Perfect state transfer on weighted graphs of the Johnson scheme
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Perfect state transfer on weighted graphs of the Johnson scheme Luc Vinet1 · Hanmeng Zhan1,2 Received: 4 May 2019 / Revised: 16 February 2020 / Accepted: 6 June 2020 © Springer Nature B.V. 2020
Abstract We characterize quantum perfect state transfer on real-weighted graphs of the Johnson scheme J (n, k), which represent spin networks with non-nearest neighbor couplings. Given J (n, k) = {A1 , A2 , . . . , Ak } and A(X ) = w0 A0 + · · · + wm Am , we show that X has perfect state transfer at time τ if and only if n = 2k, m ≥ 2log2 (k) , and there are integers c1 , c2 , . . . , cm such that (i) c j is odd if and only if j is a power of 2, and (ii) for r = 1, 2, . . . , m, m π cj k − r . wr = 2 j τ j −r j=r
j
We then characterize perfect state transfer on unweighted graphs of J (n, k). In particular, we obtain a simple construction that generates all graphs of J (n, k) with perfect state transfer at time π/2. Keywords Quantum walks · Perfect state transfer · Beyond nearest neighbor couplings · Johnson scheme · Dual Hahn polynomial Mathematics Subject Classification 05C50 · 05E30 · 11A07 · 11C20
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Hanmeng Zhan [email protected]
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Centre de Recherches Mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal, QC H3C 3J7, Canada
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Present Address: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J 1P3, Canada
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L. Vinet, H. Zhan
1 Introduction Quantum spin networks that manifest perfect state transfer between two qubits are important devices in implementing quantum information tasks. There are two main objects of studies in the literature: spin chains with modulated nearest-neighbor couplings, and spin networks with uniform nearest-neighbor couplings. The first situation is represented by a weighted path. Here, one aims to determine weightings that allow perfect state transfer between the endpoint or internal vertices, and there have been experimental results, as well as analytic solutions based on orthogonal polynomials (see, for example [2,18]). In comparison, the second situation is represented by an unweighted graph. Without the freedom to change weights, the goal is to characterize graphs that allow perfect state transfer between some pair of vertices, and many results follow from algebraic graph theory (see, for example [11,13]). Given a distance regular graph, one can “quotient” it down to a weighted path by the distance partition relative to a vertex. Hence, constructions of both types of spin networks may be related. For instance, the quotient of a hypercube is a Krawtchouk chain, and perfect state transfer between antipodal vertices of the hypercube is equivalent to perfect state transfer between the endpoints of the Krawtchouk chain (see Christandl et al. [9] and Albanese et al. [2] for such networks that admit perfect state transfer). As another example, a Johnson graph on more than two vertices can be quotient down to a dual Hahn chain, but not one of those exhibiting perfect state transfer [2]. In fact, Ahmadi et al
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