  We introduce $\varepsilon$-projectors, using which we can sample from
limiting distributions of continuous-time quantum walks. The standard algorithm
for sampling from a distribution that is close to the limiting distribution of
a given quantum walk is to run the quantum walk for a time chosen uniformly at
random from a large interval, and measure the resulting quantum state. This
approach usually results in an exponential running time.
  We show that, using $\varepsilon$-projectors, we can sample exactly from the
limiting distribution. In the black-box setting, where we only have query
access to the adjacency matrix of the graph, our sampling algorithm runs in
time proportional to $\Delta^{-1}$, where $\Delta$ is the minimum spacing
between the distinct eigenvalues of the graph. In the non-black-box setting, we
give examples of graphs for which our algorithm runs exponentially faster than
the standard sampling algorithm.
