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Perfect State Transfer in Laplacian Quantum Walk

Abstract

For a graph GG and a related symmetric matrix MM, the continuous-time quantum walk on GG relative to MM is defined as the unitary matrix U(t)=exp(itM)U(t) = \exp(-itM), where tt varies over the reals. Perfect state transfer occurs between vertices uu and vv at time τ\tau if the (u,v)(u,v)-entry of U(τ)U(\tau) 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 nn-vertex graph has perfect state transfer at time τ\tau relative to the Laplacian, then so does its complement if nτn\tau is an integer multiple of 2π2\pi. As a corollary, the double cone over any mm-vertex graph has perfect state transfer relative to the Laplacian if and only if m2(mod4)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 GG has perfect state transfer at time τ\tau relative to the normalized Laplacian, then so does the weak product G×HG \times H if for any normalized Laplacian eigenvalues λ\lambda of GG and μ\mu of HH, we have μ(λ1)τ\mu(\lambda-1)\tau is an integer multiple of 2π2\pi. As a corollary, a weak product of P3P_{3} 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 P3P_{3} 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).Comment: 26 pages, 5 figures, 1 tabl

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