Properties and Application of Nondeterministic Quantum Query Algorithms

Many quantum algorithms can be analyzed in a query model to compute Boolean functions where input is given by a black box. As in the classical version of decision trees, different kinds of quantum query algorithms are possible: exact, zero-error, bounded-error and even nondeterministic. In this paper, we study the latter class of algorithms. We introduce a fresh notion in addition to already studied nondeterministic algorithms and introduce dual nondeterministic quantum query algorithms. We examine properties of such algorithms and prove relations with exact and nondeterministic quantum query algorithm complexity. As a result and as an example of the application of discovered properties, we demonstrate a gap of n vs. 2 between classical deterministic and dual nondeterministic quantum query complexity for a specific Boolean function.Comment: 12 pages, 9 figure

Hardness Results for Dynamic Problems by Extensions of Fredman and Saks’ Chronogram Method

We introduce new models for dynamic computation based on the cell probe model of Fredman and Yao. We give these models access to nondeterministic queries or the right answer +-1 as an oracle. We prove that for the dynamic partial sum problem, these new powers do not help, the problem retains its lower bound of  Omega(log n/log log n). From these results we easily derive a large number of lower bounds of order Omega(log n/log log n) for conventional dynamic models like the random access machine. We prove lower bounds for dynamic algorithms for reachability in directed graphs, planarity testing, planar point location, incremental parsing, fundamental data structure problems like maintaining the majority of the prefixes of a string of bits and range queries. We characterise the complexity of maintaining the value of any symmetric function on the prefixes of a bit string

Lower bound techniques for data structures

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (p. 135-143).We describe new techniques for proving lower bounds on data-structure problems, with the following broad consequences: * the first [omega](lg n) lower bound for any dynamic problem, improving on a bound that had been standing since 1989; * for static data structures, the first separation between linear and polynomial space. Specifically, for some problems that have constant query time when polynomial space is allowed, we can show [omega](lg n/ lg lg n) bounds when the space is O(n - polylog n). Using these techniques, we analyze a variety of central data-structure problems, and obtain improved lower bounds for the following: * the partial-sums problem (a fundamental application of augmented binary search trees); * the predecessor problem (which is equivalent to IP lookup in Internet routers); * dynamic trees and dynamic connectivity; * orthogonal range stabbing. * orthogonal range counting, and orthogonal range reporting; * the partial match problem (searching with wild-cards); * (1 + [epsilon])-approximate near neighbor on the hypercube; * approximate nearest neighbor in the l[infinity] metric. Our new techniques lead to surprisingly non-technical proofs. For several problems, we obtain simpler proofs for bounds that were already known.by Mihai Pǎtraşcu.Ph.D

Nondeterministic Query Algorithms

Query algorithms are used to compute Boolean functions. The definition of the function is known, but input is hidden in a black box. The aim is to compute the function value using as few queries to the black box as possible. As in other computational models, different kinds of query algorithms are possible: deterministic, probabilistic, as well as nondeterministic. In this paper, we present a new alternative definition of nondeterministic query algorithms and study algorithm complexity in this model. We demonstrate the power of our model with an example of computing the Fano plane Boolean function. We show that for this function the difference between deterministic and nondeterministic query complexity is 7N versus O(3N)

COMPLEXITY OF QUANTUM ALGORITHMS AND COMMUNICATION PROTOCOLS

Anotacija Kvantu skaitošana ir datorzinatnes apakšnozare, kas balstas uz kvantu mehanikas likumiem, kuru iespejas un priekšrocibas tiek izmantotas, lai efektivak risinatu skaitošanas uzdevumus. Galvenais darba petamais objekts ir kvantu vaicajošo algoritmu modelis. Petijuma galvenais meris ir efektivo kvantu vaicajošo algoritmu atrašana konkretam problemam, ka ari visparigo algoritmu konstruešanas metožu izstradašana. Ir ieguti rezultati attieciba uz dažadiem kvantu vaicajošo algoritmu tipiem, ka eksaktais algoritms, algoritms ar kudas varbutibu un nedeterminetais algoritms. Promocijas darba pirmaja daa ir apskatiti kvantu eksaktie algoritmi un kvantu algoritmi ar kudas varbutibu Bula funkciju reinašanai. Otraja daa vaicajošo algoritmu modelis ir pielietots daudzvertigu funkciju reinašanai. Treša daa ir veltita nedeterminetiem vaicajošiem algoritmiem. Atslegas vardi: kvantu skaitošana, vaicajošais modelis, algoritmu izstrade, algoritmu sarežitibaAbstract Quantum computing is a method of computation based on the laws of quantum mechanics. This subfield of computer science aims to employ quantum mechanical effects for an efficient performance of computational tasks. The main research object of this effort is the quantum query model. The aim of the present research study is to discover efficient quantum query algorithms for certain problems and develop general techniques for designing algorithms. The study has produced several outcomes regarding different kinds of quantum algorithms: exact, bounded-error, and nondeterministic. In the first part of the thesis, exact and bounded-error quantum query algorithms for computing Boolean functions are presented. In the second part, a query model is applied for computing multivalued functions. The third part is devoted to nondeterministic query algorithms. Keywords: quantum computing, query model, algorithm design, algorithm complexit