  Based on studies on four specific networks, we conjecture a general relation
between the walk dimensions $d_{w}$ of discrete-time random walks and quantum
walks with the (self-inverse) Grover coin. In each case, we find that $d_{w}$
of the quantum walk takes on exactly half the value found for the classical
random walk on the same geometry. Since walks on homogeneous lattices satisfy
this relation trivially, our results for heterogeneous networks suggests that
such a relation holds irrespective of whether translational invariance is
maintained or not. To develop our results, we extend the renormalization group
analysis (RG) of the stochastic master equation to one with a unitary
propagator. As in the classical case, the solution $\rho(x,t)$ in space and
time of this quantum walk equation exhibits a scaling collapse for a variable
$x^{d_{w}}/t$ in the weak limit, which defines $d_{w}$ and illuminates
fundamental aspects of the walk dynamics, e.g., its mean-square displacement.
We confirm the collapse for $\rho(x,t)$ in each case with extensive numerical
simulation. The exact values for $d_{w}$ in themselves demonstrate that RG is a
powerful complementary approach to study the asymptotics of quantum walks that
weak-limit theorems have not been able to access, such as for systems lacking
translational symmetries beyond simple trees.
