I am reading Childs' paper on discrete and continuous quantum walks. I do not really understand why quantum walks are useful--- as implementing the quantum walk operator requires knowing the principal eigenvector of a Hamiltonian (The eigenvector with eigenvalue equal to the norm of the Hamiltonian).
The paper does mention this:
However, it is straightforward to implement the walk for many cases of interest, such as for an unweighted regular graph.
But it is confusing to me. What is the graph meant to represent in this case? The Hamiltonian? If not the Hamiltonian, how could we simulate Hamiltonian evolution using this unweighted regular graph method. And how would we construct the walk operator from this?