Quantum Weak Parity Problem

In this work we shall investigate quantum query complexity of weak PARITY problem. Weak Parity is the following natural query problem: What is the minimum number of queries necessary to compute PARITY(x1, x2, x3, . . . , xn) on at least 1 2 + fraction of the inputs. For randomized classical or exact quantum machines this relaxed PARITY problem remains as hard as original PARITY problem. Although Bounded error quantum machines needed almost as many queries as classical machines for original PARITY problem, they outperform classical algorithms in this relaxed problem. We will show upper O( n √ log(1/ ) ) and lower bound of Ω(max{ n log(1/ ) , √ log n}). After this, we shall investigate the connection of this problem in the setting of = 2−n with important extremal problems over boolean hypercube and we shall give some improvements in that case.