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：本报告提出一种在经典计算机上模拟由多个开放量子子系统组成的复合开放量子系统的数值方法。每个子系统被假设为强指数稳定地收敛至无退相干子空间，并受到轻微退相干通道影响，且与其他子系统弱耦合。该数值方法基于摄动分析，采用原创渐近展开，利用连续时间或离散时间下动力学的海森堡表述。其核心依赖于子系统局域标称耗散动力学的不变算符。研究表明，仅需局域计算即可完成二阶展开，避免对整个希尔伯特空间的全局计算。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。这些比较基于三种猫量子比特门：单猫量子比特上的Z门、双猫量子比特上的ZZ门、三猫量子比特上的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，而二阶海森堡模拟在机器精度范围内仍易于实现。数值研究表明，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随α²从1到16变化的指数衰减。该方法还直接提供了量子过程层析成像的数值途径，即所谓的χ矩阵，可完整表征不同错误通道及其概率。