Lower Bounds and Hierarchies for Quantum Memoryless Communication Protocols and Quantum Ordered Binary Decision Diagrams with Repeated Test

Abstract

Abstract. 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 Readk-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 kQOBDDs 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.

Classical and Quantum Computations with Restricted Memory

“…The hierarchies in [3] are the first such hierarchies for bounded-error quantum k-OBDDs, and these results show that unbounded-error and bounded-error models have different properties in this point of view.…”

“…In [3] we also showed hierarchies for sublinear width and the natural order of the input variables: Theorem 5 (Ablayev et al [3]).…”

“…The shuffled address function from [41] is used. The results also can be shown using the modification of the pointer jumping function from [19,57] that is called matrix XOR pointer jumping function [3]. We use lower bounds for memoryless communication protocols from [11,42] and prove upper bound for the shuffled address function.…”

“…Homeister and Waack [34] have shown equality of these two classes. Ablayev, Ambainis, Khadiev and Khadieva [3] modified the technique from [44] and proved hierarchies for quantum k-OBDDs with bounded error.…”

Quantum Algorithm for Dynamic Programming Approach for DAGs. Applications for Zhegalkin Polynomial Evaluation and Some Problems on DAGs

“…There are many problems where quantum algorithms outperform the best known classical algorithms [18,28]. superior of quantum over classical was shown for different computing models like query model, streaming processing models, communication models and others [6,5,4,3,2,1,30,29,27,31].In this paper, we present the quantum algorithm for the class of problems on directed acyclic graphs (DAGs) that uses a dynamic programming approach. The dynamic programming approach is one of the most useful ways to solve problems in computer science [17].…”