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

ABSTRACT

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 $n\ge 3$ 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.

KEYWORDS:

• Quantum state transfer
• perfect state transfer
• orthogonal polynomials
• weighted paths
• energy potential

AMS SUBJECT CLASSIFICATIONS:

• 05C22
• 05C50
• 15A18
• 42C05
• 81P45

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

S.K., R.P., and S.P. were supported by NSERC [Discovery Grant RGPIN/6123-2014], [Discovery Grant 400550], [Discovery Grant 1174582], respectively; S.P. is also supported by the Canada Foundation for Innovation and the Canada Research Chairs Program. D.M. was supported through a NSERC Undergraduate Student Research Award; X.Z. was supported by the University of Manitoba’s Faculty of Science and Faculty of Graduate Studies.