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The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings. (2002)
Vancouver, BC, Canada
Nov. 16, 2002 to Nov. 19, 2002
ISSN: 0272-5428
ISBN: 0-7695-1822-2
pp: 513
Yaoyun Shi , University of Michigan
ABSTRACT
Given a function f as an oracle, the collision problem is to find two distinct inputs i and j such that f(i) = f(j) under the promise that such inputs exist. In this paper, we prove that any quantum algorithm for finding a collision in an r -to-one function must evaluate the function \Omega (({n \mathord{\left/ {\vphantom {n r}} \right. \kern-\nulldelimiterspace} r}){1 \mathord{\left/ {\vphantom {1 {3)}}} \right. \kern-\nulldelimiterspace} {3)}} times, where n is the size of the domain and {r \mathord{\left/ {\vphantom {r n}} \right. \kern-\nulldelimiterspace} n}. This lower bound matches, up to a constant factor, the upper bound of Brassard, H?yer, and Tapp [ACM SIGACT News, 28:14-19, 1997], which uses the quantum algorithm of Grover [Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, pages 212-219, 1996] in a novel way. The previously best quantum lower bound is \Omega (({n \mathord{\left/ {\vphantom {n {r)^{{1 \mathord{\left/ {\vphantom {1 5}} \right. \kern-\nulldelimiterspace} 5}} }}} \right. \kern-\nulldelimiterspace} {r)^{{1 \mathord{\left/ {\vphantom {1 5}} \right. \kern-\nulldelimiterspace} 5}} }}) evaluations, due to Aaronson [Proceedings of the Thirty-Fourth Annual ACM Symposium on the Theory of Computing, pages 635-642, 2002]. Our result implies a quantum lower bound of \Omega (n^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}}) queries to the inputs for another well studied problem, the element distinctness problem, which is to determine whether or not the given n real numbers are distinct. The previous best lower bound is \Omega (\sqrt n ) queries in the black-box model; and \Omega (\sqrt n \log n) comparisons in the comparisons-only model, due to H?yer, Neerbek, and Shi [Lecture Notes in Computer Science, Vol. 2076, pp. 346-357, 2001].
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CITATION

Y. Shi, "Quantum Lower Bounds for the Collision and the Element Distinctness Problems," The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.(FOCS), Vancouver, BC, Canada, 2002, pp. 513.
doi:10.1109/SFCS.2002.1181975
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