复合开放量子系统的绝热消除:海森堡表述和数值模拟

本报告提出了一种数值方法,用于在经典计算机上模拟由几个开放量子子系统组成的开放量子系统。假设每个子系统都强烈地指数稳定向一些无相干自由子空间,稍微受到一些相干通道的影响,并且与其他子系统弱耦合。这种数值方法基于利用动力学的海森堡公式的原始渐近展开的一种扰动分析,无论是在连续时间还是离散时间中。它基于子系统的局部和名义耗散动力学的不变算子。证明了二阶展开可以通过只进行本地计算来计算,从而避免对整个希尔伯特空间进行全局计算。该算法特别适用于自治量子纠错方案的仿真,例如具有Schr\”odinger猫态的玻色子代码。这些二阶海森堡模拟已与完整的Schr\”odinger模拟和通过二阶绝热消除获得的解析公式进行了比较。这些比较已对三个猫量子比特门进行了执行:单猫量子比特上的Z门;两个猫量子比特上的ZZ门;三个猫量子比特上的ZZZ门。对于ZZZ门,当每个猫量子比特的能量$\alpha^2$超过8时,完整的Schr\”odinger模拟几乎是不可能的,而二阶海森堡模拟仍然容易达到机器精度。这些数值研究表明,二阶海森堡动力学捕捉了极小的比特翻转误差概率及其指数下降,其$\alpha^2$变化范围从1到16。它们还提供了对量子过程拓扑的直接数值访问,即所谓的$\chi$矩阵,它提供了不同误差通道及其概率的完整特征。
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.
论文链接:http://arxiv.org/pdf/2303.05089v1


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