamics simulation and numerical analysis of arbitrary time-dependent $\mathcal{PT}$-symmetric system based on density operators

$\mathcal{PT}$-symmetric system has attracted extensive attention in recent years because of its unique properties and applications. How to simulate $\mathcal{PT}$-symmetric system in traditional quantum mechanical system has not only fundamental theoretical significance but also practical value. We propose a dynamics simulation scheme of arbitrary time-dependent $\mathcal{PT}$-symmetric system based on density operators, and the results are compatible with previous methods based on pure-state vectors. Based on the above, we are able to study the influence of quantum noises on the simulation results with the technique of vectorization of density operators and matrixization of superoperators (VDMS), and we show the depolarizing (Dep) noise is the most fatal and should be avoided as much as possible. Meanwhile, we also give a numerical analysis. We find that the problem of chronological product usually has to be solved not only in the numerical calculation, but also even in the experiment, because the dilated higher-dimensional Hamiltonian is usually time-dependent. Through theoretical analysis and numerical calculation, we find that on the premise of meeting the goal of calculation accuracy and saving computing resources, the time step of calculation and the cut-off term of Magnus series have to be carefully balanced.