  A controlled quantum system possesses a search landscape defined by the
target physical objective as a function of the controls. This paper focuses on
the landscape for the transition probability Pif between the states of a finite
level quantum system. Traditionally, the controls are applied fields; here we
extend the notion of control to also include the Hamiltonian structure, in the
form of time independent matrix elements. Level sets of controls that produce
the same transition probability value are shown to exist at the bottom Pif =
0.0 and top Pif = 1.0 of the landscape with the field and/or Hamiltonian
structure as controls. We present an algorithm to continuously explore these
level sets starting from an initial point residing at either extreme value of
Pif . The technique can also identify control solutions that exhibit the
desirable properties of (a) robustness at the top and (b) the ability to
rapidly rise towards an optimal control from the bottom. Numerical simulations
are presented to illustrate the varied control behavior at the top and bottom
of the landscape for several simple model systems.
