Quantum walks are quantum counterparts of random walks. In the last 5 years, they have become one of main methods of designing quantum algorithms. Quantum walk based algorithms include element distinctness, spatial search, quantum speedup of Markov chains, evaluation of Boolean formulas and search on "glued trees" graph. In this talk, I will describe the quantum walk method for designing search algorithms and show several of its applications.

Quantum random walks - new method for designing quantum algorithms

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 the efficient performance of computational tasks. In this article, we review the work by the author in a field of quantum algorithms development. In the first part of the article, 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.

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Quantum Query Algorithms

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 = (x 1 , x 2 , …, x n ). Black box can be accessed by querying x i values. 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.

An improved quantum query algorithm for computing AND Boolean function

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 7 N versus O(3 N ).

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Nondeterministic Query Algorithms

Quantum algorithms can be analyzed in a query model to compute Boolean functions. Function input is provided in a black box, and the aim is to compute the function value using as few queries to the black box as possible. The complexity of the algorithm is measured by the number of queries on the worst-case input. In this paper we consider computing AND Boolean function. First, we present a quantum algorithm for AND of two bits. Our algorithm uses one quantum query and correct result is obtained with a probability p=4/5, that improves previous results. The main result is generalization of our approach to design efficient quantum algorithms for computing composite function AND(f1,f2) where fi is a Boolean function. Finally, we demonstrate another kind of an algorithm for AND of two variables, that has a correct answer probability p=9/10, but cannot be extended to compute AND(f1,f2).

High precision quantum query algorithm for computing AND-based boolean functions

Quantum algorithms can be analyzed in a query model to compute Boolean functions where input is given in a black
box, but the aim is to compute function value for arbitrary input using as few queries as possible. In this paper we
concentrate on quantum query algorithm designing tasks. The main aim of research was to find new efficient algorithms
and develop general algorithm designing techniques. 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 sets of exactly computable functions. Finally, we propose quantum algorithm designing methods.
Given algorithms for the set of sub-functions, our methods use them to design 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.

Quantum query algorithms for certain functions and general algorithm construction techniques

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.

Alina Dubrovska

Papers

Quantum query algorithms for certain functions and general algorithm construction techniques

Properties and application of nondeterministic quantum query algorithms

Properties and Application of Nondeterministic Quantum Query Algorithms.