  A logical qubit is a two-dimensional subspace of a higher dimensional system,
chosen such that it is possible to detect and correct the occurrence of certain
errors. Manipulation of the encoded information generally requires arbitrary
and precise control over the entire system. Whether based on multiple physical
qubits or larger dimensional modes such as oscillators, the individual elements
in realistic devices will always have residual interactions which must be
accounted for when designing logical operations. Here we demonstrate a holistic
control strategy which exploits accurate knowledge of the Hamiltonian to
manipulate a coupled oscillator-transmon system. We use this approach to
realize high-fidelity (99%, inferred), decoherence-limited operations on a
logical qubit encoded in a superconducting cavity resonator using
four-component cat states. Our results show the power of applying numerical
techniques to control linear oscillators and pave the way for utilizing their
large Hilbert space as a resource in quantum information processing.
