Constructing parsimonious control functions using B-splines with carrier waves

We consider the optimal control problem for realizing logical gates in closed quantum systems, where the evolution of the state vector is governed by the time-dependent Schroedinger equation. The number of parameters in the control functions is made independent of the number of time steps by expanding them in terms of B-spline basis functions, with and without carrier waves. We use an interior point gradient-based technique from the IPOPT package to minimize the gate infidelity subject to amplitude constraints on the control functions. The symplectic Stromer-Verlet scheme is used to integrate a real-valued formulation of Schroedinger's equation in time and the gradient of the gate infidelity is obtained by solving the corresponding adjoint equation. This allows all components of the gradient to be calculated at the cost of solving 3 Schroedinger systems, independently of the number of parameters in the control functions. The method is applied to Hamiltonians that model the dynamics of several coupled super-conducting qubits. We find that including judiciously chosen frequencies in the carrier waves of the basis functions can significantly reduce the number of parameters and lead to smoother control functions.