Uncomputability and complexity of quantum control

In laboratory and numerical experiments, physical quantities are known with a fnite precision and described by rational numbers. Based on this, we deduce that quantum control problems both for open and closed systems are in general not algorithmically solvable, i.e., there is no algorithm that can decide whether dynamics of an arbitrary quantum system can be manipulated by accessible external interactions (coherent or dissipative) such that a chosen target reaches a desired value. This conclusion holds even for the relaxed requirement of the target only approximately attaining the desired value. These fndings do not preclude an algorithmic solvability for a particular class of quantum control problems. Moreover, any quantum control problem can be made algorithmically solvable if the set of accessible interactions (i.e., controls) is rich enough. To arrive at these results, we develop a technique based on establishing the equivalence between quantum control problems and Diophantine equations, which are polynomial equations with integer coefcients and integer unknowns. In addition to proving uncomputability, this technique allows to construct quantum control problems belonging to diferent complexity classes. In particular, an example of the control problem involving a two-mode coherent feld is shown to be NP-hard, contradicting a widely held believe that two-body problems are easy.