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International Telecommunications Symposium (ITS)

Quantum search algorithm using quantum bit string comparator

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

Published Sep 1, 2006• 12 citations
Publisher

Abstract

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.

DOI
10.1109/its.2006.4433341

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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.
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.
Quantum Bit String Comparators (QBSC) operate on two sequences of n-qubits, enabling the determination of their relationships, such as equality, greater than, or less than. This is analogous to the way conditional statements are used in programming languages. Consequently, QBSCs play a crucial role in various algorithms that can be executed or adapted for quantum computers. The development of efficient and generalized comparators for any nnn-qubit length has long posed a challenge, as they have a high-cost footprint and lead to quantum delays. Comparators that are efficient are associated with inputs of fixed length. As a result, comparators without a generalized circuit cannot be employed at a higher level, though they are well-suited for problems with limited size requirements. In this paper, we introduce a generalized design for the comparison of two nnn-qubit logic states using just two ancillary bits. The design is examined on the basis of qubit requirements, ancillary bit usage, quantum cost, quantum delay, gate operations, and circuit complexity, and is tested comprehensively on various input lengths. The work allows for sufficient flexibility in the design of quantum algorithms, which can accelerate quantum algorithm development.
Considering the essential difference between quantum bit and classical bit,this paper proposes a quantum binary searching algorithm.Using the quantum parallel process and quantum permutation operations,the circuit implementation of quantum binary searching algorithm is presented.The example shows that the algorithm can greatly reduce the implementation time and steps compared with classical binary search algorithm.This scheme has important applicable potential in quantum information processing within decoherence time.
We propose a quantum algorithm for closest pattern matching which allows us to search for as many distinct patterns as we wish in a given string (database), requiring a query function per symbol of the pattern alphabet. This represents a significant practical advantage when compared to Grover's search algorithm as well as to other quantum pattern matching methods, which rely on building specific queries for particular patterns. Our method makes arbitrary searches on long static databases much more realistic and implementable. Our algorithm, inspired by Grover's, returns the position of the closest substring to a given pattern of size MMM with non-negligible probability in O(N)O(\sqrt{N})O(N​) queries, where NNN is the size of the string. Furthermore, we give the full recipe to implement our algorithm (together with its total circuit complexity), thus offering an oracle-based quantum algorithm ready to be implemented.
This paper presents a new technique of magnitudecomparator for quantum bit string comparison. In the proposedmethod, the comparison between two quantum bit strings isperformed with the optimum number of operations. We have alsoshown that the proposed technique has time complexityO(( + )), whereas the best known existing technique hasO(nlog n),where n is the number of quantum bits. In addition, wehave proposed another technique to produce a compact quantumcomparator circuit. We have also introduced three new quantumgates with unique unitary matrices which represent the proposedquantum circuit more compactly than the existing quantumgates. The proposed comparator circuit has been designed usingthe proposed quantum bit comparator circuit named MidpointQubits Comparison (MQC) Circuit and another proposedquantum bit comparator circuit named Rest Qubits Comparison(RQC) Circuit. The comparative study shows that the proposedcomparator circuit outperforms the existing comparators; e.g.,the proposed 64-qubit comparator improves 7.19% on number ofquantum gates, area and delay; and 50.39% on garbage outputsover the existing best one
The nature of the search Suppose you know that exactly one n -bit integer satisfies a certain condition, and suppose you have a black-boxed subroutine that acts on the N = 2 n different n -bit integers, outputting 1 if the integer satisfies the condition and 0 otherwise. In the absence of any other information, to find the special integer you can do no better with a classical computer than to apply the subroutine repeatedly to different random numbers until you hit on the special one. If you apply it to M different integers the probability of your finding the special number is M/N . You must test ½ N different integers to have a 50% chance of success. If, however, you have a quantum computer with a subroutine that performs such a test, then you can find the special integer with a probability that is very close to 1 when N is large, using a method that calls the subroutine a number of times no greater than. This very general capability of quantum computers was discovered by Lov Grover, and goes under the name of Grover's search algorithm . Shor's period-finding algorithm and Grover's search algorithm, together with their various modifications and extensions, constitute the two masterpieces of quantum-computational software. One can think of Grover's black-boxed subroutine in various ways. The subroutine might perform a mathematical calculation to determine whether the input integer is the special one. Here is a simple example.