Spatial search for topological defects

Evolution under Dirac Hamiltonian is known to provide an almost-Grover speedup (i.e. search takes O(\sqrt(N)) time up to polylog(N) corrections) for quantum spatial search on a 2D lattice. We consider a special case when the marked location is created by a topological defect. The unusual feature of this problem is that the topological defect cannot be introduced to the Hamiltonian by any local perturbation.

We introduce a 6-8 topological defect (dislocation) in the honeycomb lattice and argue that this leads to a similar almost-O(\sqrt(N)) search time. Additionally, we investigate the quasi-bound states on such defects that are crucial for this quantum search. Interestingly, topological defects, such as the 6-8 defect, are known to occur in graphene systems in experiment.