Iterative Phase Optimisation of Elementary Quantum Error Correcting Codes

Performing experiments on small-scale quantum computers is certainly a challenging endeavor. Many parameters need to be optimized to achieve high-fidelity operations. This can be done efficiently for operations acting on single qubits as errors can be fully characterized. For multi-qubit operations, though, this is no longer the case as in the most general case analyzing the effect of the operation on the system requires a full state tomography for which resources scale exponentially with the system size. Furthermore, in recent experiments additional electronic levels beyond the two-level system encoding the qubit have been used to enhance the capabilities of quantum information processors, which additionally increases the number of parameters that need to be controlled. For the optimization of the experimental system for a given task (e.g.~a quantum algorithm), one has to find a satisfactory error model and also efficient observables to estimate the parameters of the model. In this manuscript we demonstrate a method to optimize the encoding procedure for a small quantum error correction code in the presence of unknown but constant phase shifts. The method, which we implement here on a small-scale linear ion-trap quantum computer, is readily applicable to other AMO platforms for quantum information processing.