The Power of Various Real-Valued Quantum Queries

The computation of combinatorial and numerical problems on quantum computers is often much faster than on a classical computer in numbers of queries. A query is a procedure by which the quantum computer gains information about the specific problem. Different query definitions were given and our aim is to review them and to show that these definitions are not equivalent. To achieve this result we will study the simulation and approximation of one query type by another. While approximation is easy in one direction, we will show that it is hard in the other direction by a lower bound for the numbers of queries needed in the simulation. The main tool in this lower bound proof is a relationship between quantum algorithms and trigonometric polynomials that we will establish