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JUCS - Journal of Universal Computer Science

Nondeterministic Query Algorithms

Alina Vasilieva, Rūsiņš Freivalds

Published Apr 7, 2020• 2 citations
Publisher

Abstract

Abstract: 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 com-pute 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).

DOI
10.3217/jucs-017-06-0859

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Abstract: 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 com-pute 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).
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, with bounded error and even nondeterministic. In this paper, we study the latter class of algorithms. We introduce a new 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. Finally, we show an approach how to construct examples where quantum nondeterministic complexity of an algorithm is O(1), however classical deterministic algorithm for the same function would require O(n) queries.
We consider the quantum query model for computing Boolean functions. The definition of the function is known, but a black box contains the input X = (x1, x2, ..., xn). Black box can be accessed by querying xivalues. The goal is to develop an algorithm, which would compute the function value for arbitrary input using as few queries to the black box as possible. We present two different quantum query algorithms for computing the basic Boolean function - logical AND of two bits. Both algorithms use only one query to determine the function value. Correct answer probability for the first algorithm is 80%, but for the second algorithm it is 90%. To compute this function with the same probability in the classical model, two queries are necessary. We analyze algorithms implementation differences and propose a way to extend the first algorithm to compute AND of two functions.
In this thesis we study various models of query complexity. A query algorithm computes a function under the restriction that the input can be accessed only by making probes to the the bits of the input. The query complexity of a function f is the minimum number of probes made by any query algorithm that computes f. In this thesis, we consider three different models of query complexity, (1) deterministic decision tree complexity (query complexity when the underlying algorithm is deterministic), (2) approximate decision tree complexity aka. property testing (query complexity when the underlying algorithm is probabilistic and only expected to ”approximately ” compute f) and quantum query complexity (query complexity when the underlying algorithm is allowed to make quantum queries). The main results in this thesis are: • We study the relation between deterministic decision tree complexity and other combinatorial measures of complexity measures like sensitivity and block sensitivity. We prove that for minterm-transitive functions the sensitivity is quadratically related to block sensitivity which is polynomially
The query model (or black-box model) has attracted much attention from the communities of both classical and quantum computing. Usually, quantum advantages are revealed by presenting a quantum algorithm that has a better query complexity than its classical counterpart. For example, the well-known quantum algorithms including Deutsch-Jozsa algorithm, Simon algorithm and Grover algorithm all show a considerable advantage of quantum computing from the viewpoint of query complexity. Recently we have considered in (Phys. Rev. A. {\bf 101}, 02232 (2020)) the problem: what functions can be computed by an exact one-query quantum algorithm? This problem has been addressed for total Boolean functions but still open for partial Boolean functions. Thus, in this paper we continue to characterize the computational power of exact one-query quantum algorithms for partial Boolean functions by giving several necessary and sufficient conditions. By these conditions, we construct some new functions that can be computed exactly by one-query quantum algorithms but have essential difference from the already known ones. Note that before our work, the known functions that can be computed by exact one-query quantum algorithms are all symmetric functions, whereas the ones constructed in this papers are generally asymmetric.
Quantum algorithms can be analyzed in a query model to compute Boolean functions where input is given in a black box and the aim is to compute function value for arbitrary input using as few queries as possible. We concentrate on quantum query algorithm designing tasks in this paper. The main aim of the research was to find new efficient algorithms and develop general algorithm designing techniques. First, we present several exact quantum query algorithms for certain problems that are better than classical counterparts. Next, we introduce algorithm transformation methods that allow significant enlarging of exactly computable functions sets. Finally, we propose quantum algorithm designing methods. Given algorithms for the set of sub-functions, our methods use them to design a more complex one, based on algorithms described before. Methods are applicable for input algorithms with specific properties and preserve acceptable error probability and number of queries. Methods offer constructions for computing AND, OR and MAJORITY kinds of Boolean functions.
Quantum algorithms can be analyzed in a query model to compute Boolean functions where input is given in a black box and the aim is to compute function value for arbitrary input using as few queries as possible. We concentrate on quantum query algorithm designing tasks in this paper. The main aim of the research was to find new efficient algorithms and develop general algorithm designing techniques. First, we present several exact quantum query algorithms for certain problems that are better than classical counterparts. Next, we introduce algorithm transformation methods that allow significant enlarging of exactly computable functions sets. Finally, we propose quantum algorithm designing methods. Given algorithms for the set of sub-functions, our methods use them to design a more complex one, based on algorithms described before. Methods are applicable for input algorithms with specific properties and preserve acceptable error probability and number of queries. Methods offer constructions for computing AND, OR and MAJORITY kinds of Boolean functions.