Quantum error correction under numerically exact open-quantum-system dynamics

The known quantum error-correcting codes are typically built on approximative open-quantum-system models such as Born-Markov master equations. However, it is an open question how such codes perform in actual physical systems that, to some extent, necessarily exhibit phenomena beyond the limits of these models. To this end, we employ numerically exact open-quantum-system dynamics to analyze the performance of a five-qubit error-correction code where each qubit is coupled to its own bath. We first focus on the performance of a single error-correction cycle covering timescales and coupling strengths beyond those of Born-Markov models. We observe distinct power-law behavior of the error-corrected channel infidelity ∝t2a: a≲2 in the ultrashort times t<3/ωc and a≈1/2 in the short-time range 3/ωc<t<30/ωc, where ωc is the cutoff angular frequency of the bath. Furthermore, the observed scaling of the performance of error correction is found to be robust against various imperfections in physical qubit systems, such as perturbations in qubit-qubit coupling strengths and parametric disorder. For repeated error correction, we demonstrate the breaking of the five-qubit error-correction code and of the Born-Markov models if the repetition rate of the error-correction cycles exceeds 2π/ω or the coupling strength κ≳ω/10, where ω is the angular frequency of the qubit. Our results provide bounds of validity for the standard quantum error-correction codes and pave the way for applying numerically exact open-quantum-system methods for further studies of error correction beyond simple error models and for other strongly coupled many-body models.