  We consider the optimal control problem of determining electromagnetic pulses
for implementing logical gates in a closed quantum system, where the
Hamiltonian models the dynamics of coupled superconducting qudits. The quantum
state is governed by Schr\"odinger's equation, which we formulate in terms of
the real and imaginary parts of the state vector and solve by the
St\"ormer-Verlet scheme, which is a symplectic partitioned Runge-Kutta method.
A novel parameterization of the control functions based on B-splines with
carrier waves is introduced. The carrier waves are used to trigger the resonant
frequencies in the system Hamiltonian, and the B-spline functions specify their
amplitude and phase. This approach allows the number of control parameters to
be independent of, and significantly smaller than, the number of time steps for
integrating Schr\"odinger's equation.
  We present numerical examples of how the proposed technique can be combined
with an interior point L-BFGS algorithm for realizing quantum gates, and
generalize our approach to calculate risk-neutral controls that are resilient
to noise in the Hamiltonian model. The proposed method is also shown to compare
favorably with QuTiP/pulse\_optim and Grape-Tensorflow.
