TY - JOUR

T1 - Trapping and spreading properties of quantum walk in homological structure

AU - Machida, Takuya

AU - Segawa, Etsuo

N1 - Funding Information:
TM is grateful to the Japan Society for the Promotion of Science for the support and to the Math. Dept. UC Berkeley for hospitality. 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, Springer Science+Business Media New York.

PY - 2015/5/1

Y1 - 2015/5/1

N2 - We attempt to extract a homological structure of two kinds of graphs by the Grover walk. The first one consists of a cycle and two semi-infinite lines, and the second one is assembled by a periodic embedding of the cycles in $$\mathbb {Z}$$Z. We show that both of them have essentially the same eigenvalues induced by the existence of cycles in the infinite graphs. The eigenspace of the homological structure appears as so called localization in the Grover walks, in which the walk is partially trapped by the homological structure. On the other hand, the difference of the absolutely continuous part of spectrum between them provides different behaviors. We characterize the behaviors by the density functions in the weak convergence theorem: The first one is the delta measure at the bottom, while the second one is expressed by two kinds of continuous functions, which have different finite supports $$(-1/\sqrt{10},1/\sqrt{10})$$(-1/10,1/10) and $$(-2/7,2/7)$$(-2/7,2/7), respectively.

AB - We attempt to extract a homological structure of two kinds of graphs by the Grover walk. The first one consists of a cycle and two semi-infinite lines, and the second one is assembled by a periodic embedding of the cycles in $$\mathbb {Z}$$Z. We show that both of them have essentially the same eigenvalues induced by the existence of cycles in the infinite graphs. The eigenspace of the homological structure appears as so called localization in the Grover walks, in which the walk is partially trapped by the homological structure. On the other hand, the difference of the absolutely continuous part of spectrum between them provides different behaviors. We characterize the behaviors by the density functions in the weak convergence theorem: The first one is the delta measure at the bottom, while the second one is expressed by two kinds of continuous functions, which have different finite supports $$(-1/\sqrt{10},1/\sqrt{10})$$(-1/10,1/10) and $$(-2/7,2/7)$$(-2/7,2/7), respectively.

KW - Homological structure

KW - Linear spreading

KW - Localization

KW - Quantum walk

KW - Spectral mapping

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U2 - 10.1007/s11128-014-0819-6

DO - 10.1007/s11128-014-0819-6

M3 - Article

AN - SCOPUS:84929096231

VL - 14

SP - 1539

EP - 1558

JO - Quantum Information Processing

JF - Quantum Information Processing

SN - 1570-0755

IS - 5

ER -