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<p>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 <i>d</i>-dimensional torus <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40509_2025_361_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">\((d \ge 2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>≥</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. 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.</p>

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QW-search/zeta correspondence

  • 期刊论文

  • 发表日期：

    2025年3月11日

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  • Taisuke Hosaka,
  • Norio Konno,
  • Etsuo Segawa

摘要

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 \((d \ge 2)\) ( d ≥ 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.