  In variational quantum algorithms (VQAs), the most common objective is to
find the minimum energy eigenstate of a given energy Hamiltonian. In this
paper, we consider the general problem of finding a sufficient control
Hamiltonian structure that, under a given feedback control law, ensures
convergence to the minimum energy eigenstate of a given energy function. By
including quantum non-demolition (QND) measurements in the loop, convergence to
a pure state can be ensured from an arbitrary mixed initial state. Based on
existing results on strict control Lyapunov functions, we formulate a
semidefinite optimization problem, whose solution defines a non-unique control
Hamiltonian, which is sufficient to ensure almost sure convergence to the
minimum energy eigenstate under the given feedback law and the action of QND
measurements. A numerical example is provided to showcase the proposed
methodology.
