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·xis at least 1(1 + ε). 2 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 Ω(n/ε2) such queries. We show that the above quantum algorithm is optimal in terms of both EQ and IP queries. Specifically, Ω(1/ε) EQ queries are necessary, and Ω(1/ε) IP queries are necessary if the number of EQ queries o ( √ 2n)