Grover's quantum search algorithm can be formulated as a quantum particle randomly walking on the (highly symmetric) complete graph, with one vertex marked by a nonzero potential. From an initial equal superposition, the state evolves in a two-dimensional subspace. Strongly regular graphs have a local symmetry that ensures that the state evolves in a three-dimensional subspace, but most have no global symmetry. Using degenerate perturbation theory, we show that quantum random walk search on known families of strongly regular graphs nevertheless achieves the full quantum speedup of Θ( √ N ), disproving the intuition that fast quantum search requires global symmetry.