  On thinking of a Grover-type quantum search algorithm for an ordered tuple of
multi-qubit states, a gradient system associated with the negative von-Neumann
entropy is studied on the space of regular relative-configurations of
multi-qubit states (SR${}^2$CMQ). The SR${}^2$CMQ emerges, through a geometric
procedure, from the space of ordered tuples of multi-qubit states for the
quantum search. The aim of this paper is to give a brief report on the
integrability of the gradient dynamical system together with quantum
information geometry of the underlying space, SR${}^2$CMQ, of that system.
