Nondeterministic complexity in quantum and classical models of computation

Giuseppe Castagnoli, Dalida Monti

François Le Gall

François Le Gall

Aida Gainutdinova

In this paper we investigate ordered binary decision diagrams (OBDD) – a model for computing Boolean functions. The aim of this work is a comparative complexity analysis of quantum and classical nondeterministic OBDDs of large width. We study the complexity of computing the Boolean function "Equality" in nondeterministic quantum OBDDs for different order of reading variables in comparison with the classical complexity. We show that when using the order of reading for which the width of the classical nondeterministic OBDD is constant, the width of the quantum model is linear and the proved lower bound is tight. We define a Boolean function for which the width of nondeterministic quantum OBDD is exponential for any order of reading variables. We construct a quantum algorithm for computing this function with zero error. We present a result on the relationship between complexity classes for quantum and classical nondeterministic OBDDs of large width.

Richard Jozsa

Alina Dubrovska

Many quantum algorithms can be analyzed in a query model to compute Boolean functions where input is given by a black box. As in the classical version of decision trees, different kinds of quantum query algorithms are possible: exact, with bounded error and even nondeterministic. In this paper, we study the latter class of algorithms. We introduce a new notion in addition to already studied nondeterministic algorithms and introduce dual nondeterministic quantum query algorithms. We examine properties of such algorithms and prove relations with exact and nondeterministic quantum query algorithm complexity. As a result and as an example of the application of discovered properties, we demonstrate a gap of n vs. 2 between classical deterministic and dual nondeterministic quantum query complexity for a specific Boolean function. Finally, we show an approach how to construct examples where quantum nondeterministic complexity of an algorithm is O(1), however classical deterministic algorithm for the same function would require O(n) queries.