  We show how to extract the scaling behavior of quantum walks using the
renormalization group (RG). We introduce the method by efficiently reproducing
well-known results on the one-dimensional lattice. As a nontrivial model, we
apply this method to the dual Sierpinski gasket and obtain its exact, closed
system of RG-recursions. Numerical iteration suggests that under rescaling the
system length, $L^{\prime}=2L$, characteristic times rescale as
$t^{\prime}=2^{d_{w}}t$ with the exact walk exponent
$d_{w}=\log_{2}\sqrt{5}=1.1609\ldots$. Despite the lack of translational
invariance, this is very close to the ballistic spreading, $d_{w}=1$, found for
regular lattices. However, we argue that an extended interpretation of the
traditional RG formalism will be needed to obtain scaling exponents
analytically. Direct simulations confirm our RG-prediction for $d_w$ and
furthermore reveal an immensely rich phenomenology for the spreading of the
quantum walk on the gasket. Invariably, quantum interference localizes the walk
completely with a site-access probability that declines with a powerlaw from
the initial site, in contrast with a classical random walk, which would pass
all sites with certainty.
