  Superconducting Josephson junction qubits constitute the main current
technology for many applications, including scalable quantum computers and
thermal devices. Theoretical modeling of such systems is usually done within
the two-level approximation. However, accurate theoretical modeling requires
taking into account the influence of the higher excited states without limiting
the system to the two-level qubit subspace. Here, we study the dynamics and
control of a superconducting transmon using the numerically exact stochastic
Liouville-von Neumann equation approach. We focus on the role of state leakage
from the ideal two-level subspace for bath induced decay and single-qubit gate
operations. We find significant short-time state leakage due to the strong
coupling to the bath. We quantify the leakage errors in single-qubit gates and
demonstrate their suppression with DRAG control for a five-level transmon in
the presence of decoherence. Our results predict the limits of accuracy of the
two-level approximation and possible intrinsic constraints in qubit dynamics
and control for an experimentally relevant parameter set.
