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 characterize the graphs that are diagonalizable by the standard Hadamard matrix, showing a direct relationship to cubelike graphs. We give some example constructions illustrating our results. ${\bf Co-author(s):}$ N. Johnston (Mount Allison University), S. Kirkland (University of Manitoba), R. Storey (Brandon University), and X. Zhang (University of Manitoba).