  In the emerging domain of quantum algorithms, the Grover's quantum search is
certainly one of the most significant. It is relatively simple, performs a
useful task and more importantly, does it in an optimal way. However, due to
the success of quantum walks in the field, it is logical to study quantum
search variants over several kind of walks. In this paper, we propose an
in-depth study of the quantum search over a hypercube layout. First, through
the analysis of elementary walk operators restricted to suitable eigenspaces,
we show that the acting component of the search algorithm takes place in a
small subspace of the Hilbert workspace that grows linearly with the problem
size. Subsequently, we exploit this property to predict the exact evolution of
the probability of success of the quantum search in polynomial time.
