  We give an estimate of the quantum variance for $d$-regular graphs quantised
with boundary scattering matrices that prohibit back-scattering. For families
of graphs that are expanders, with few short cycles, our estimate leads to
quantum ergodicity for these families of graphs. Our proof is based on a
uniform control of an associated random walk on the bonds of the graph. We show
that recent constructions of Ramanujan graphs, and asymptotically almost
surely, random $d$-regular graphs, satisfy the necessary conditions to conclude
that quantum ergodicity holds.
