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Quantum Algorithm for Triangle Finding in Sparse Graphs

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Abstract

This paper presents a quantum algorithm for triangle finding over sparse graphs that improves over the previous best quantum algorithm for this task by Buhrman et al. (SIAM J Comput 34(6):1324–1330, 2005). Our algorithm is based on the recent \({\tilde{O}}(n^{5/4})\)-query algorithm given by Le Gall (Proceedings of the 55th IEEE annual symposium on foundations of computer science, pp 216–225, 2014) for triangle finding over dense graphs (here n denotes the number of vertices in the graph). We show in particular that triangle finding can be solved with \(O(n^{5/4-\epsilon })\) queries for some constant \(\epsilon >0\) whenever the graph has at most \(O(n^{2-c})\) edges for some constant \(c>0\).

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Notes

  1. In this paper the notation \({\tilde{O}}(\cdot )\) removes \(\mathrm {polylog}n\) factors.

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Acknowledgements

The authors are grateful to Mathieu Laurière, Frédéric Magniez, Keiji Matsumoto, Harumichi Nishimura and Seiichiro Tani for helpful comments. This work is supported by the Grant-in-Aid for Young Scientists (B) No. 24700005 and the Grant-in-Aid for Scientific Research (A) No. 24240001 of the Japan Society for the Promotion of Science, and the Grant-in-Aid for Scientific Research on Innovative Areas No. 24106009 of the Ministry of Education, Culture, Sports, Science and Technology in Japan.

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  1. Department of Computer Science, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo, Japan

    François Le Gall & Shogo Nakajima

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  1. François Le Gall
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  2. Shogo Nakajima
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Correspondence to Shogo Nakajima.

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Le Gall, F., Nakajima, S. Quantum Algorithm for Triangle Finding in Sparse Graphs. Algorithmica 79, 941–959 (2017). https://doi.org/10.1007/s00453-016-0267-z

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