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QW-Search/Zeta Correspondence

Taisuke Hosaka, Norio Konno, Etsuo Segawa

Abstract

We consider the connection between this zeta function and quantum search via quantum walk. First, we give an explicit expression of the zeta function on the one-dimensional torus in the general case of the number and position of marked vertices. Moreover, we deal with the two special cases of the position of the marked vertices on the $d$-dimensional torus $(d \ge 2)$. Additionally, we treat the property of the zeta function by using the Mahler measure. Our results show the relationship between the zeta function and quantum search algorithms for the first time.

QW-Search/Zeta Correspondence

Abstract

We consider the connection between this zeta function and quantum search via quantum walk. First, we give an explicit expression of the zeta function on the one-dimensional torus in the general case of the number and position of marked vertices. Moreover, we deal with the two special cases of the position of the marked vertices on the -dimensional torus . Additionally, we treat the property of the zeta function by using the Mahler measure. Our results show the relationship between the zeta function and quantum search algorithms for the first time.
Paper Structure (6 sections, 4 theorems, 37 equations, 1 figure)

This paper contains 6 sections, 4 theorems, 37 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The model of quantum search
  3. One-dimensional torus
  4. $d$-dimensional torus
  5. Comparison between search and non-search cases
  6. Conclusion

Key Result

Proposition 1

(Konno, Sato and Segawa KSS) Let G be a connected graph with N vertices and $\epsilon$ edges. $W'$ is the time evolution matrix with search algorithm on $G$. Then we get Here, $P_{M}$ is an $(N-m) \times (N-m)$ matrix describing $RW$ with the Dirichlet boundary condition at $M$, that is, for $v,x \in M$.

Figures (1)

  • Figure 1: The solid and dot curves correspond to $\mathcal{L}(T^{d}_{\infty},u)$ and $\mathcal{L}(W'_{2},T^{d}_{\infty},u)$, respectively.

Theorems & Definitions (12)

  • Definition 1: Walk/Zeta
  • Proposition 1
  • Theorem 1
  • Remark 1
  • proof
  • Theorem 2
  • Remark 2
  • Remark 3
  • Remark 4
  • proof
  • ...and 2 more