Quantum lower bounds for the Goldreich--Levin problem

Authors:
Mark Adcock;Richard Cleve;Kazuo Iwama;Raymond Putra;Shigeru Yamashita
Affiliations:
Department of Computer Science, University of Calgary, Canada;Department of Computer Science, University of Calgary, Canada and School of Computer Science and Institute for Quantum Computing, University of Waterloo, Canada;Graduate School of Informatics, Kyoto University/QCI, ERATO, JST, Japan;Graduate School of Informatics, Kyoto University/QCI, ERATO, JST, Japan and School of Computer Science, McGill University, Canada;Nara Institute of Science and Technology, Japan
Venue:
Information Processing Letters
Year:
2006

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Abstract

At the heart of the Goldreich-Levin theorem is the problem of determining an n-bit string a by making queries to two oracles, referred to as IP (inner product) and EQ (equivalence). The IP oracle, on input x, returns a bit that is biased towards a@?x (the modulo two inner product of a with x) in the following sense. For a random x, the probability that IP(x)=a@?x is at least 12(1+@?). The EQ oracle, on input x, returns a bit specifying whether or not x=a. It has been shown that a quantum algorithm can solve this problem with O(1/@?) IP and EQ queries, whereas any classical algorithm requires @W(n/@?^2) such queries. Also, the quantum algorithm requires only O(n/@?) auxiliary one- and two-qubit gates in addition to its queries. We show that the above quantum algorithm is optimal in terms of both EQ and IP queries. Specifically, @W(1/@?) EQ queries are necessary, and @W(1/@?) IP queries are necessary if the number of EQ queries is o(2^n).