TY - JOUR

T1 - Spectral and asymptotic properties of Grover walks on crystal lattices

AU - Higuchi, Yusuke

AU - Konno, Norio

AU - Sato, Iwao

AU - Segawa, Etsuo

N1 - Funding Information:
We thank the anonymous referee for valuable comments. YuH's work was supported in part by JSPS Grant-in-Aid for Scientific Research (C) 20540113 , 25400208 and (B) 24340031 . NK and IS also acknowledge financial supports of the Grant-in-Aid for Scientific Research (C) from Japan Society for the Promotion of Science (Grants No. 24540116 and No. 23540176 , respectively). ES thanks to the financial support of the Grant-in-Aid for Young Scientists (B) of Japan Society for the Promotion of Science (Grant No. 25800088 ).
Publisher Copyright:
© 2014 Elsevier Inc.

PY - 2014/12/1

Y1 - 2014/12/1

N2 - We propose a twisted Szegedy walk for estimating the limit behavior of a discrete-time quantum walk on a crystal lattice, an infinite abelian covering graph, whose notion was introduced by [14]. First, we show that the spectrum of the twisted Szegedy walk on the quotient graph can be expressed by mapping the spectrum of a twisted random walk onto the unit circle. Secondly, we show that the spatial Fourier transform of the twisted Szegedy walk on a finite graph with appropriate parameters becomes the Grover walk on its infinite abelian covering graph. Finally, as an application, we show that if the Betti number of the quotient graph is strictly greater than one, then localization is ensured with some appropriated initial state. We also compute the limit density function for the Grover walk on Zd with flip flop shift, which implies the coexistence of linear spreading and localization. We partially obtain the abstractive shape of the limit density function: the support is within the d-dimensional sphere of radius 1/d, and 2d singular points reside on the sphere's surface.

AB - We propose a twisted Szegedy walk for estimating the limit behavior of a discrete-time quantum walk on a crystal lattice, an infinite abelian covering graph, whose notion was introduced by [14]. First, we show that the spectrum of the twisted Szegedy walk on the quotient graph can be expressed by mapping the spectrum of a twisted random walk onto the unit circle. Secondly, we show that the spatial Fourier transform of the twisted Szegedy walk on a finite graph with appropriate parameters becomes the Grover walk on its infinite abelian covering graph. Finally, as an application, we show that if the Betti number of the quotient graph is strictly greater than one, then localization is ensured with some appropriated initial state. We also compute the limit density function for the Grover walk on Zd with flip flop shift, which implies the coexistence of linear spreading and localization. We partially obtain the abstractive shape of the limit density function: the support is within the d-dimensional sphere of radius 1/d, and 2d singular points reside on the sphere's surface.

KW - Crystal lattice

KW - Quantum walks

KW - Spectral mapping theorem

KW - Weak limit theorem

UR - http://www.scopus.com/inward/record.url?scp=84908191745&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84908191745&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2014.09.003

DO - 10.1016/j.jfa.2014.09.003

M3 - Article

AN - SCOPUS:84908191745

VL - 267

SP - 4197

EP - 4235

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 11

ER -