Relative Complexity

Lila Fontes, Rahul Jain, Iordanis Kerenidis +3 more

Does the information complexity of a function equal its communication complexity? We examine whether any currently known techniques might be used to show a separation between the two notions. Ganor et al. [2014] recently provided such a separation in the distributional case for a specific input distribution. We show that in the non-distributional setting, the relative discrepancy bound is smaller than the information complexity; hence, it cannot separate information and communication complexity. In addition, in the distributional case, we provide a linear program formulation for relative discrepancy and relate it to variants of the partition bound, resolving also an open question regarding the relation of the partition bound and information complexity. Last, we prove the equivalence between the adaptive relative discrepancy and the public-coin partition, implying that the logarithm of the adaptive relative discrepancy bound is quadratically tight with respect to communication.

Fabio Benatti

Robert Cobb Myers, Shan-Ming Ruan

Ronald de Wolf

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.

Alina Vasilieva

This paper investigates the computability of mathematical relations in a quantum query model. The important task in complexity theory is to find examples with a large gap between classical and quantum algorithm complexity of the same computational problem. We present new results in quantum query algorithm design that allow achieving a large separation between classical and quantum query complexity of a specific relation. We demonstrate an example where quantum query algorithm for a finite relation needs more than two times fewer queries than the best possible classical analogue. We also show that relation can be extended to infinite family of relations with an input of general size N.

Jack D. Hidary