Optimal control of a leaking qubit

Physical implementations of quantum bits can contain coherent transitions to energetically close nonqubit states. In particular, for inharmonic-oscillator systems such as the superconducting phase qubit and the transmon, a two-level approximation is insufficient. We apply optimal control theory to the envelope of a resonant Rabi pulse in a qubit in the presence of a single weakly off-resonant leakage level. The gate error of a spin-flip (NOT) operation reduces by orders of magnitude compared to simple pulse shapes. Near-perfect gates can be achieved for any pulse duration longer than an intrinsic limit given by the nonlinearity. The pulses can be understood as composite sequences that refocus the leakage transition. We also discuss ways to improve the pulse shapes.