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Trapping and spreading properties of quantum walk in homological structure

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Abstract

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 \(\mathbb {Z}\). 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 \((-1/\sqrt{10},1/\sqrt{10})\) and \((-2/7,2/7)\), respectively.

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Notes

  1. The one-to-one correspondence between \(V(C_4')\) and \(\mathbb {Z}\times V(\mathcal {D})\) is denoted by \(v_j\leftrightarrow (j,v)\), \((v\in \{0,0',u,d\})\) and one between \(A(C_4')\) and \(\mathbb {Z}\times A(\mathcal {D})\) is \((v_j,w_j)\leftrightarrow (j,(v,w))\) for \((v_j,w_j)\in A(C_4^{(j)})\), \((0_j,0'_{j+1})\leftrightarrow (j,(0,0'))\) and \((0_j',0_{j-1})\leftrightarrow (j,(0',0))\).

  2. More precisely, we impose the following assumptions to \(h(k)\). (i) \(h(k)=h(k+2\pi )\) for all \(k\in \mathbb {R}\), (ii) we permit discontinuity of \(h(k)\) only at \(\{2\pi n+k_j\}_{j=0}^{s-1}\), \(n\in \mathbb {N}\). (iii) for any interval, \(h(k)\) does not take a constant value.

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Acknowledgments

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).

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Authors and Affiliations

  1. Japan Society for the Promotion of Science, Hongo, Bunkyo, Tokyo, 113-8654, Japan

    Takuya Machida

  2. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Takuya Machida

  3. Graduate School of Information Sciences, Tohoku University, Aoba, Sendai, 980-8579, Japan

    Etsuo Segawa

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  1. Takuya Machida
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  2. Etsuo Segawa
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Correspondence to Etsuo Segawa.

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Machida, T., Segawa, E. Trapping and spreading properties of quantum walk in homological structure. Quantum Inf Process 14, 1539–1558 (2015). https://doi.org/10.1007/s11128-014-0819-6

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