Pretty good state transfer between internal nodes of paths

Coutinho, Gabriel; Guo, Krystal; Bommel, Christopher M. van

doi:10.26421/qic17.9-10-5

Cited by 19 publications

(21 citation statements)

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“…It is an open question if the converse holds even if there are eigenvectors with zeros in the jth place. We note that in the more relaxed setting of PGST, this more general conjectured version of the converse fails to be true [25]: PGST can occur between internal vertices of paths in the absence of PGST between the end vertices.…”

Section: Preliminariesmentioning

confidence: 85%

Perfect quantum state transfer in weighted paths with potentials (loops) using orthogonal polynomials

Kirkland

McLaren

Pereira

et al. 2018

Linear and Multilinear Algebra

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A simple method for transmitting quantum states within a quantum computer is via a quantum spin chain-that is, a path on n vertices. Unweighted paths are of limited use, and so a natural generalization is to consider weighted paths; this has been further generalized to allow for loops (potentials in the physics literature). We study the particularly important situation of perfect state transfer with respect to the corresponding adjacency matrix or Laplacian through the use of orthogonal polynomials. Low-dimensional examples are given in detail. Our main result is that PST with respect to the Laplacian matrix cannot occur for weighted paths on n ≥ 3 vertices nor can it occur for certain symmetric weighted trees. The methods used lead us to a conjecture directly linking the rationality of the weights of weighted paths on n > 3 vertices, with or without loops, with the capacity for PST between the end vertices with respect to the adjacency matrix.2010 Mathematics Subject Classification. 05C22; 05C50; 15A18; 42C05; 81P45 .

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Section: Preliminariesmentioning

confidence: 85%

Perfect quantum state transfer in weighted paths with potentials (loops) using orthogonal polynomials

Kirkland

McLaren

Pereira

et al. 2018

Linear and Multilinear Algebra

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“…Coutinho, Guo, and van Bommel [13] determined an infinite family of paths which exhibit pretty good state transfer between internal vertices but not between the end vertices, and van Bommel [14] completed the characterization by showing there were no additional examples, leading to the following result.…”

Section: Introductionmentioning

confidence: 90%

Pretty Good State Transfer and Minimal Polynomials

van Bommel

2020

Preprint

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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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“…Coutinho, Guo, and van Bommel [13] determined an infinite family of paths which exhibit pretty good state transfer between internal vertices but not between the end vertices, and van Bommel [21] completed the characterization by showing there were no additional examples, leading to the following result.…”

Section: Introductionmentioning

confidence: 90%