Characterization of non-deterministic quantum query and quantum communication complexity

It is known that the classical and quantum query complexities of a total Boolean function f are polynomially related to the degree of its representing polynomial, but the optimal exponents in these relations are unknown. We show that the non-deterministic quantum query complexity of f is linearly related to the degree of a “non-deterministic” polynomial for f. We also prove a quantum-classical gap of 1 vs. N for non-deterministic query complexity for a total f. In the case of quantum communication complexity there is a (partly undetermined) relation between the complexity of f and the logarithm of the rank of its communication matrix. We show that the non-deterministic quantum communication complexity of f is linearly related to the logarithm of the rank of a non-deterministic version of the communication matrix and that it can be exponentially smaller than its classical counterpart