  We show that combining ideas from the fields of quantum invariants and of
optimal control can be used to design quantum control of quantum systems
without explicit reference to quantum states. The scaling in numerical effort
of the resultant approach is given by commutation relations of system
operators, and it can be polynomial in the number of subsystems despite the
general quantum mechanical exponential scaling of the Hilbert space. As
explicit applications, we discuss state preparation and quantum simulation with
Hamiltonians including three-body and many-body interactions with spin chains
of up to 50 constituents, and the perspective of use for topologically
protected quantum information processing.
