Quantum algorithms for hidden nonlinear structures

Attempts to find new quantum algorithms that outperform classical computation have focused primarily on the nonabelian hidden subgroup problem, which generalizes the central problem solved by Shor's factoring algorithm. We suggest an alternative generalization, namely to problems of finding hidden nonlinear structures over finite fields. We give examples of two such problems that can be solved efficiently by a quantum computer, but not by a classical computer. We also give some positive results on the quantum query complexity of finding hidden nonlinear structures.Comment: 13 page

The cubic moment of central values of automorphic L-functions

The authors study the central values of L-functions in certain families; in particular they bound the sum of the cubes of these values.Contents:Comment: 42 pages, published versio

The square root law and structure of finite rings

Let $R$ be a finite ring and define the hyperbola $H=\{(x,y) \in R \times R: xy=1 \}$. Suppose that for a sequence of finite odd order rings of size tending to infinity, the following "square root law" bound holds with a constant $C>0$ for all non-trivial characters $\chi$ on $R^2$: $$\left| \sum_{(x,y)\in H}\chi(x,y)\right|\leq C\sqrt{|H|}.$$ Then, with a finite number of exceptions, those rings are fields. For rings of even order we show that there are other infinite families given by Boolean rings and Boolean twists which satisfy this square-root law behavior. We classify the extremal rings, those for which the left hand side of the expression above satisfies the worst possible estimate. We also describe applications of our results to problems in graph theory and geometric combinatorics. These results provide a quantitative connection between the square root law in number theory, Salem sets, Kloosterman sums, geometric combinatorics, and the arithmetic structure of the underlying rings