  Continuous-time quantum walks provide a natural framework to tackle the
fundamental problem of finding a node among a set of marked nodes in a graph,
known as spatial search. Whether spatial search by continuous-time quantum walk
provides a quadratic advantage over classical random walks has been an
outstanding problem. Thus far, this advantage is obtained only for specific
graphs or when a single node of the underlying graph is marked. In this
article, we provide a new continuous-time quantum walk search algorithm that
completely resolves this: our algorithm can find a marked node in any graph
with any number of marked nodes, in a time that is quadratically faster than
classical random walks. The overall algorithm is quite simple, requiring time
evolution of the quantum walk Hamiltonian followed by a projective measurement.
A key component of our algorithm is a purely analog procedure to perform
operations on a state of the form $e^{-tH^2}|\psi\rangle$, for a given
Hamiltonian $H$: it only requires evolving $H$ for time scaling as $\sqrt{t}$.
This allows us to quadratically fast-forward the dynamics of a continuous-time
classical random walk. The applications of our work thus go beyond the realm of
quantum walks and can lead to new analog quantum algorithms for preparing
ground states of Hamiltonians or solving optimization problems.
