  In this Letter we make progress on a longstanding open problem of Aaronson
and Ambainis [Theory of Computing 1, 47 (2005)]: we show that if A is the
adjacency matrix of a sufficiently sparse low-dimensional graph then the
unitary operator e^{itA} can be approximated by a unitary operator U(t) whose
sparsity pattern is exactly that of a low-dimensional graph which gets more
dense as |t| increases. Secondly, we show that if U is a sparse unitary
operator with a gap \Delta in its spectrum, then there exists an approximate
logarithm H of U which is also sparse. The sparsity pattern of H gets more
dense as 1/\Delta increases. These two results can be interpreted as a way to
convert between local continuous-time and local discrete-time processes. As an
example we show that the discrete-time coined quantum walk can be realised as
an approximately local continuous-time quantum walk. Finally, we use our
construction to provide a definition for a fractional quantum fourier
transform.
