  We consider quantum state transfer relative to the Laplacian matrix of a
graph. Let $N(u)$ denote the set of all neighbors of a vertex $u$ in a graph
$G$. A pair of vertices $u$ and $v$ are called twin vertices of $G$ provided
$N(u)\setminus\{v \}=N(v)\setminus\{u \}$. We investigate the existence of
quantum state transfer between a pair of twin vertices in a graph when the edge
between the vertices is perturbed. We find that removal of any set of pairwise
non-adjacent edges from a complete graph with a number of vertices divisible by
$4$ results Laplacian perfect state transfer (or LPST) at $\frac{\pi}{2}$
between the end vertices of every edge removed. Further, we show that all
Laplacian integral graphs with a pair of twin vertices exhibit LPST when the
edge between the vertices is perturbed. In contrast, we conclude that LPST can
be achieved in every complete graph between the end vertices of any number of
suitably perturbed non-adjacent edges. The results are further generalized to
obtain a family of edge perturbed circulant graphs exhibiting Laplacian pretty
good state transfer (or LPGST) between twin vertices. A subfamily of which is
also identified to admit LPST at $\frac{\pi}{2}$.
