Quantum Algorithms for Similarity Measurement Based on Euclidean Distance

Similarity measurement is a fundamental problem that arise both on its own and as a key subroutine in more complex tasks, such as machine learning. However, in classical algorithms, the time used to similarity measurement usually increases exponentially as the amount of data and the number of data dimensions increase. In this paper, we presented three quantum algorithms based on Euclidean distance to measure the similarity between data sets. In the proposed algorithms, some special unitary operations are utilized to construct imperative quantum states from quantum random access memory. Then, a badly needed result for estimating the similarity between data sets, can be got by performing projective measurements. Furthermore, it is shown that these algorithms can achieve the exponential acceleration of the classical algorithm in the quantity or the dimension of data.

[1]  Zhiqiang Zhang,et al.  An efficient storage and service method for multi-source merging meteorological big data in cloud environment , 2019, EURASIP Journal on Wireless Communications and Networking.

[2]  Masoud Mohseni,et al.  Quantum support vector machine for big feature and big data classification , 2013, Physical review letters.

[3]  Guy W. Mineau,et al.  A simple KNN algorithm for text categorization , 2001, Proceedings 2001 IEEE International Conference on Data Mining.

[4]  Davide Anguita,et al.  Quantum optimization for training support vector machines , 2003, Neural Networks.

[5]  Hamid Seifoddini,et al.  Single linkage versus average linkage clustering in machine cells formation applications , 1989 .

[6]  M. Schuld,et al.  Prediction by linear regression on a quantum computer , 2016, 1601.07823.

[7]  Philip N. Klein,et al.  Local Search Yields Approximation Schemes for k-Means and k-Median in Euclidean and Minor-Free Metrics , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[8]  Aristides Gionis,et al.  Clustering Aggregation (long version) , 2007 .

[9]  A. Harrow,et al.  Quantum algorithm for linear systems of equations. , 2008, Physical review letters.

[10]  Yilin Bei,et al.  Medical Health Big Data Classification Based on KNN Classification Algorithm , 2020, IEEE Access.

[11]  Seth Lloyd,et al.  Quantum algorithm for data fitting. , 2012, Physical review letters.

[12]  Aristides Gionis,et al.  Clustering aggregation , 2005, 21st International Conference on Data Engineering (ICDE'05).

[13]  vCaslav Brukner,et al.  Quantum circuits cannot control unknown operations , 2013, 1309.7976.

[14]  Seth Lloyd,et al.  Quantum algorithm for data fitting. , 2012, Physical review letters.

[15]  Zhang Wei,et al.  Investigation of the evaluation system of SMEs’ industrial cluster management performance based on wireless network development , 2019, EURASIP Journal on Wireless Communications and Networking.

[16]  Patricia D Franklin,et al.  Using Cluster Analysis to Identify Patient Factors Linked to Differential Functional Gains After Total Knee Arthroplasty. , 2019, The Journal of arthroplasty.

[17]  Juan Wang,et al.  Quantum Voting Scheme Based on Locally Indistinguishable Orthogonal Product States , 2020 .

[18]  Tanvir Habib Sardar,et al.  An analysis of MapReduce efficiency in document clustering using parallel K-means algorithm , 2018, Future Computing and Informatics Journal.

[19]  S. Lloyd,et al.  Quantum algorithms for supervised and unsupervised machine learning , 2013, 1307.0411.

[20]  Gilles Brassard,et al.  Quantum speed-up for unsupervised learning , 2012, Machine Learning.

[21]  Miguel-Angel Martin-Delgado,et al.  Google in a Quantum Network , 2011, Scientific Reports.

[22]  G. Brassard,et al.  Quantum Amplitude Amplification and Estimation , 2000, quant-ph/0005055.

[23]  Yue Ruan,et al.  Quantum Algorithm for K-Nearest Neighbors Classification Based on the Metric of Hamming Distance , 2017, International Journal of Theoretical Physics.

[24]  Wei Zhang,et al.  Investigation of the evaluation system of SMEs' industrial cluster management performance based on wireless network development , 2019, EURASIP J. Wirel. Commun. Netw..

[25]  Jesús Gómez-Gardeñes,et al.  Quantum Navigation and Ranking in Complex Networks , 2012, Scientific Reports.

[26]  L. Wossnig,et al.  Quantum Linear System Algorithm for Dense Matrices. , 2017, Physical review letters.

[27]  Luming Duan,et al.  Quantum discriminant analysis for dimensionality reduction and classification , 2015, 1510.00113.

[28]  Chao-Hua Yu,et al.  Quantum algorithms for ridge regression , 2017 .

[29]  Lov K. Grover.  Quantum Mechanics Helps in Searching for a Needle in a Haystack , 1997, quant-ph/9706033.

[30]  Iordanis Kerenidis,et al.  Quantum Recommendation Systems , 2016, ITCS.

[31]  Seth Lloyd,et al.  Quantum random access memory. , 2007, Physical review letters.

[32]  Daniel A. Lidar,et al.  Adiabatic quantum algorithm for search engine ranking. , 2011, Physical review letters.

[33]  Maria Schuld,et al.  The quest for a Quantum Neural Network , 2014, Quantum Information Processing.

[34]  S. Lloyd,et al.  Quantum principal component analysis , 2013, Nature Physics.

[35]  Dan Qu,et al.  An online log template extraction method based on hierarchical clustering , 2019, EURASIP J. Wirel. Commun. Netw..

[36]  ZhaoXia,et al.  Hierarchical and Complex System Entropy Clustering Analysis Based Validation for Traditional Chinese Medicine Syndrome Patterns of Chronic Atrophic Gastritis. , 2019 .

[37]  Ola Svensson,et al.  Better Guarantees for k-Means and Euclidean k-Median by Primal-Dual Algorithms , 2020, SIAM J. Comput..

[38]  Peter W. Shor,et al.  Algorithms for quantum computation: discrete logarithms and factoring , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[39]  Scott Aaronson,et al.  BQP and the polynomial hierarchy , 2009, STOC '10.

[40]  Miguel-Angel Martin-Delgado,et al.  Quantum Google in a Complex Network , 2013, Scientific Reports.

[41]  R. Cleve,et al.  Efficient Quantum Algorithms for Simulating Sparse Hamiltonians , 2005, quant-ph/0508139.