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The so-called Hamming distance measures the difference between two binary strings A and B. In simplified form, it measures the number of changes in A to get B. This type of distance is very useful in classical computing in applications such as error correction. It is also advantageous in quantum computing, being for example widely used in quantum machine learning. Since current quantum computers have limited resources, this type of distance is particularly attractive because it can be computed using fewer qubits and operations than other distances such as Euclidean or Manhattan distances. In this paper, two circuits for calculating Hamming distances using exclusively Clifford+T gates are presented. The aim of both circuits is to reduce the quantum cost and number of T gates needed to compute the Hamming distance. The T gate is more expensive than the other gates, so this reduction will have a significant impact on the total cost of the circuits. Furthermore, the proposed circuits are implemented using only Clifford+T gates. The circuits implemented exclusively with this group of gates are compatible with proven error detection and correction codes. [ABSTRACT FROM AUTHOR]

Generalized hesitant fuzzy numbers (GHFNs) are able to directly manage situations in which we may encounter a finite set of known values with a finite set of degrees of doubt as quantitative approximations of an uncertain situation/quantification of a linguistic expression. They are new extensions of hesitant fuzzy sets, which have been considered in this paper. In fact, in this paper, GHFNs will be utilized to model the uncertainty of the assessment values of options against criteria in multi-attribute decision making (MADM) problems. It means that all of the elements of decision matrix are GHFNs. Then, the technique for order of preference by similarity to ideal solution (TOPSIS) method, as a very successful method in solving MADM problems, will be updated to be used with GHFNs. To this end, the distance between GHFNs must be defined to obtain the distances between given alternatives from each of two subjective alternatives (positive/negative ideal solutions). Thus, three existing famous distance measures, i.e., general distance ( d g ), Hamming distance ( d h ), and Euclidean distance ( d e ) measures, have been updated for GHFNs firstly. Then, the new TOPSIS method will be proposed based on GHFNs. Finally, the numerical examples have been appointed to illustrate the proposed method, analyze comparatively and validate it. [ABSTRACT FROM AUTHOR]

Image representation and approximate query is always a research challenge and is affected greatly by the dimension and size of images. Since hash-based methods and binary encodings in combination with other techniques, such as kernel tricks, a longer binary code and mapping vectors rotation, can maintain a linear query time and query accuracy, they have been used in this area broadly. This paper develops principal component analysis hashing (PCAH) and unequal length of binary coding to divide images into more categories, denoted as PCA-MD, to improve accuracy of the representation and lookup of images. This paper firstly proves that the eigenvector mapping is locality sensitive, which is the basis for more classes division. For the anisotropy of the eigenvectors, PCA-MD utilizes an unequal length of binary coding and fewer eigenvectors, rather than an equal code, to divide the images mapped on every eigenvector to more categories. Moreover, L1-norm distance is applied to measure the distances of images to avoid the enormous computation of Euclidean distance. Theoretical analysis and extensive experimental results demonstrate that the PCA-MD has a higher query performance and a slight longer run time than the state-of-the-art approaches based on the Hamming distance. This in turn verifies that PCAH is a locality sensitive hash and that partitioning into more categories rather than only two categories is reasonable. [ABSTRACT FROM AUTHOR]

In this study, hard turning tests are carried out on the hardened AISI 4140 steel material. Power consumption, sound intensity, processing time and surface roughness values are measured for all combinations of cutting conditions such as three different cutting speeds, three different feed rates, three different depths of cut and two different tool cutting edge angle. According to these measurements, the ideal cutting condition is found by using the TOPSIS method which is integrated by fuzzy Hamming distance, Euclidean distance and Hausdorff distance measure methods. Also in case that machinability parameters have different importance degrees, ideal cutting condition is determined by using the weighted fuzzy distance measure method. According to the experimental results obtained, it is observed that the surface roughness value increased with the increasing feed rate. Roughness values obtained at 90 degree tool cutting edge angle are lower than 62.5 degrees. It is observed that the sound intensity increases with the increasing feed rate. The machining time is reduced with increasing feed rate, depth of cut and cutting speed. Thus, total power consumption is reduced. [ABSTRACT FROM AUTHOR]

The concept of distance between Pythagorean fuzzy sets (PFSs) has been proven to be relevant in the applications of PFSs as seen in the literature. The main purpose of this paper is to show that Zhang and Xu's distance measure between PFSs fails the conditions of distance measure; hence, it is not an appropriate distance measure for PFSs. Some numerical examples are used to validate this stance. In order to remedy this shortcoming, Zhang and Xu's distance measure for PFSs is normalised/modified to cater for the limitation by employing the technique used to normalise both Hamming and Euclidean distances between intuitionistic fuzzy sets by Szmidt and Kacprzyk. The modified Zhang and Xu's distance measure for PFSs satisfies the conditions of the axiomatic definition of distance measure for PFSs; hence, it is an appropriate/reliable distance measure for PFSs. Finally, the modified Zhang and Xu's distance measure for PFSs is applied to pattern recognition problems of classification of building materials and mineral fields. [ABSTRACT FROM AUTHOR]

Compared with Boolean functions, multi-output Boolean functions (a.k.a. vectorial Boolean functions) are commonly used in classical cryptography. More generally, many cryptographic primitives can be treated as multi-output Boolean functions. Hence, the research on property testing of multi-output Boolean functions is meaningful for the design and cryptanalysis of symmetric cryptography. This paper mainly focuses on the cryptographic property testing of multi-output Boolean functions in the quantum world. Firstly, the generalized Deutsch–Jozsa algorithm is proposed to distinguish balanced multi-output Boolean functions from constant ones with a single query. This algorithm has a wider scope of applications with arbitrary ancillary inputs. The first generalized Bernstein–Vazirani algorithm suitable for multi-output Boolean functions is presented to recover the linear coefficients of linear functions. Then, combined with the generalized Deutsch–Jozsa algorithm, the quantum algorithm for estimating Walsh coefficients of multi-output Boolean functions is proposed with the same idea of quantum approximate counting algorithm, accompanied with an algorithm for computing the Walsh coefficient at a specified point based on quantum exact counting algorithm. Finally, with the usage of algorithms mentioned above, the cryptographic property testing of multi-output Boolean functions is studied. In order to describe the distances from having the certain properties, Euclidean distance and Manhattan distance are introduced as complements of Hamming distance. According to the definition, the first balance testing of multi-output Boolean functions is presented by testing the uniformity of images. The second algorithm exploits the relationship between balance and the Walsh coefficients at the point 0 which could be easily extended to k-order resiliency testing. We also briefly analyze the query complexities of strict avalanche criterion testing and k-order propagation criteria testing. The linearity testing algorithm for multi-output Boolean functions based on the generalized Bernstein–Vazirani algorithm can be adapted to Boolean functions achieving a further speedup. The non-junta testing algorithm is proposed with lower query complexity. [ABSTRACT FROM AUTHOR]