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Lobachevskii Journal of Mathematics

Very narrow quantum OBDDs and width hierarchies for classical OBDDs

Farid Mansurovich Ablayev, Aida Gainutdinova, Kamil Ravilevich Khadiev, Abuzer Yakaryılmaz

Published Nov 1, 2016• 37 citations
Publisher

Abstract

In the paper we investigate Ordered Binary Decision Diagrams (OBDDs)–a model for computing Boolean functions. We present a series of results on the comparative complexity for several variants of OBDDmodels. • We present results on the comparative complexity of classical and quantum OBDDs. We consider a partial function depending on a 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 nondeterminismcan bemore efficient than classical nondeterminism. In particular, an explicit function is presented that is computed by a quantum nondeterministic OBDD of 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.

DOI
10.1134/s199508021606007x

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In the paper we investigate Ordered Binary Decision Diagrams (OBDDs)–a model for computing Boolean functions. We present a series of results on the comparative complexity for several variants of OBDDmodels. • We present results on the comparative complexity of classical and quantum OBDDs. We consider a partial function depending on a 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 nondeterminismcan bemore efficient than classical nondeterminism. In particular, an explicit function is presented that is computed by a quantum nondeterministic OBDD of 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 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 2k+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 non-deterministic OBDDs. We focus both on small and large widths.
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.
We consider a model of computation for discrete functions—Ordered Binary Decision Diagrams (OBDD).We investigate comparative complexity of quantum, deterministic, probabilistic and nondeterministic (quantum and classical) OBDDs for total and partial functions. The measure of complexity is a width of OBDD. It is known that for total functions bounded error quantum OBDDs can be exponentially more effective than deterministic and bounded error probabilistic OBDDs. We show that such quantum OBDDs also can be exponentially more effective than nondeterministic OBDDs (both quantum and classical). For partial functions the gap can be more significant. For partial function depending on a parameter k exact quantum OBDD has the width two. Deterministic and bounded error probabilistic OBDD for this function must have a width exponentially depending on k .
We consider quantum, nondterministic and probabilistic versions of known computational model Ordered Read-kkk-times Branching Programs or Ordered Binary Decision Diagrams with repeated test (kkk-QOBDD, kkk-NOBDD and kkk-POBDD). We show width hierarchy for complexity classes of Boolean function computed by these models and discuss relation between different variants of kkk-OBDD.
An ordered binary decision diagram (OBDD) is a directed acyclic graph that represents a Boolean function.Since OBDDs have many nice properties as data structures,they have been extensively studied for decades in both theoretical and practical fields,such as VLSI (Very Large Scale Integration) design, formal verification, machine learning, and combinatorial problems.Arguably, the most crucial problem in using OBDDs is thatthey may vary exponentially in size depending on their variable ordering (i.e., the order in which the variables are to be read) when they represent the same function.Indeed, it is NP hard to find an optimal variable ordering that minimizes an OBDDfor a given function.Friedman and Supowit provided a clever deterministic algorithm with time/space complexityO*(3n), where nnn is the number of variables of the function,which is much better than the trivial brute-force bound O*(n!2n)$.This paper shows that a further speedup is possible with quantum computersby presenting a quantum algorithmthat produces a minimum OBDD together with the corresponding variable orderingin O*(2.77286n) time and space with an exponentially small error probability. Moreover, this algorithm can be adapted toconstructing other minimum decision diagrams such as zero-suppressed BDDs (ZBDDs or ZDDs).