International Journal of Image, Graphics and Signal Processing (IJIGSP)
IJIGSP Vol. 10, No. 7, 8 Jul. 2018
Cover page and Table of Contents: PDF (size: 803KB)
Quantum Wavelet Transforms Generated by the Product of the Sine Polynomial and the Gaussian Envelope on the Tetrahedral Graph
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Author(s)
Index Terms
Graphs Theory, Schrödinger Equation, Sine Polynomial, Haar Wavelets, Quantum Gates, Spectral Analysis, Yang-Baxter Equation, Gamma Matrices, Hamiltonian, Hadamard-Walsh Transform
Abstract
In this paper we present a novel technique that permits to extract the essential on information embedded in the product of sine polynomial and Gaussian envelope by simply knowing the vertices of the tetrahedral graph. The study proves that the matrix of vertices of the tetrahedral graph and its variants are the building block of both Haar wavelets, Hadamard-Walsh transform, wavelets sets and tight frames. We also prove that the Berkeley B Gate is a function of the degree matrix and the adjacency matrix of the tetrahedral graph. The latter is the Hermitian part of the unitary polar decomposition in terms of elementary gates for quantum computation [68] which reveals interesting properties of the tetrahedral graph in both quantum group, Lie group and Pauli group for wavelets sets, quantum image processing and quantum data compression. We explore the connection existing among graphs theory, wavelets, tight frames and quantum logic gates.
Cite This Paper
Jean Bosco Mugiraneza, "Quantum Wavelet Transforms Generated by the Product of the Sine Polynomial and the Gaussian Envelope on the Tetrahedral Graph", International Journal of Image, Graphics and Signal Processing(IJIGSP), Vol.10, No.7, pp. 11-24, 2018. DOI: 10.5815/ijigsp.2018.07.02
Reference
[1]Nora Leonardi, Dimitri Van De Ville, “Wavelet Frames on Graphs Defined by FMRI Functional Connectivity”, IEEE, ISBI, 2011.
[2]Akie Sakiyama and Yuichi Tanaka, “Spectral Graph Wavelets and Filter Banks With Low Approximation Error”, IEEE Transactions on Signal and Information Processing Over Networks, Vol. 2, No 3, June 2016., “doi10.1109/TSIPN.2016. 2581303
[3]Nicolas Tremblay and Pierre Borgnat,” Subgraph Base Filter banks for Graphs Signals”, Dec 2015, HAL Id: hal-0124388.
[4]D. Shuman, S. Narang, P. Frossard, A. Ortega, and P. Vandergheynst, “The emerging field of signal processing on graphs: Extending highdimensional data analysis to networks and other irregular domains,” Signal Processing Magazine, IEEE, Vol. 30, no. 3, pp. 83–98, May 2013.
[5]R. Vilela Mendes, Hugo C. Mendes, Tanya Ara´ujo,” Signal Processing on Graphs: Transforms and Tomograms”, June 2014.
[6]D. I. Shuman, S. Narang, P. Frossard, A. Ortega, and P. Vandergheynst, “The Emerging Field of Signal Processing on Graphs: Extending Highdimensional Data Analysis to Networks and Other Irregular Domains,” IEEE Signal Process. Mag., vol. 30, No. 3, pp. 83–98, 2013.
[7]A. Sandryhaila and J. M. F. Moura, “Discrete Signal Processing ongraphs,” IEEE Trans. Signal Process., Vol. 61, No. 7, pp. 1644–1656,2013.
[8]David K. Hammond, Pierre Vandergheynst and Remi Gribonval, Wavelets on Graphs via Spectral Graph Theory, Applied and Computational Harmonics Analysis, Vol. 30, pp. 129-150, 2011
[9]C. Godsil and G. F. Royle, Algebraic Graph Theory. Springer, 2001.
[10]F. K. Chung, Spectral Graph Theory. Vol. 92 of the CBMS Regional Conference Series in Mathematics, AMS Bokstore, 1997.
[11]D. Spielman, “Spectral graph theory,” in Combinatorial Scientific Computing. Chapman and Hall / CRC Press, 2012.
[12]Sunkyu Yu, Xianji Piao, Jiho Hong and Namkyoo Park Inter-dimensional Optical Isospectrality Inspired by Graph Networks, 2016,https://arxiv.org/ftp/arxiv/papers/1601/1601.05635.pdf
[13]Chensheng Wu, Jonathan Ko, Christopher C. Davis, Using Graph Theory and a Plenoptic Sensor to Recognize Phase Distortions of a Laser Beam, https://arxiv.org/ftp/arxiv/papers/1506/1506.00310.pdf
[14]Kan Wu, Javier Garcıa de Abajo, Cesare Soci, Perry Ping Shum, and Nikolay I Zheludev, An Optical Fiber Network Oracle for NP-Complete Problems, Light: Science & Applications (2014) 3,e147; doi:10.1038/lsa.2014.28
[15]Mario Krenn, Xuemei Gu and Anton Zeilinger, Quantum Experiments and Graphs: Multiparty States as Coherent Superpositions of Perfect Matchings, May 2017.
[16]M. Krenn, M. Malik, R. Fickler, R. Lapkiewicz and A. Zeilinger, Automated Search for New Quantum Experiments. Physical review letters116, 09040, 5 2016.
[17]M. Hein, J. Eisert and H.J. Briegel, Multiparty Entanglement in Graph States. Physical Review A69,062311, 2004.
[18]C.Y. Lu, X.Q. Zhou, O. G ̈uhne, W.B. Gao, J. Zhang, Z.S. Yuan, A. Goebel, T. Yang and J.W. Pan, Experimental Entanglement of Six Photons in Graph States. Nature Physics 3, 91–95, 2007.
[19]S.L. Braunstein, S. Ghosh and S. Severini, The Laplacian of a Graph as a Density Matrix: A Basic Combinatorial Approach to Separability of Mixed States.Annals of Combinatorics 10, 291–317, 2006.
[20]Dragan Stevanovic, Applications of Graph Spectra in Quantum Mechanics, Mathematics Subject Classification, 2010.
[21]R. Raussendorf and H.J. Briegel, A one-way quantum computer. Physical Review Letters86, 5188, 2001.
[22]Ashok Muthukrishnan, Classical and Quantum Logic Gates: An Introduction to Quantum Computing, Quantum Information Seminar, September 1999.
[23]L. Pauling, J. Chem. Phys. 4, 673 (1936).
[24]H. Kuhn, Helv. Chim. Acta, 31, 1441 (1948).
[25]C. Flesia, R. Johnston, and H. Kunz, Europhys. Lett. 3, 497 (1987).
[26]R. Mitra and S. W. Lee, Analytical techniques in the Theory of Guided Waves (Macmillan, New York, 1971).
[27]Y. Imry, Introduction to Mesoscopic Systems (Oxford, New York, 1996).
[28]D. Kowal, U. Sivan, O. Entin-Wohlman, Y. Imry, Phys. Rev. B 42, 9009 (1990).
[29]E. L. Ivchenko, A. A. Kiselev, JETP Lett. 67, 43 (1998).
[30]J.A. Sanchez-Gil, V. Freilikher, I. Yurkevich, and A. A. Maradudin, Phys. Rev. Lett. 80 , 948(1998).
[31]Y. Avishai and J.M. Luck, Phys. Rev. B 45, 1074 (1992).
[32]T. Nakayama, K. Yakubo, and R. L. Orbach, Rev. Mod. Phys. 66, 381 (1994).
[33]T. Kottos and H. Schanz, Physica E 9, 523 (2003).
[34]T. Kottos and U. Smilansky, J. Phys. A 36, 3501 (2003).
[35]Oleh Hul, Oleg Tymoshchuk, Szymon Bauch, Peter M. Koch and Leszek Sirko, Experimental Investigation of Wigner’s Reaction Matrix for Irregular Graphs with Absorption, March 2009.
[36]Trystan Koch, Ziyuan FU, Electrical Networks and the Graph Model, 2016
[37]Darwin Gosal and Wayne Lawton, Quantum Haar Wavelet Transforms and Their Applications, November 2001.
[38]C.P. Williams, Explorations in Quantum Computing, Texts in Computer Science, Chapter2: Quantum Gates, DOI 10.1007/978-1-84628-887-6_2, Springer-Verlag London Limited, 2011.
[39]Amir Fijany and Colin P. Williams, Quantum Wavelet Transforms: Fast Algorithms and Complete Circuits, 1st NASA Int. Conf. on Quantum Computing and Communication, September 1998.
[40]P. Shor, Algorithms for Quantum Computation: Discrete Logarithms and Factoring, Proc. 35th Annual Symposium on Foundations of Computer Science, p. 124, 1994.
[41]L. K. Grover, A Fast Quantum Mechanical Algorithm for Database Search, Proc. 28th Annual ACM Symposium on the Theory of Computing, Philadelphia, p. 212, 1996.
[42]R. Jozsa, Quantum Algorithms and the Fourier Transform, Los Alamos preprint archive, http://xxx.lanl.gov/archive/quant-ph/9707033, 1997.
[43]C. Van Loan, Computational Frameworks for the Fast Fourier Transform, SIAM Publications, Philadelphia, 1992.
[44]John Ashmead, Morlet Wavelets in Quantum Mechanics, Quanta, Vol.1, Issue 1, pp:58-78, November 2012, doi: 10.12743/quanta.v1i1.5
[45]Georgiev L. S., Topologically Protected Quantum Gates for Computation with NonAbelian Anyons in the Pfaffian Quantum Hall State, Phys. Rev. B 74, 235112 [arXiv:condmat/0607125 [cond-mat.mes-hall],2006.
[46]Georgiev L. S., Towards a Universal Set of Topologically Protected Gates for Quantum Computation with Pfaffian qubits, Nucl. Phys. B 789, 552 [arXiv:hep-th/0611340], 2008
[47]Georgiev L. S., Ultimate Braid-Group Generators for Coordinate Exchanges of Ising Anyons from the Multi-Anyon Pfaffian wave functions, J. Phys. A: Math. Theor. 42, 225203 [arXiv:0812.2334 [math-ph], 2009.
[48]Ovidiu Racorean, Quantum Gates and Quantum Circuits of Stock Portfoliofile:///F:/STOCK%20QUANTUM.pdf
[49]Christopher L. Mueller, Techniques for Resonant Optical Interferometry with Applications to the Advanced Ligo Gravitational Wave Detectors, PhD Thesis, 2014
[50]H.Kogelnik and T.Li, Laser Beams and Resonators, 1150 Applied Physics, Vol.5, No 10, October 1966.
[51]Greyson Gilson, Unified Theory of Wave-Particle Duality, the Schröinger Equations and Quantum Diffraction, September 2014.
[52]Martin Aigner, On the Tetrahedral Graph, Pacific Journal of Mathematics, Vol.25, No 2, October 1968.
[53]H.S.M. Coxeter, Regular Polytopes, Dover, New York, 1973.
[54]Lois H. Kauffman and Samuel J Lomonaco Jr, Braiding Operators are Universal Quantum Gates, New Journal of Physics 6 (2004) 134, October 2004. doi:10.1088/1367-2630/6/1/134
[55]Florin Nichita, Introduction to the Yang-Baxter Equation with Open Ploblems, Axioms 2012, 1, 33-37; doi:10.3390/axioms1010033, April 2012.
[56]Michio Jimbo, Introduction to Yang-bater Equation, Internation Journal of Modern Physics A, Vol.4, No.15 (1989), pp.3759-3777, March 1989.
[57]Dye H, Quantum Inform Process. 2 117 (Preprint quant-ph/0211050v3), August 2003.
[58]Magnus Karlsson, The Connection between Polarization Calculus and Four-Dimensional Rotations, March 2013.
[59]Jean Krommweh, Tight Frame Characterization of Multiwalet Vector Function in Trems of the Polyphase Matrix, January 2008. file:///F:/QUANTUM%20QUINCUNX%203.pdf
[60]Wen-Ran Zhang, Bipolar Quantum Logic Gates and Quantum Cellular Combinatorics – A Logical Extension to Quantum Entanglement, June 2013. file:///F:/Drive/Bipolar%20Quantum%20Logic%20Gates%20and%20Quantum%20Cellular%20Combi natorics%20%20.pdf
[61]Strang, G., Linear Algebra and Its Applications,4th Edn (Thomson/Brooks/Cole), 2006.
[62]Theodor Banica, Ion Netchita and Karol Zyczkowski, Almost Hadamard Matrices: General Theory and Examples, 2000.
[63]Theodor Banica, Ion Netchita and Karol Zyczkowski, Almost Hadamard Matrices: General Theory and Examples, 2000.
[64]C.P. Williams, Explorations in Quantum Computing, Texts in Computer Science, Chapter2: Quantum Gates, DOI 10.1007/978-1-84628-887-6_2, Springer-Verlag London Limited, 2011.
[65]Stephen S. Bullock and Igor L. Markov, An Arbitrary Two-qubit Computation In 23 Elementary Gates, Feb. 2008.
[66]Ashok Muthukrishnan, Classical and Quantum Logic Gates: An Introduction to Quantum Computing, Quantum Information Seminar, September 1999.
[67]Ovidiu Racorean, Quantum Gates and Quantum Circuits of Stock Portfoliofile:///F:/STOCK%20QUANTUM.pdf
[68]Chapter 2: Matrix Description of Wave Propagation and Polarization,file:///F:/QUANTUM%20OPTIC%20BEAM%20SPLITTER.pdf
[69]L.D. Baumert and Marshall Hall, Jr., Hadamard Matrices of the Williamson Type, http://mathscinet.ru/files/Williamson_1965.pdf
[70]http://www4.ncsu.edu/~franzen/public_html/CH736/lecture/Particle_in_a_box.pdf
[71]Y. Ordokhani, An Application of Walsh Functions for of Fredholm-Hammerstein Integro-Differential Equation, Int.J. Contemp.Math Sciences, Vol.5, No22, pp.1055-1063, 2010.
[72]Ashish M. Kothari and Ved Vyas Dwivedi, Video Watermaking- Combination of Discrete Wavelet and Cosine Transform to Achieve Extra Robustness, IJIGSP, Vol.5, No. 3, pp. 36-41, March 2013, DOI: 10.5815/ijigsp.2013.03.05
[73]J.M De Freitas and M.A Player, Polarization Effect in Heterodyne Interferometry, Journal of Modern Optics, Vol.42, No. 9, pp. 1875-1899, 1995.
[74]R. J. Duffin and A. C. Schaeffer, “A class of nonharmonic Fourier series,” Trans. Amer. Math. Soc., vol. 72, pp. 314–366, 1952.
[75]R. M. Young, An introduction to nonharmonic Fourier series, New York: Academic Press, 1980.
[76]A. Aldroubi and K. Gro¨chenig, “Non-uniform sampling and reconstruction in shift-invariant spaces,” Siam Rev., to appear.
[77]Y. C. Eldar and A. V. Oppenheim, “Orthogonal and projected orthogonal matched-filter detection,” submitted to IEEE Trans. Signal Processing, Jan. 2001.
[78]Y. C. Eldar and A. M. Chan, “Orthogonal and projected orthogonal multiuser detection,” submitted to IEEE Trans. Inform. Theory, May 2001.
[79]V. K. Goyal, J. Kovaˇcevi´c, and M. Vetterli, “Multiple description transform coding: Robustness to erasures using tight frame expansions,” in Proc. IEEE Int. Symp. Inform. Theory, Cambridge, MA, Aug. 1998, p. 408.
[80]H. B¨olcskei, Oversampled filter banks and predictive subband coders, Ph.D. thesis, Vienna University of Technology, Nov. 1997.
[81]H. B¨olcskei, F. Hlawatsch, and H. G. Feichtinger, “Frame-theoretic analysis of oversampled filter banks,” IEEE Trans. on Signal Processing, vol. 46, no. 12, pp. 3256–3268, Dec. 1998.
[82]P. J. S. G. Ferreira, “Mathematics for multimedia signal processing II — discrete finite frames and signal reconstruction,” in Signal processing for multimedia, J. S. Byrnes, Ed., pp. 35–54. IOC Press, 1999.
[83]Yonina C. Eldar and G. David Forney, Jr., Optimal Tight Frames and Quantum Measurement, March 2018.
[84]Reinhold A. Bertlmann, and Philipp Krammer, Bloch Vectors for Qudits, June 2008.