This report proposes a numerical method for simulating on a classical computer an open quantum system composed of several open quantum subsystems. Each subsystem is assumed to be strongly stabilized exponentially towards a decoherence free sub-space, slightly impacted by some decoherence channels and weakly coupled to the other subsystems. This numerical method is based on a perturbation analysis with an original asymptotic expansion exploiting the Heisenberg formulation of the dynamics, either in continuous time or discrete time. It relies on the invariant operators of the local and nominal dissipative dynamics of the subsystems. It is shown that second-order expansion can be computed with only local calculations avoiding global computations on the entire Hilbert space. This algorithm is particularly well suited for simulation of autonomous quantum error correction schemes, such as in bosonic codes with Schr\"odinger cat states. These second-order Heisenberg simulations have been compared with complete Schr\"odinger simulations and analytical formulas obtained by second order adiabatic elimination. These comparisons have been performed three cat-qubit gates: a Z-gate on a single cat qubit; a ZZ-gate on two cat qubits; a ZZZ-gate on three cat qubits. For the ZZZ-gate, complete Schr\"odinger simulations are almost impossible when $\alpha^2$, the energy of each cat qubit, exceeds 8, whereas second-order Heisenberg simulations remain easily accessible up to machine precision. These numerical investigations indicate that second-order Heisenberg dynamics capture the very small bit-flip error probabilities and their exponential decreases versus $\alpha^2$ varying from 1 to 16. They also provides a direct numerical access to quantum process tomography, the so called $\chi$ matrix providing a complete characterization of the different error channels with their probabilities.

翻译：本文提出了一种在经典计算机上模拟由多个开放量子子系统组成的开放量子系统的数值方法。假设每个子系统被指数级强稳定化至无退相干子空间，受轻微退相干通道影响，并与其他子系统弱耦合。该方法基于摄动分析，利用海森堡动力学表述（连续时间或离散时间）进行原创渐近展开，并依赖于子系统局部标称耗散动力学的不变算符。研究表明，二阶展开可通过仅涉及局部计算实现，无需对整个希尔伯特空间进行全局计算。该算法特别适用于自主量子纠错方案的模拟，例如包含薛定谔猫态的玻色子编码。我们将二阶海森堡模拟与完整薛定谔模拟以及通过二阶绝热消除获得的解析公式进行了比较，这些比较在三种猫量子比特门上进行：单猫量子比特的Z门、双猫量子比特的ZZ门以及三猫量子比特的ZZZ门。对于ZZZ门，当每个猫量子比特的能量α²超过8时，完整薛定谔模拟几乎不可行，而二阶海森堡模拟仍可轻松达到机器精度。这些数值研究揭示了二阶海森堡动力学能够捕捉极小的比特翻转错误概率及其在α²从1到16变化时的指数衰减特性，并提供了直接数值访问量子过程断层扫描的途径，即通过χ矩阵完整表征不同错误通道及其概率。