  There are well developed theoretical tools to analyse how quantum dynamics
can solve computational problems by varying Hamiltonian parameters slowly, near
the adiabatic limit. On the other hand, there are relatively few tools to
understand the opposite limit of rapid quenches, as used in quantum annealing
and (in the limit of infinitely rapid quenches) in quantum walks. In this
paper, we develop several tools which are applicable in the rapid quench
regime. Firstly, we analyse the energy expectation value of different elements
of the Hamiltonian. From this, we show that monotonic quenches, where the
strength of the problem Hamiltonian is consistently increased relative to
fluctuation (driver) terms, will yield a better result on average than random
guessing. Secondly, we develop methods to determine whether dynamics will occur
locally under rapid quench Hamiltonians, and identify cases where a rapid
quench will lead to a substantially improved solution. In particular, we find
that a technique we refer to as "pre-annealing" can significantly improve the
performance of quantum walks. We also show how these tools can provide
efficient heuristic estimates for Hamiltonian parameters, a key requirement for
practical application of quantum annealing.
