  Quantum state transfer within a quantum computer can be achieved by using a
network of qubits, and such a network can be modelled mathematically by a
graph. Here, we focus on the corresponding Laplacian matrix, and those graphs
for which the Laplacian can be diagonalized by a Hadamard matrix. We give a
simple eigenvalue characterization for when such a graph has perfect state
transfer at time $\pi /2$; this characterization allows one to choose the
correct eigenvalues to build graphs having perfect state transfer. We
characterize the graphs that are diagonalizable by the standard Hadamard
matrix, showing a direct relationship to cubelike graphs. We then give a number
of constructions producing a wide variety of new graphs that exhibit perfect
state transfer, and we consider several corollaries in the settings of both
weighted and unweighted graphs, as well as how our results relate to the notion
of pretty good state transfer. Finally, we give an optimality result, showing
that among regular graphs of degree at most $4$, the hypercube is the sparsest
Hadamard diagonalizable connected unweighted graph with perfect state transfer.
