Numerical Optimal Control of Open Quantum Systems*

We consider the ground state initialization problem for a superconducting (multi-level) qudit in a 3-D cavity, where the quantum system is modeled by Lindblad's master equation. This is an optimal control problem, which aims to find control pulses that passively drive the qudit and the cavity to the ground state, independent of the initial state. To represent the control waveforms we use a flexible ansatz consisting of B-spline basis functions combined with carrier waves that trigger the known transition frequencies of the Hamiltonian. The objective of the optimization is to minimize a linear convex combination of the expected energy levels in each of the subsystems at the final time. Based on linearity, we can drive any initial state towards the desired final ground state through one superposition of basis states that spans all possible initial conditions; thus reducing the computational burden drastically. The optimization problem is solved iteratively using the L-BFGS algorithm combined with a projected line-search algorithm to satisfy amplitude bounds on the control functions. Numerical examples indicate that the reset time for a qudit can be significantly reduced compared to previous techniques.