Abstract
We present two quantum walk algorithms for 3-Distinctness. Both algorithms have time complexity \(\tilde{O}(n^{5/7})\), improving the previous \(\tilde{O}(n^{3/4})\) and matching the best known upper bound for query complexity (obtained via learning graphs) up to log factors. The first algorithm is based on a connection between quantum walks and electric networks. The second algorithm uses an extension of the quantum walk search framework that facilitates quantum walks with nested updates.
This paper is a merge of two submitted papers, whose full versions are available at http://arxiv.org/abs/1302.3143 and http://arxiv.org/abs/1302.7316 . Support for this work was provided by European Social Fund within the project “Support for Doctoral Studies at University of Latvia”, the European Commission IST STREP project 25596 (QCS), NSERC, the Ontario Ministry of Research and Innovation, the US ARO, and the French ANR Blanc project ANR-12-BS02-005 (RDAM).
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Belovs, A., Childs, A.M., Jeffery, S., Kothari, R., Magniez, F. (2013). Time-Efficient Quantum Walks for 3-Distinctness. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds) Automata, Languages, and Programming. ICALP 2013. Lecture Notes in Computer Science, vol 7965. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39206-1_10
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DOI: https://doi.org/10.1007/978-3-642-39206-1_10
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