  Motivated by the immense success of random walk and Markov chain methods in
the design of classical algorithms, we consider_quantum_ walks on graphs. We
analyse in detail the behaviour of unbiased quantum walk on the line, with the
example of a typical walk, the ``Hadamard walk''. We show that after t time
steps, the probability distribution on the line induced by the Hadamard walk is
almost uniformly distributed over the interval [-t/sqrt(2),t/sqrt(2)]. This
implies that the same walk defined on the circle mixes in_linear_ time. This is
in direct contrast with the quadratic mixing time for the corresponding
classical walk. We conclude by indicating how our techniques may be applied to
more general graphs.
