Reordering method and hierarchies for quantum and classical ordered binary decision diagrams

© Springer International Publishing AG 2017.We consider Quantum OBDD model. It is restricted version of read-once Quantum Branching Programs, with respect to “width” complexity. It is known that maximal complexity gap between determin-istic and quantum model is exponential. But there are few examples of such functions. We present method (called “reordering”), which allows to build Boolean function g from Boolean Function f, such that if for f we have gap between quantum and deterministic OBDD complexity for natural order of variables, then we have almost the same gap for function g, but for any order. Using it we construct the total function REQ which deterministic OBDD complexity is 2Ω(n/logn) and present quantum OBDD of width O(n2). It is bigger gap for explicit function that was known before for OBDD of width more than linear. Using this result we prove the width hierarchy for complexity classes of Boolean functions for quantum OBDDs. Additionally, we prove the width hierarchy for complexity classes of Boolean functions for bounded error probabilistic OBDDs. And using “reordering” method we extend a hierarchy for k-OBDD of polynomial size, for k = o(n/log3n). Moreover, we proved a similar hierarchy for bounded error probabilistic k-OBDD. And for deterministic and proba-bilistic k-OBDDs of superpolynomial and subexponential size

Lower bounds and hierarchies for quantum memoryless communication protocols and quantum ordered binary decision diagrams with repeated test

© 2018, Springer International Publishing AG. We explore multi-round quantum memoryless communication protocols. These are restricted version of multi-round quantum communication protocols. The “memoryless” term means that players forget history from previous rounds, and their behavior is obtained only by input and message from the opposite player. The model is interesting because this allows us to get lower bounds for models like automata, Ordered Binary Decision Diagrams and streaming algorithms. At the same time, we can prove stronger results with this restriction. We present a lower bound for quantum memoryless protocols. Additionally, we show a lower bound for Disjointness function for this model. As an application of communication complexity results, we consider Quantum Ordered Read-k-times Branching Programs (k-QOBDD). Our communication complexity result allows us to get lower bound for k-QOBDD and to prove hierarchies for sublinear width bounded error k-QOBDDs, where k=o(n). Furthermore, we prove a hierarchy for polynomial size bounded error k-QOBDDs for constant k. This result differs from the situation with an unbounded error where it is known that an increase of k does not give any advantage