Quantum control with noisy fields: Computational complexity versus sensitivity to noise

A closed quantum system is defined as completely controllable if an arbitrary unitary transformation can be executed using the available controls. In practice, control fields are a source of unavoidable noise, which has to be suppressed to retain controllability. Can one design control fields such that the effect of noise is negligible on the time-scale of the transformation? This question is intimately related to the fundamental problem of a connection between the computational complexity of the control problem and the sensitivity of the controlled system to noise. The present study considers a paradigm of control, where the Lie-algebraic structure of the control Hamiltonian is fixed, while the size of the system increases with the dimension of the Hilbert space representation of the algebra. We find two types of control tasks, easy and hard. Easy tasks are characterized by a small variance of the evolving state with respect to the operators of the control operators. They are relatively immune to noise and the control field is easy to find. Hard tasks have a large variance, are sensitive to noise and the control field is hard to find. The influence of noise increases with the size of the system, which is measured by the scaling factor N of the largest weight of the representation. For fixed time and control field the ability to control degrades as O(N) for easy tasks and as O(N²) for hard tasks. As a consequence, even in the most favorable estimate, for large quantum systems, generic noise in the controls dominates for a typical class of target transformations, i.e. complete controllability is destroyed by noise.