Control and tomography of bosonic qubit encodings in a multi-level transmon device

Bosonic qubit encodings have attracted significant interest as a potential path towards fault-tolerant quantum computation, with recent error correction experiments surpassing the break-even point. Such encodings typically use the infinite-dimensional Hilbert space of a harmonic oscillator, generally realized in a three-dimensional superconducting cavity, with an ancillary nonlinearity for control, syndrome measurement, and logical readout. In this work, we propose to encode a logical qubit directly in the multi-dimensional Hilbert space of a transmon. We simultaneously increase the number of levels confined in the transmon potential and decrease the effect of charge dispersion in the higher levels by designing our device to operate with a large EJ/EC. We take the first step towards error correction in this system by demonstrating the initialization and control of logical states, implementing a truncated displacement operation, and directly measuring a finite-dimensional phase space representation analogous to the Wigner function. Importantly, the natural anharmonicity of the transmon potential permits resonant driving of transitions and dispersive readout of the state without the need for an ancillary nonlinearity, thus minimizing a major source of error and simplifying control.