Quantum algorithms for weighing matrices and quadratic residues

In this article we investigate how we can employ the structure of combinatorial objects like Hadamard matrices and weighing matrices to devise new quantum algorithms. We show how the properties of a weighing matrix can be used to construct a problem for which the quantum query complexity is significantly lower than the classical one. It is pointed out that this scheme captures both Bernstein and Vazirani's inner-product protocol, as well as Grover's search algorithm. In the second part of the article we consider Paley's construction of Hadamard matrices, which relies on the properties of quadratic characters over finite fields. We design a query problem that uses the Legendre symbol chi (which indicates if an element of a finite field F-q is a quadratic residue or not). It is shown how for a shifted Legendre function f(s) (i) = chi (i + s), the unknown s is an element of F-q can be obtained exactly with only two quantum calls to f(s). This is in sharp contrast with the observation that any classical, probabilistic procedure requires more than log q + log((1 - epsilon)/2) queries to solve the same problem.