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Continuous time quantum walks provide an important framework for designingnew algorithms and modelling quantum transport and state transfer problems.Often, the graph representing the structure of a problem contains certainsymmetries that confine the dynamics to a smaller subspace of the full Hilbertspace. In this work, we use invariant subspace methods, that can be computedsystematically using Lanczos algorithm, to obtain the reduced set of statesthat encompass the dynamics of the problem at hand without the specificknowledge of underlying symmetries. First, we apply this method to obtain newinstances 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 arenot needed to achieve optimal spatial search. We also show that this methodconsiderably simplifies the calculation of quantum transport efficiencies.Furthermore, we observe improved efficiencies by removing a few links fromhighly symmetric graphs. Finally, we show that this reduction method alsoallows us to obtain an upper bound for the fidelity of a single qubit transferon an XY spin network.

Systematic Dimensionality Reduction for Quantum Walks: Optimal Spatial Search and Transport on Non-Regular Graphs