Polynomial degree vs. quantum query complexity

AbstractThe degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of f. This observation has been a source of many lower bounds on quantum algorithms. It has been an open problem whether this lower bound is tight.We exhibit a function with polynomial degree M and quantum query complexity Ω(M1.321…). This is the first superlinear separation between polynomial degree and quantum query complexity. The lower bound is shown by a generalized version of the quantum adversary method

Weak Parity

We study the query complexity of Weak Parity: the problem of computing the parity of an n-bit input string, where one only has to succeed on a 1/2+eps fraction of input strings, but must do so with high probability on those inputs where one does succeed. It is well-known that n randomized queries and n/2 quantum queries are needed to compute parity on all inputs. But surprisingly, we give a randomized algorithm for Weak Parity that makes only O(n/log^0.246(1/eps)) queries, as well as a quantum algorithm that makes only O(n/sqrt(log(1/eps))) queries. We also prove a lower bound of Omega(n/log(1/eps)) in both cases; and using extremal combinatorics, prove lower bounds of Omega(log n) in the randomized case and Omega(sqrt(log n)) in the quantum case for any eps>0. We show that improving our lower bounds is intimately related to two longstanding open problems about Boolean functions: the Sensitivity Conjecture, and the relationships between query complexity and polynomial degree.Comment: 18 page