Abstract: I will present recent experiments demonstrating a qubit encoded in the harmonic motion of a single trapped 40Ca+ ion . The usage of the oscillator allows to study a logical qubit with a single quantum system, while in contrast commonly used error-correction schemes are based on arrays of many physical qubits. The approximate logical code states are formed from a periodically spaced superposition of displaced squeezed components, which has theoretically been shown to have optimal performance for a large set of errors [2, 3]. Our first time demonstration of these qubits is based on coupling the ion motional oscillator to an internal state ancillary qubit, which we can subsequently readout. This indirect readout of the oscillator via the ancillary qubit we have previously interpreted as a modular position or momentum measurement and explored the relations between sequences of these measurements . Such sequences allow us to create the logical codes states as well as measure their spatial and momentum probability densities, revealing the non-local features simultaneously present in both densities. Using the modular measurements we further implement logical state readout in the Pauli basis which we demonstrate on the six cardinal states of the Bloch sphere for which we reach an average square fidelity of 87.3 ± 0.7%. We implement the logical Pauli gates by displacements of the oscillator and realize arbitrary single qubit operations by modifying the modular measurements slightly. We analyze the performance of a universal single logical qubit gate set by performing process tomography, for Pauli gates we reach process fidelities of ≈ 97%, while for continuous rotations we achieve fidelities of ≈ 89%.
 C. Flühmann, T. L. Nguyen, M. Marinelli, V. Negnevitsky, K. Mehta, and J. Home, (2018), arXiv:1807.01033.
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 C. Flühmann, V. Negnevitsky, M. Marinelli, and J. P. Home, Phys. Rev. X 8, 021001 (2018).