  We present new intuition behind Grover's quantum search algorithm by means of
a Hamiltonian. Given a black-box Boolean function f mapping strings of length n
into {0,1} such that f(w) = 1 for exactly one string w, L. K. Grover describes
a quantum algorithm that finds w in O(2^{n/2}) time. Farhi & Gutmann show that
w can also be found in the same amount time by letting the quantum system
evolve according to a simple Hamiltonian depending only on f. Their system
evolves along a path far from that taken by Grover's original algorithm,
however. The current paper presents an equally simple Hamiltonian matching
Grover's algorithm step for step. The new Hamiltonian is similar in appearance
from that of Farhi & Gutmann, but has some important differences, and provides
new intuition for Grover's algorithm itself. This intuition both contrasts with
and supplements other explanations of Grover's algorithm as a rotation in two
dimensions, and suggests that the Hamiltonian-based approach to quantum
algorithms can provide a useful heuristic for discovering new quantum
algorithms.
