Quantum lower bounds for the Goldreich–Levin problem

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 1 2 ( 1 + ɛ ) . 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 / ɛ ) IP and EQ queries, whereas any classical algorithm requires Ω ( n / ɛ 2 ) such queries. Also, the quantum algorithm requires only O ( n / ɛ ) 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, Ω ( 1 / ɛ ) EQ queries are necessary, and Ω ( 1 / ɛ ) IP queries are necessary if the number of EQ queries is o ( 2 n ) .