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Results in Physics

A quantum algorithm based on entanglement measure for classifying Boolean multivariate function into novel hidden classes

Mohammed Zidan, Abdel‐Haleem Abdel‐Aty, Duc Manh Nguyen, Ahmed S.A. Mohamed, Yazeed A. Al-Sbou, Hichem Eleuch, Mahmoud Abdel‐Aty

Published Aug 9, 2019• 58 citations
Publisher

Abstract

In this paper, we propose a novel algorithm that solves a generalized version of the Deutsch-Jozsa problem. The proposed algorithm has the potential to classify an oracle U F , that represents an unknown Boolean function on n Boolean variables, to one of 2 n different classes instead of only two classes which are constant and balanced classes in the case of Deutsch-Jozsa algorithm. The proposed algorithm is based on the use of entanglement measure to explore 2 n - 2 additional classes compared to the standard Deutsch-Jozsa algorithm. In addition, the comparison between the proposed quantum algorithm and the classical one is investigated in details. The comparison shows that the proposed algorithm is faster when the number of Boolean variables exceed 14 variables.

DOI
10.1016/j.rinp.2019.102549

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In this paper, we propose a novel algorithm that solves a generalized version of the Deutsch-Jozsa problem. The proposed algorithm has the potential to classify an oracle U F , that represents an unknown Boolean function on n Boolean variables, to one of 2 n different classes instead of only two classes which are constant and balanced classes in the case of Deutsch-Jozsa algorithm. The proposed algorithm is based on the use of entanglement measure to explore 2 n - 2 additional classes compared to the standard Deutsch-Jozsa algorithm. In addition, the comparison between the proposed quantum algorithm and the classical one is investigated in details. The comparison shows that the proposed algorithm is faster when the number of Boolean variables exceed 14 variables.
We present a generalized Deutsch-Jozsa (DJ) quantum algorithm that not only determines both the global type of an unknown Boolean function (constant or balanced) but also determines explicit output values of the function in a single oracle query. Unlike the original DJ algorithm, which identifies only whether a function is constant or balanced, our generalization retrieves actual function output values at the same time with using a Bell state as ancilla. This makes a richer function characterization with minimal queries to have practical quantum advantages, e.g. data classification, logistic regression, and quantum cryptography.
Boolean functions are important building blocks in cryptography for their wide application in both stream and block cipher systems. For cryptanalysis of such systems, one tries to find out linear functions that are correlated to the Boolean functions used in the crypto system. Let f be an n-variable Boolean function and its Walsh spectra is denoted by Wf(ω) at the point ω ∈ {0, 1}n. The Boolean function is available in the form of an oracle. We like to find a ω such that Wf(ω) ≠ 0 as this will provide one of the linear functions which are correlated to f. We show that the quantum algorithm proposed by Deutsch and Jozsa7 solves this problem in constant time. However, the best known classical algorithm to solve the problem requires exponential time in n. We also analyze certain classes of cryptographically significant Boolean functions and highlight how the basic Deutsch–Jozsa algorithm performs on them.
We present a novel quantum algorithm to evaluate the hamming distance between two unknown oracles via measuring the degree of entanglement between two ancillary qubits. In particular, we use the power of the entanglement degree based quantum computing model that preserves at most the locality of interactions within the quantum model structure. This model uses one of two techniques to retrieve the solution of a quantum computing problem at hand. In the first technique, the solution of the problem is obtained based on whether there is an entanglement between the two ancillary qubits or not. In the second, the solution of the quantum computing problem is obtained as a function in the concurrence value, and the number of states that can be generated from the Boolean variables. The proposed algorithm receives two oracles, each oracle represents an unknown Boolean function, then it measures the hamming distance between these two oracles. The hamming distance is evaluated based on the second technique. It is shown that the proposed algorithm provides exponential speedup compared with the classical counterpart for Boolean functions that have large numbers of Boolean variables. The proposed algorithm is explained via a case study. Finally, employing recently developed experimental techniques, the proposed algorithm has been verified using IBM's quantum computer simulator.
Deutsch-Jozsa (DJ) problem is one of the most important problems demonstrating the power of quantum algorithm. DJ problem can be described as a Boolean function fff: {0,1}n→{0,1}\{0,1\}^n\rightarrow \{0,1\}{0,1}n→{0,1} with promising it is either constant or balanced, and the purpose is to determine which type it is. DJ algorithm can solve it exactly with one query. In this paper, we first discover the inherent structure of DJ problem in distributed scenario by giving a number of equivalence characterizations between fff being constant (balanced) and some properties of fff's subfunctions, and then we propose three distributed exact quantum algorithms for solving DJ problem. Our algorithms have essential acceleration over distributed classical deterministic algorithm, and can be extended to the case of multiple computing nodes. Compared with DJ algorithm, our algorithms can reduce the number of qubits and the depth of circuit implementing a single query operator. Therefore, we find that the structure of problem should be clarified for designing distributed quantum algorithm to solve it.