  Continuous time quantum walks provide an important framework for designing
new algorithms and modelling quantum transport and state transfer problems.
Often, the graph representing the structure of a problem contains certain
symmetries that confine the dynamics to a smaller subspace of the full Hilbert
space. In this work, we use invariant subspace methods, that can be computed
systematically using Lanczos algorithm, to obtain the reduced set of states
that encompass the dynamics of the problem at hand without the specific
knowledge of underlying symmetries. First, we apply this method to obtain new
instances of graphs where the spatial quantum search algorithm is optimal:
complete graphs with broken links and complete bipartite graphs, in particular,
the star graph. These examples show that regularity and high-connectivity are
not needed to achieve optimal spatial search. We also show that this method
considerably simplifies the calculation of quantum transport efficiencies.
Furthermore, we observe improved efficiencies by removing a few links from
highly symmetric graphs. Finally, we show that this reduction method also
allows us to obtain an upper bound for the fidelity of a single qubit transfer
on an XY spin network.
