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arXiv:2010.06779

Pretty Good State Transfer and Minimal Polynomials

Christopher M. van Bommel

Published Oct 14, 2020• 2 citations
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Abstract

We examine conditions for a pair of strongly cospectral vertices to have pretty good quantum state transfer in terms of minimal polynomials, and provide cases where pretty good state transfer can be ruled out. We also provide new examples of simple, unweighted graphs exhibiting pretty good state transfer. Finally, we consider modifying paths by adding symmetric weighted edges, and apply these results to this case.

DOI
10.48550/arxiv.2010.06779

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We examine conditions for a pair of strongly cospectral vertices to have pretty good quantum state transfer in terms of minimal polynomials, and provide cases where pretty good state transfer can be ruled out. We also provide new examples of simple, unweighted graphs exhibiting pretty good state transfer. Finally, we consider modifying paths by adding symmetric weighted edges, and apply these results to this case.
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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.
Abstract We establish the theory for pretty good state transfer in discrete-time quantum walks. For a class of walks, we show that pretty good state transfer is characterized by the spectrum of certain Hermitian adjacency matrix of the graph; more specifically, the vertices involved in pretty good state transfer must be strongly cospectral relative to this matrix, and the arccosines of its eigenvalues must satisfy some number theoretic conditions. Using normalized adjacency matrices, cyclic covers, and the theory on linear relations between geodetic angles, we construct several infinite families of walks that exhibits this phenomenon.
We consider pretty good state transfer in coined quantum walks between antipodal vertices on the hypercube [Formula: see text]. When [Formula: see text] is a prime, this was proven to occur in the arc-reversal walk with Grover coins. We extend this result by constructing weighted Grover coins that enable pretty good state transfer on every [Formula: see text]. Our coins are real and require modification of the weight on only one arc per vertex. We also generalize our approach and establish a sufficient condition for pretty good state transfer to occur on other graphs.This article is part of the theme issue 'Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms'.