arXiv:quant-ph/0605003

Solving mathematical problems with quantum search algorithm

Rubens Viana Ramos, Paulo Benicio Melo de Sousa, David Sena Oliveira

Published Apr 29, 2006• 3 citations

Abstract

Quantum computation has attracted much attention since it was shown by Shor and Grover the possibility to implement quantum algorithms able to realize, respectively, factoring and searching in a faster way than any other known classical algorithm. It is possible to use Grover algorithm, taking profit of its ability to find a specific value in a unordered database, to find, for example, the zero of a logical function; the minimal or maximal value in a database or to recognize if an odd number is prime or not. Here we show quantum algorithms to solve those cited mathematical problems. The solution requires the use of a quantum bit string comparator being used as oracle. This quantum circuit compares two quantum states and identifies if they are equal or, otherwise, which of them is the largest. Moreover, we also show the quantum bit string comparator allow us to implement conditional statements in quantum computation, a fundamental structure for designing of algorithms.

DOI: 10.48550/arxiv.quant-ph/0605003

Related papers

Quantum search algorithm using quantum bit string comparator

David Sena Oliveira, Paulo Benicio Melo de Sousa, Rubens Viana Ramos

In this work, a quantum circuit able to compare if two bit strings are equal or not and, in this last case, identify which of them is the largest, is proposed. It is named quantum bit string comparator. Further, we present two applications of the quantum bit string comparator working together with Grover quantum search algorithm: implementation of conditional statements in quantum algorithms and a finder of the zero of a logical function.

International Telecommunications Symposium (ITS)Sep 1, 200612 citations

Quantum Boolean circuit approach for searching an unordered database

I-Ming Tsai, Sy‐Yen Kuo, David S. L. Wei

The discovery of polynomial time prime factorization, secure key distribution, and fast database search algorithm have recently made quantum computing the most rapidly expanding research field. For a quantum algorithm to be useful, it is essential that the algorithm can be implemented using quantum circuits. Nanotechnology, in particular quantum mechanics based devices, can be used to realize such an algorithm. In this paper, we show how quantum Boolean circuits can be used to implement the oracle circuit and the inversion-about-average function in Grover's search algorithm. We also show that a slight modification of the oracle circuit can be used to search multiple targets.

IEEE Conference on Nanotechnology (IEEE NANO)Jun 25, 20037 citations

Quantum Computing Algorithms for Solving Complex Mathematical Problems

Et al. Lalit Mohan Trivedi

The power of quantum mechanics, that is too complex for conventional computers, can be solved by an innovative model of computing known as quantum computing. Quantum algorithms can provide exponential speedups for some types of problems, such as many difficult mathematical ones. In this paper, we review some of the most important quantum algorithms for hard mathematical problems. When factoring large numbers, Shor's algorithm is orders of magnitude faster than any other known classical algorithm. The Grover's algorithm, which searches unsorted databases much more quickly than conventional algorithms, is then discussed.

International Journal on Recent and Innovation Trends in Computing and CommunicationNov 7, 20231 citation

Implementation of Grover's Algorithm based on Quantum Reservoir Computing

Shivani Mehta, Vidya Praveen Bhallamudi, Arige Sumanth +1 more

Quantum computing represents the leading edge of computational technology, leveraging the principles of quantum mechanics to execute targeted computations much faster than classical computers. In contrast to classical bits, which are limited to representing either 0 or 1, qubits, or quantum bits, exhibit the extraordinary property of superposition. This distinctive characteristic enables qubits to simultaneously occupy multiple states, empowering quantum computers to explore numerous potential solutions to a problem concurrently. This feature makes quantum computing particularly potent for specific tasks. Recent research endeavors have been sparked by the potential of advanced quantum computing technology, leading to the creation of simulations of quantum computers using classical hardware. Grover's quantum search algorithm serves as a notable illustration of quantum computing application, enabling quantum computers to conduct a database search within an unsorted array with a quadratic speedup in time efficiency compared to classical computers. This document presents the quantum Grover search algorithm and its application through 5-qubit quantum circuits, as well as a design framework to simplify the creation of an oracle for a greater number of qubits.

International Conference for Emerging Technology (INCET)May 24, 20242 citations

Classical Search and Quantum Search Algorithms for Synthesis of Quantum Circuits and Optimization of Quantum Oracles

Sazzad Hossain

We observe an enormous increase in the computational power of digital computers. This was due to the revolution in manufacturing processes and controlling semiconductor structures on submicron scale, ultimately leading to the control of individual atoms. Eventually, the classical electric circuits encountered the barrier of quantum mechanics and its effects. However, the laws of quantum mechanics can be also used to produce computational devices that lead to extraordinary speed increases over classical computers. Thus quantum computing becomes a very promising and attractive research area. The Computer Aided Design for Quantum circuits becomes an essential ingredient for such emerging research which may lead to these powerful computers to be realized—an era of Quantum computing. This thesis presents an integrated theoretical study of software algorithms to design circuits of quantum oracles as well as methods for designing quantum oracles for Grover algorithm to solve combinatorial problems. An implementation of quantum algorithm involves the initialization of the input state and its manipulation with quantum gates followed by the measurements. In Grover algorithm the problem to be solved is specified by a permutative logic oracle – the fundamental problem is then how to build this oracle from quantum logic circuits and how to optimize these circuits. These problems are NP-hard and require search algorithms. In future, the search will be also done in quantum and this thesis leads to quantum algorithms to design quantum circuits more efficiently.

Jan 1, 20091 citation

A Method for Application of a Quantum Search Algorithm to Classical Databases

David Jones, Benjamin T. H. Varcoe

Grover's algorithm is normally presented as a method of searching a database, however it would be more accurately described as a method of identifying elements of an interval of the integers which satisfy some logical clause - an example might be identifying binary strings which correspond to the solutions of a Sudoku problem. In this paper we present the first method of performing a true database search using Grover's search algorithm, by first creating a mapping from a set of indices in the range 0:2^n-1 to a set of database elements, then applying the clause to these elements. We then demonstrate the feasibility of an attack against the Diffie-Hellman cryptosystem based on a Grover's search of a database of candidate solutions generated via the number field sieve algorithm.

arXiv:2206.03938Jun 8, 20220 citations