TY - JOUR

T1 - Irrational Quantum Walks

AU - Coutinho, Gabriel

AU - Baptista, Pedro Ferreira

AU - Godsil, Chris

AU - Spier, Thomás Jung

AU - Werner, Reinhard

N1 - Funding Information: Acknowledgments. The authors acknowledge the hospitality of the Banff International Research Station during the 2019 Quantum Walks and Information Tasks workshop, when seeds for this work were planted. Gabriel Coutinho acknowledges the support of CNPq and FAPEMIG, Pedro Baptista acknowledges the support of CAPES and Chris Godsil acknowledges the support of NSERC. Funding Information: *Received by the editors September 9, 2022; accepted for publication (in revised form) May 15, 2023; published electronically July 25, 2023. https://doi.org/10.1137/22M1521262 Funding: The work of the first author was supported by Fapemig and CNPq (Conselho Nacional de Desenvolvi-mento Cient\{\i'}fico e Tecnolo\'gico). The work of the second author was supported by a CAPES master's scholarship. The work of the third author was supported by the Natural Sciences and Engineering Council of Canada (NSERC) through grant RGPIN-9439. The work of the fourth author was supported by Fapemig. \dagger Department of Computer Science, Universidade Federal de Minas Gerais, Belo Horizonte, MG 31270-901, Brazil (gabriel@dcc.ufmg.br, pedro.baptista@dcc.ufmg.br, thomasjung@dcc.ufmg.br). \ddagger Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON N2L 3G1, Canada (cgodsil@uwaterloo.ca). \S Institut fu\"r Theoretische Physik, Leibniz Universit\a"t Hannover, Hannover, Germany (reinhard.werner@itp.uni-hannover.de). Funding Information: The work of the first author was supported by Fapemig and CNPq (Conselho Nacional de Desenvolvimento Cient\'{\i}fico e Tecnol\' ogico). The work of the second author was supported by a CAPES master's scholarship. The work of the third author was supported by the Natural Sciences and Engineering Council of Canada (NSERC) through grant RGPIN-9439. The work of the fourth author was supported by Fapemig. The authors acknowledge the hospitality of the Banff International Research Station during the 2019 Quantum Walks and Information Tasks workshop, when seeds for this work were planted. Gabriel Coutinho acknowledges the support of CNPq and FAPEMIG, Pedro Baptista acknowledges the support of CAPES and Chris Godsil acknowledges the support of NSERC.

PY - 2023/9

Y1 - 2023/9

N2 - The adjacency matrix of a graph G is the Hamiltonian for a continuous-time quantum walk on the vertices of G. Although the entries of the adjacency matrix are integers, its eigenvalues are generally irrational and, because of this, the behavior of the walk is typically not periodic. In this paper, we develop a theory to exactly study any quantum walk generated by an integral Hamiltonian, and we put emphasis on those with irrational eigenvalues-what we call irrational quantum walks. As a result, we provide exact methods to compute the average of the mixing matrices, and to decide whether pretty good (or almost perfect) state transfer occurs in a given graph. We also use our methods to study geometric properties of beautiful curves arising from entries of the quantum walk matrix and discuss possible applications of these results. Throughout the paper, we emphasize the interplay between different fields of mathematics applied to the study of quantum walks.

AB - The adjacency matrix of a graph G is the Hamiltonian for a continuous-time quantum walk on the vertices of G. Although the entries of the adjacency matrix are integers, its eigenvalues are generally irrational and, because of this, the behavior of the walk is typically not periodic. In this paper, we develop a theory to exactly study any quantum walk generated by an integral Hamiltonian, and we put emphasis on those with irrational eigenvalues-what we call irrational quantum walks. As a result, we provide exact methods to compute the average of the mixing matrices, and to decide whether pretty good (or almost perfect) state transfer occurs in a given graph. We also use our methods to study geometric properties of beautiful curves arising from entries of the quantum walk matrix and discuss possible applications of these results. Throughout the paper, we emphasize the interplay between different fields of mathematics applied to the study of quantum walks.

KW - average mixing matrix

KW - continuous-time quantum walk

KW - pretty good state transfer

UR - http://www.scopus.com/inward/record.url?scp=85170268546&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2208.08971

DO - 10.48550/arXiv.2208.08971

M3 - Article

AN - SCOPUS:85170268546

VL - 7

SP - 567

EP - 584

JO - SIAM Journal on Applied Algebra and Geometry

JF - SIAM Journal on Applied Algebra and Geometry

IS - 3

ER -