Very narrow quantum OBDDs and width hierarchies for classical OBDDs

Abstract

Abstract. We present several results on comparative complexity for different variants of OBDD models.-We present some results on comparative complexity of classical and quantum OBDDs. We consider a partial function depending on parameter k such that for any k > 0 this function is computed by an exact quantum OBDD of width 2 but any classical OBDD (deterministic or stable bounded error probabilistic) needs width 2 k+1 . -We consider quantum and classical nondeterminism. We show that quantum nondeterminism can be more efficient than classical one. In particular, an explicit function is presented which is computed by a quantum nondeterministic OBDD with constant width but any classical nondeterministic OBDD for this function needs non-constant width. -We also present new hierarchies on widths of deterministic and nondeterministic OBDDs. We focus both on small and large widths.

Reordering Method and Hierarchies for Quantum and Classical Ordered Binary Decision Diagrams

“…And third one is for width less than 2 O(n) and here gap is the biggest. Similar hierarchy is already known for deterministic and nondeterministic OBDD [AGKY14], [AGKY16], for deterministic k-OBDD [Kha15]. And we present not tight width hierarchy for bounded error probabilistic OBDD in the paper.…”

“…F. Ablayev, A. Khasianov and A. Vasiliev [AKV08] presented the quantum OBDD of width O(n log n) for P ERM . Let us note that for partial functions the gap between widths of quantum and deterministic OBDDs can be more than exponential [AGKY14], [Gai15], [AGKY16].…”

Quantum Online Streaming Algorithms with Logarithmic Memory

“…There is a problem that has a quantum online streaming algorithm with singel qubit with a better competitive ratio than any classical online streaming algorithm. The problem is based on the P artialM OD function from [7,5,4] and the "Black Hats Method". The paper is organized in the following way.…”

“…Here we claim, that we can construct algorithm with single qubit, but not two qubits as in Theorem 4. The idea of the algorithm is based on ides from [5,4,7,20]. Let us describe an algorithm B for BH t k,r,w (f ), f = P artialM OD β m .…”

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].…”