  We investigate a set of discrete-time quantum search algorithms on the
n-dimensional hypercube following a proposal by Shenvi, Kempe and Whaley. We
show that there exists a whole class of quantum search algorithms in the
symmetry reduced space which perform a search of a marked vertex in time of
order $\sqrt{N}$ where $N = 2^n$, the number of vertices. In analogy to
Grover's algorithm, the spatial search is effectively facilitated through a
rotation in a two-level sub-space of the full Hilbert space. In the hypercube,
these two-level systems are introduced through avoided crossings. We give
estimates on the quantum states forming the 2-level sub-spaces at the avoided
crossings and derive improved estimates on the search times.
