  Pumped at approximately twice the natural frequency, a Josephson parametric
oscillator called parametron or Kerr parametric oscillator shows
self-oscillation. Quantum annealing and universal quantum computation using
self-oscillating parametrons as qubits were proposed. However, controls of
parametrons under the pump field are degraded by unwanted rapidly oscillating
terms in the Hamiltonian, which we call non-resonant rapidly oscillating terms
(NROTs) coming from the violation of the rotating wave approximation.
Therefore, the pump field can be an intrinsic origin of the imperfection of
controls of parametrons. Here, we theoretically study the influence of the
NROTs on the accuracy of controls of a parametron: a cat-state creation and a
single-qubit gate. It is shown that there is a trade-off relationship between
the suppression of the nonadiabatic transitions and the validity of the
rotating wave approximation in a conventional approach. We also show that the
tailored time dependence of the detuning of the pump field can suppress both of
the nonadiabatic transitions and the disturbance of the state of the parametron
due to the NROTs.
