bibkey: connelly2017universality authors: Erin Connelly, Nathaniel Grammel, Michael Kraut, Luis Serazo, Christino Tamon year: 2017 title: “Universality in perfect state transfer” doi: 10.1016/j.laa.2017.06.015 url: https://arxiv.org/abs/1701.04145v2 claim: “For the continuous-time quantum walk U(t) = exp(-i t A) of a graph with Hermitian adjacency matrix A, universal perfect state transfer means that for all vertices u, v some time gives |U(t)_{v,u}| = 1; a graph has property T if every nonzero entry of A has modulus 1. The paper notes that K_2 and Circ(0,-i,i) are the only known complex unit gain graphs with universal perfect state transfer and conjectures that Circ(0,-i,i) is the only circulant with property T which has universal perfect state transfer.” strata_touched:

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Universality in perfect state transfer

E. Connelly, N. Grammel, M. Kraut, L. Serazo, C. Tamon, arXiv:1701.04145 (v1 2017-01-16, v2 2017-01-19, the latest); Linear Algebra Appl. 531 (2017) 516–532. Subjects: quant-ph, math.CO.

The paper studies graphs whose continuous-time quantum walk U(t) = exp(−i t A) has perfect state transfer between every pair of vertices (universal perfect state transfer). It characterizes such graphs through switching equivalence (MA = BM for a monomial matrix M) and the circulants Circ(a_0, …, a_{n−1}), C_{jk} = a_{k−j}, and quotes the eigenvalue criterion of Cameron et al. for circulants. The concluding section considers complex unit gain graphs and states:

We say a graph has {\em property $\TT$} if all of the nonzero coefficients
in its adjacency matrix are complex numbers with unit magnitude.
\begin{conjecture} $\Circ(0,-\ii,\ii)$ is the only circulant with property
$\TT$ which has universal perfect state transfer. \end{conjecture}

The introduction adds:

The only known examples of complex unit gain graphs
with the universal property are the circulants $K_{2}$ and $\Circ(0,-\ii,\ii)$.
We conjecture that this set is unique.

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• DOI: https://doi.org/10.1016/j.laa.2017.06.015 (the journal text was not read).
• URL: https://arxiv.org/abs/1701.04145v2 (source retrieved 2026-10-01): main.tex, the quantum walk and perfect state transfer (l. 150–156), the universal property (abstract, l. 128–130), the circulant convention (l. 230–231), monomial matrices and switching equivalence (l. 241–243), the introduction’s two examples (l. 210–212), the Cameron et al. criterion (l. 566–577), and property 𝕋 with the conjecture (l. 1150–1155).