Computational complexity of quantum optimal control landscapes

We study the Hamiltonian-independent contribution to the complexity of quantum optimal control problems. The optimization of controls that steer quantum systems to desired objectives can itself be considered a classical dynamical system that executes an analog computation. The systemindependent component of the equations of motion of this dynamical system can be integrated analytically for various classes of discrete quantum control problems. For the maximization of observable expectation values from an initial pure state and the maximization of the fidelity of quantum gates, the time complexity of the corresponding computation belongs to the class continuous log (CLOG), the lowest analog complexity class, equivalent to the discrete complexity class NC. By contrast, the optimization of arbitrary nondegenerate observables from mixed initial states belongs to the class continuous P (CP), equivalent to the discrete complexity class P. Observables of different rank and degeneracy classes display qualitatively distinct polynomial scalings. These results indicate that the optimization via gradient flows of quantum observables from pure initial states is inherently less complex than that from mixed initial states, and suggest that local search algorithms will be most effective when the quantum state is pure.