Abstract
We characterize quantum perfect state transfer on real-weighted graphs of the Johnson scheme \({\mathcal {J}}(n,k)\), which represent spin networks with non-nearest neighbor couplings. Given \({\mathcal {J}}(n,k)=\{A_{1},A_{2},\ldots ,A_{k}\}\) and \(A(X) = w_0A_0 + \cdots + w_m A_m\), we show that X has perfect state transfer at time \(\tau \) if and only if \(n=2k\), \(m\ge 2^{\lfloor {\log _2(k)} \rfloor }\), and there are integers \(c_{1},c_{2},\ldots ,c_{m}\) such that
-
(i)
\(c_j\) is odd if and only if j is a power of 2, and
-
(ii)
for \(r=1,2,\ldots ,m\),
$$\begin{aligned} w_r = \frac{\pi }{\tau } \sum _{j=r}^m \frac{c_j}{\left( {\begin{array}{c}2j\\ j\end{array}}\right) } \left( {\begin{array}{c}k-r\\ j-r\end{array}}\right) . \end{aligned}$$
We then characterize perfect state transfer on unweighted graphs of \({\mathcal {J}}(n,k)\). In particular, we obtain a simple construction that generates all graphs of \({\mathcal {J}}(n,k)\) with perfect state transfer at time \(\pi /2\).
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We would like to thank the anonymous referees for their helpful comments.
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Vinet, L., Zhan, H. Perfect state transfer on weighted graphs of the Johnson scheme. Lett Math Phys 110, 2491–2504 (2020). https://doi.org/10.1007/s11005-020-01298-6
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DOI: https://doi.org/10.1007/s11005-020-01298-6
Keywords
- Quantum walks
- Perfect state transfer
- Beyond nearest neighbor couplings
- Johnson scheme
- Dual Hahn polynomial