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Quantum circuits for new applications of the Deutch-Jozsa algorithm

José Manuel Bravo

Published Apr 25, 2018• 0 citations
Publisher

Abstract

In this paper a serial of new applications on binary strings based on the Deutch-Jozsa algorithm are presented. These new applications can be used to calculate the Hamming weight and the Hamming distance of a binary string or to compare if two binary strings have the same Hamming weight or the same Hamming distance. To test the applications the quantum computer prototype that IBM has given open access to on the cloud has been used to test the results.

DOI
10.31219/osf.io/ewdn8

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In this paper a serial of new applications on binary strings based on the Deutch-Jozsa algorithm are presented. These new applications can be used to calculate the Hamming weight and the Hamming distance of a binary string or to compare if two binary strings have the same Hamming weight or the same Hamming distance. To test the applications the quantum computer prototype that IBM has given open access to on the cloud has been used to test the results.
In this brief paper a quantum algorithm to calculate the Hamming distance of two binary strings of equal length (or messages in theory information) is presented. The algorithm calculates the Hamming weight of two binary strings in one query of an oracle. To calculate the hamming distance of these two strings we only have to calculate the Hamming weight of the xor operation of both strings. To test the algorithms the quantum computer prototype that IBM has given open access to on the cloud has been used to test the results.
In this brief paper a quantum algorithm to calculate the Hamming distance of two binary strings of equal length (or messages in theory information) is presented. The algorithm calculates the Hamming weight of two binary strings in one query of an oracle. To calculate the hamming distance of these two strings we only have to calculate the Hamming weight of the xor operation of both strings. To test the algorithms the quantum computer prototype that IBM has given open access to on the cloud has been used to test the results.
The Hamming distance is ubiquitous in computing. Its computation gets expensive when one needs to compare a string against many strings. Quantum computers (QCs) may speed up the comparison. In this paper, we extend an existing algorithm for computing the Hamming distance. The extension can compare strings with symbols drawn from an arbitrary-long alphabet (which the original algorithm could not). We implement our extended algorithm using the QisKit framework to be executed by a programmer without the knowledge of a QC (the code is publicly available). We then provide four pedagogical examples: two from the field of bioinformatics and two from the field of software engineering. We finish by discussing resource requirements and the time horizon of the QCs becoming practical for string comparison.
Abstract In the coding theory, it is one of the most important research problems to find an efficient algorithm for the decoding process. A quantum computer is an innovative computing device that performs calculations using the principles of quantum mechanics. In this paper, we find a decoding algorithm for binary linear codes using quantum circuits; this is the first time that quantum circuits are used as a main tool for decoding binary linear codes. Our decoding algorithm is mainly based on Grover’s search algorithm. In detail, we study the process of implementing the exhaustive search using quantum circuits for decoding binary linear codes. For our purpose, we design a modified Grover’s algorithm (algorithm 1) using three tools: the matrix multiplier, the Hamming weight counter, and the state indicator. We present algorithm 2, which is a quantum decoding algorithm for binary linear codes. We illustrate algorithm 1 using the [7, 4, 3] Hamming code, and algorithm 2 is implemented by IBM Quantum Composer.
Quantum Computation is seen as an alternative paradigm for computing as the physical implentations for the basic information unit, the binary digit, approach the atomic level. As the laws of quantum mechanics take over at this level, quantum phenomena such as superposition, the ensuing parallelism and entanglement can be harnessed to offer faster algorithms. Deutch- Jozsa was the first problem for such an algorithm was exhibited that was provably faster than any deterministic classical algorithm. Here we make an attempt at a generalization of this algorithm, hoping that it could lead to similar generalizations for other quantum algorithms.