  We analyse the eigenvalue and eigenvector structure of the flip-flop quantum
walk on regular graphs, explicitly demonstrating how it is quadratically faster
than the classical random walk. Then we use it in a controlled spatial search
algorithm with multiple target states, and determine the oracle complexity as a
function of the spectral gap and the number of target states. The oracle
complexity is optimal as a function of the graph size and the number of target
states, when the spectral gap of the adjacency matrix is $\Theta(1)$. It is
also optimal for spatial search on $D>4$ dimensional hypercubic lattices.
Otherwise it matches the best result available in the literature, with a much
simpler algorithm. Our results also yield bounds on the classical hitting time
of random walks on regular graphs, which may be of independent interest.
