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arXiv:2609.16549

Pretty good quantum state transfer via transcendental edge weights

Addison Ballif, Mark Kempton, James B. Larsen, Kellon Sandall, Christino Tamon, Trevor Wai

quant-phmath.COSubmitted Sep 15, 2026• Announced Sep 16, 2026• 0 citations
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Abstract

We prove that if we take a rooted product of a circulant graph with universal perfect state transfer with a path of fixed length whose end edge is weighted with a transcendental number, then there is pretty good state transfer between any pair of endpoints of these paths. As a consequence, in a path with an even number of vertices with transcendental weights on the two edges incident to the endpoints, there is pretty good state transfer.

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We prove that if we take a rooted product of a circulant graph with universal perfect state transfer with a path of fixed length whose end edge is weighted with a transcendental number, then there is pretty good state transfer between any pair of endpoints of these paths. As a consequence, in a path with an even number of vertices with transcendental weights on the two edges incident to the endpoints, there is pretty good state transfer.
We study pretty good quantum state transfer (i.e., state transfer that becomes arbitrarily close to perfect) between vertices of graphs with an involution in the presence of an energy potential. In particular, we show that if a graph has an involution that fixes at least one vertex or at least one edge, then there exists a choice of potential on the vertex set of the graph for which we get pretty good state transfer between symmetric vertices of the graph. We show further that in many cases, the potential can be chosen so that it is only non-zero at the vertices between which we want pretty good state transfer. As a special case of this, we show that such a potential can be chosen on the endpoints of a path to induce pretty good state transfer in paths of any length. This is in contrast to the result of [6], in which the authors show that there cannot be perfect state transfer in paths of length 4 or more, no matter what potential is chosen.
We study the existence of quantum state transfer in Q\mathcal{Q}Q-graphs in this paper. The Q\mathcal{Q}Q-graph of a graph GGG, denoted by Q(G)\mathcal{Q}(G)Q(G), is the graph derived from GGG by plugging a new vertex to each edge of GGG and joining two new vertices which lie on adjacent edges of GGG by an edge. We show that, if all eigenvalues of a regular graph GGG are integers, then its Q\mathcal{Q}Q-graph Q(G)\mathcal{Q}(G)Q(G) has no perfect state transfer. In contrast, we also prove that the Q\mathcal{Q}Q-graph of a regular graph has pretty good state transfer under some mild conditions. Finally, applying the obtained results, we also exhibit many new families of Q\mathcal{Q}Q-graphs having no perfect state transfer, but admitting pretty good state transfer.
We give a complete characterization of pretty good state transfer on paths between any pair of vertices with respect to the quantum walk model determined by the XY-Hamiltonian. If nnn is the length of the path, and the vertices are indexed by the positive integers from 1 to nnn, with adjacent vertices having consecutive indices, then the necessary and sufficient conditions for pretty good state transfer between vertices aaa and bbb are that (a) a+b=n+1a + b = n + 1a+b=n+1, (b) n+1n + 1n+1 has at most one odd non-trivial divisor, and (c) if n=2tr−1n = 2^t r - 1n=2tr−1, for rrr odd and r≠1r \neq 1r=1, then aaa is a multiple of 2t−12^{t - 1}2t−1.
We give a complete characterization of pretty good state transfer on paths between any pair of vertices with respect to the quantum walk model determined by the XY-Hamiltonian. If n is the length of the path, and the vertices are indexed by the positive integers from 1 to n, with adjacent vertices having consecutive indices, then the necessary and sufficient conditions for pretty good state transfer between vertices a and b are that (a) a + b = n + 1, (b) n + 1 has at most one positive odd non-trivial divisor, and (c) if n = 2^t r - 1, for r odd and r \neq 1, then a is a multiple of 2^{t - 1}.