  Non-Markovianity may significantly speed up quantum dynamics when the system
interacts strongly with an infinite large reservoir, of which the coupling
spectrum should be fine-tuned. The potential benefits are evident in many
dynamics schemes, especially the continuous-time quantum walk. Difficulty
exists, however, in producing closed-form solutions with controllable accuracy
against the complexity of memory kernels. Here, we introduce a new
multiple-scale perturbation method that works on integro-differential equations
for general study of memory effects in dynamical systems. We propose an
open-system model in which a continuous-time quantum walk is enclosed in a
non-Markovian reservoir, that naturally corresponds to an error correction
algorithm scheme. By applying the multiple-scale method we show how emergence
of different time scales is related to transition of system dynamics into the
non-Markovian regime. We find that up to two long-term modes and two short-term
modes exist in regular networks, limited by their intrinsic symmetries. In
addition to the effective approximation by our perturbation method on general
forms of reservoirs, the speed-up of quantum walks assisted by non-Markovianity
is also confirmed, revealing the advantage of reservoir engineering in
designing time-sensitive quantum algorithms.
