  We introduce a new framework to analyze quantum algorithms with the
renormalization group (RG). To this end, we present a detailed analysis of the
real-space RG for discrete-time quantum walks on fractal networks and show how
deep insights into the analytic structure as well as generic results about the
long-time behavior can be extracted. The RG-flow for such a walk on a dual
Sierpinski gasket and a Migdal-Kadanoff hierarchical network is obtained
explicitly from elementary algebraic manipulations, after transforming the
unitary evolution equation into Laplace space. Unlike for classical random
walks, we find that the long-time asymptotics for the quantum walk requires
consideration of a diverging number of Laplace-poles, which we demonstrate
exactly for the closed form solution available for the walk on a 1d-loop. In
particular, we calculate the probability of the walk to overlap with its
starting position, which oscillates with a period that scales as
$N^{d_{w}^{Q}/d_{f}}$ with system size $N$. While the largest Jacobian
eigenvalue $\lambda_{1}$ of the RG-flow merely reproduces the fractal
dimension, $d_{f}=\log_{2}\lambda_{1}$, the asymptotic analysis shows that the
second Jacobian eigenvalue $\lambda_{2}$ becomes essential to determine the
dimension of the quantum walk via
$d_{w}^{Q}=\log_{2}\sqrt{\lambda_{1}\lambda_{2}}$. We trace this fact to
delicate cancellations caused by unitarity. We obtain identical relations for
other networks, although the details of the RG-analysis may exhibit
surprisingly distinct features. Thus, our conclusions -- which trivially
reproduce those for regular lattices with translational invariance with
$d_{f}=d$ and $d_{w}^{Q}=1$ -- appear to be quite general and likely apply to
networks beyond those studied here.
