Encoding a qubit in a trapped-ion mechanical oscillator

The stable operation of quantum computers will rely on error-correction, in which single quantum bits of information are stored redundantly in the Hilbert space of a larger system. Such encoded qubits are commonly based on arrays of many physical qubits, but can also be realized using a single higher-dimensional quantum system, such as a harmonic oscillator. A powerful encoding is formed from a periodically spaced superposition of position eigenstates. Various proposals have been made for realizing approximations to such states, but these have thus far remained out of reach. Here, we demonstrate such an encoded qubit using a superposition of displaced squeezed states of the harmonic motion of a single trapped Calcium ion, controlling and measuring the oscillator through coupling to an ancilliary internal-state qubit. We prepare and reconstruct logical states with an average square fidelity of $87.3 \pm 0.7 \%$, and demonstrate a universal logical single qubit gate set which we analyze using process tomography. For Pauli gates we reach process fidelities of $\approx 97\%$, while for continuous rotations we use gate teleportation achieving fidelities of $\approx 89 \%$. The control demonstrated opens a route for exploring continuous variable error-correction as well as hybrid quantum information schemes using both discrete and continuous variables. The code states also have direct applications in quantum sensing, allowing simultaneous measurement of small displacements in both position and momentum.