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Mohamed Sabri
Department of Mathematics, The Open University of Sri Lanka, Nugegoda, Sri Lanka
Astronomy Chemistry Earth & Planetary Sciences Mathematics Physics

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Tailed Grover Search on Cyclic Graphs Chamod Kaushalya; Mohamed Sabri; Anuradha Mahasinghe; Awansika Nimuthumana; Asanka Sayakkara; Kasun De Zoysa
Journal of Mathematical and Fundamental Sciences Vol. 57 No. 3 (2026)
Publisher : Directorate for Research and Innovation (DRI) ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/j.math.fund.sci.2026.57.3.2

Abstract

Quantum walks offer promising advantages for search algorithms over graphs. Among these, Grover’s quantum search provides a quadratic speedup with a time complexity of  for unstructured search problems. Nevertheless, Grover search performs poorly on cyclic graphs due to the dynamics of the Grover walk. This study explores the behavior of Grover’s quantum walk on cyclic graphs , analyzing the probability distribution of finding the marked vertex. To analyze this behavior, we extend each vertex of the cycle by attaching semi-infinite-length paths (tails). We develop a direct analytical approach to obtain the transition matrix , whose elements are independent of . For small cycles , we observe success probabilities exceeding 0.5. In contrast, with large cycles, the effectiveness drops rapidly. We observe that the success probability approaches zero  as the size of the graph increases , indicating the limitations of the quantum search in cyclic structures.
Co-Authors Anuradha Mahasinghe Asanka Sayakkara Awansika Nimuthumana Chamod Kaushalya Kasun De Zoysa