Constant search time algorithm via topological quantum walks

It is well-known that quantum algorithms such as Grover's can provide a quadradic speed-up for unstructured search problems. By adding topological structure to a search problem, we show that it is possible to achieve a constant search-time quantum algorithm with a constant improvement of the search probability over classical search. Specifically, we study the spatial search algorithm implemented by a two-dimensional split-step quantum random walks that realize topologically nontrivial phases and show the asymptotic search behavior is constant with growing system size. Using analytical and numerical calculations, we determine the efficient search regions in the parameter space of the quantum walker. These regions correspond to pairs of trapped states formed near a lattice defect. By studying the spectral properties of the discrete time-evolution-operators, we show that these trapped states have large overlap with the initial state. This correspondence, which is analogous to localization by constructive interference of bound states, makes it possible to reach the best possible search-time asymptotic and produce a disorder-protected fast search in quantum random walks.