Quantum computing aims to use quantum mechanical effects for the efficient performance of computational tasks. A popular research direction is enlarging the gap between classical and quantum algorithm complexity of the same computational problem. We present new results in quantum query algorithm design for multivalued functions that allow to achieve a large quantum versus classical complexity separation. To compute a basic finite multifunction in a quantum model only one query is enough while classically three queries are required. Then, we present two generalizations and a modification of the original algorithm, and obtain the following complexity gaps: QUD(M') ≤ N versus CUD(M') ≥ 3N, and QRD(M'') = 1 versus N/2 + 1 ≤ CRD(M'') ≤ N.
Quantum computing aims to use quantum mechanical effects for the efficient performance of computational tasks. A popular research direction is enlarging the gap between classical and quantum algorithm complexity of the same computational problem. We present new results in quantum query algorithm design for multivalued functions that allow to achieve a large quantum versus classical complexity separation. To compute a basic finite multifunction in a quantum model only one query is enough while classically three queries are required. Then, we present two generalizations and a modification of the original algorithm, and obtain the following complexity gaps: QUD(M') ≤ N versus CUD(M') ≥ 3N, and QRD(M'') = 1 versus N/2 + 1 ≤ CRD(M'') ≤ N.
This paper investigates the computability of mathematical relations in a quantum query model. The important task in complexity theory is to find examples with a large gap between classical and quantum algorithm complexity of the same computational problem. We present new results in quantum query algorithm design that allow achieving a large separation between classical and quantum query complexity of a specific relation. We demonstrate an example where quantum query algorithm for a finite relation needs more than two times fewer queries than the best possible classical analogue. We also show that relation can be extended to infinite family of relations with an input of general size N.
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.
Query complexity is a model of computation in which we have to compute a function f(x1,…,xN) of variables xi which can be accessed via queries. The complexity of an algorithm is measured by the number of queries that it makes. Query complexity is widely used for studying quantum algorithms, for two reasons. First, it includes many of the known quantum algorithms (including Grover's quantum search and a key subroutine of Shor's factoring algorithm). Second, one can prove lower bounds on the query complexity, bounding the possible quantum advantage. In the last few years, there have been major advances on several longstanding problems in the query complexity. In this talk, we survey these results and related work, including: - the biggest quantum-vs-classical gap for partial functions (a problem solvable with 1 query quantumly but requiring Ω(N) queries classically); - the biggest quantum-vs-determistic and quantum-vs-probabilistic gaps for total functions (for example, a problem solvable with M queries quantumly but requiring Ω~(M2.5) queries probabilistically); - the biggest probabilistic-vs-deterministic gap for total functions (a problem solvable with M queries probabilistically but requiring Ω~(M2) queries deterministically); - the bounds on the gap that can be achieved for subclasses of functions (for example, symmetric functions); - the connections between query algorithms and approximations by low-degree polynomials.
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.
The last decade witnessed a great surge of fruitful studies in the new paradigm of quantum computing. Remarkable progress has been made in all the areas including quantum algorithms, quantum complexity theory, quantum error-correcting code, quantum information theory, quantum cryptography and physical implementations of quantum computers. However, quantum algorithm design, probably the most important task in quantum computing, did not make as much progress as people expected. This thesis aims at both designing new quantum algorithms and studying why efficient quantum algorithms are so hard to design by proving many lower bounds on quantum query complexity for various problems.
We first review (and generalize) known techniques for designing quantum algorithms and for proving quantum lower bounds. Using these techniques, we then study the quantum query complexity for many problems, including some specific problems such as Graph Matching and other natural graph properties, the general tuple search, and local search. We also study the lowest possible quantum query complexity for a class of functions as a whole, such as graph properties, circular functions, and functions invariant of a transitive permutation group, where tight results are given in all these cases.
Finally for the quantum lower bounds techniques themselves, we study the main two techniques and show that they are incomparable in power. For the most widely used one, which has a lot of parameters to choose, we show limitations for it in terms of certificate complexity. This implies that we cannot use the method to prove better lower bounds for almost all known open problems in this area.