## Abstract

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 \(\tau \) if the (*u*, *v*)-entry of \(U(\tau )\) has unit magnitude. This paper studies quantum walks relative to graph Laplacians. Some main observations include the following closure properties for perfect state transfer. If an *n*-vertex graph has perfect state transfer at time \(\tau \) relative to the Laplacian, then so does its complement if \(n\tau \in 2\pi {\mathbb {Z}}\). As a corollary, the join of \(\overline{K}_{2}\) with 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 the join of \(\overline{K}_{2}\) with a clique (Bose et al. in Int J Quant Inf 7:713–723, 2009). If a graph *G* has perfect state transfer at time \(\tau \) relative to the normalized Laplacian, then so does the weak product \(G \times H\) if for any normalized Laplacian eigenvalues \(\lambda \) of *G* and \(\mu \) of *H*, we have \(\mu (\lambda -1)\tau \in 2\pi {\mathbb {Z}}\). As a corollary, a weak product of \(P_{3}\) 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 \(P_{3}\) 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 (Godsil in Discret Math 312(1):129–147, 2011).

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## Acknowledgments

We would like to thank Ada Chan, Gabriel Coutinho and Chris Godsil for their generous and helpful comments. We also thank the anonymous reviewers for comments and suggestions which improve this paper, and for pointing out the work of Bayat et al. [4]. The research of R.A., S.D., B.L., J.M. and C.T. was supported by a National Science Foundation Grant DMS-1262737 and a National Security Agency Grant H98230-14-1-0141. The research of H.Z. is supported by a Graduate Student Fellowship at the University of Waterloo while working the guidance of Chris Godsil.

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Alvir, R., Dever, S., Lovitz, B. *et al.* Perfect state transfer in Laplacian quantum walk.
*J Algebr Comb* **43, **801–826 (2016). https://doi.org/10.1007/s10801-015-0642-x

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### Keywords

- Laplacian
- Quantum walk
- Perfect state transfer
- Join
- Equitable partition
- Weak product

### Mathematics Subject Classification

- MC05C50