Exponential Separation between Quantum and Classical Ordered Binary Decision Diagrams, Reordering Method and Hierarchies

In this paper, we study quantum OBDD model, it is a restricted version of read-once quantum branching programs, with respect to “width” complexity. It is known that the maximal gap between deterministic and quantum complexities is exponential. But there are few examples of functions with such a gap. We present a method (called “reordering”), which allows us to transform a Boolean function f into a Boolean function f ′, such that if for f we have some gap between quantum and deterministic OBDD complexities for the natural order over the variables of f , then for any order we have almost the same gap for the function f ′. Using this transformation, we construct a total function REQ such that the deterministic OBDD complexity of it is at least 2 , and the quantum OBDD complexity of it is at most O(n). It is the biggest known gap for explicit functions not representable by OBDDs of a linear width. We also prove the quantum OBDD width hierarchy for complexity classes of Boolean functions. Additionally, we show that shifted equality function can also give a good gap between quantum and deterministic OBDD complexities. Moreover, we prove the bounded error probabilistic OBDD width hierarchy for complexity classes of Boolean functions. And using “reordering” method we extend a hierarchy for k-OBDD of polynomial width, for k = o(n/ log n). We prove a similar hierarchy for bounded error probabilistic k-OBDDs of polynomial, superpolynomial and subexponential width.