Quantum lower bounds for the Goldreich-Levin problem

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 2 (1+ ε). 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/ε) IP and EQ queries, whereas any classical algorithm requires (n/ε2) such queries. Also, the quantum algorithm requires only O(n/ε) auxiliary oneand 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, (1/ε) EQ queries are necessary, and (1/ε) IP queries are necessary if the number of EQ queries is o( √ 2n ).  2005 Elsevier B.V. All rights reserved.