Perfect State Transfer in Laplacian Quantum Walk

For a graph G and a related symmetric matrix M, the continuous-time quantum walk on G relative to M is defined as the unitary matrix U(t) = exp(-itM), where t varies over the reals. Perfect state transfer occurs between vertices u and v at time τ if the (u,v)-entry of U(τ) has unit magnitude. This paper studies quantum walks relative to graph Laplacians. Some main observations include the following closure properties for perfect state transfer: (1) If a n-vertex graph has perfect state transfer at time τ relative to the Laplacian, then so does its complement if nτ is an integer multiple of 2π. As a corollary, the double cone over any m-vertex graph has perfect state transfer relative to the Laplacian if and only if m equiv 2 pmod{4}. This was previously known for a double cone over a clique (S. Bose, A. Casaccino, S. Mancini, S. Severini, Int. J. Quant. Inf., 7:11, 2009). (2) If a graph G has perfect state transfer at time τ relative to the normalized Laplacian, then so does the weak product G times H if for any normalized Laplacian eigenvalues λ of G and μ of H, we have μ(λ-1)τ is an integer multiple of 2π. As a corollary, a weak product of P₃ with an even clique or an odd cube has perfect state transfer relative to the normalized Laplacian. It was known earlier that a weak product of a circulant with odd integer eigenvalues and an even cube or a Cartesian power of P₃ has perfect state transfer relative to the adjacency matrix. As for negative results, no path with four vertices or more has antipodal perfect state transfer relative to the normalized Laplacian. This almost matches the state of affairs under the adjacency matrix (C. Godsil, Discrete Math., 312:1, 2011).