Adiabatic elimination for composite open quantum systems: Heisenberg formulation and numerical simulations

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.