  Continuous-time quantum walks and adiabatic quantum evolution are two general
techniques for quantum computing, both of which are described by Hamiltonians
that govern their evolutions by Schr\"odinger's equation. In the former, the
Hamiltonian is fixed, while in the latter, the Hamiltonian varies with time. As
a result, their formulations of Grover's algorithm evolve differently through
Hilbert space. We show that this difference is fundamental; they cannot be made
to evolve along each other's path without introducing structure more powerful
than the standard oracle for unstructured search. For an adiabatic quantum
evolution to evolve like the quantum walk search algorithm, it must interpolate
between three fixed Hamiltonians, one of which is complex and introduces
structure that is stronger than the oracle for unstructured search. Conversely,
for a quantum walk to evolve along the path of the adiabatic search algorithm,
it must be a chiral quantum walk on a weighted, directed star graph with
structure that is also stronger than the oracle for unstructured search. Thus
the two techniques, although similar in being described by Hamiltonians that
govern their evolution, compute by fundamentally irreconcilable means.
