Abstract
In this paper, a new concept of the resolution map is presented to extract periodic structures that compose the signal. The resolution map is described by using the frequency–time representation of the signal, which is known as the paired transform that provides the frequency-time representation of signals. The sequential calculation of resolution maps over the signal components of large sizes allows for calculating the small periodic structures, or patterns, which can be used for signal processing, for instance filtration, and from which the signal can be reconstructed. The length of the signal is considered to be a power of two, a case that fits well with qubit processing in quantum computing. The following new results are described: (1) the quantum scheme for the 1D discrete paired transform, (2) the quantum circuit for calculating the signal resolution map, (3) the quantum circuit for signal reconstruction from the resolution map, (4) different schemes for resolution maps for processing signals, and (5) the convolution of signals by their periodic patterns.
Similar content being viewed by others
References
Burt, P.J., Adelson, E.H.: The Laplacian pyramid as a compact image code. IEEE Trans. Commun. 31(4), 532–540 (1983)
Mallat, S.: Multiresolution approximation and wavelet orthogonal bases of L2(R). Trans. Am. Math. Soc. 315(1), 69–87 (1989)
Myer, Y.: Wavelets and Operations. Advanced Mathematics. Cambridge Univ. Press, Cambridge (1992)
Gabor, D.: Theory of communication. J. IEE 93, 429–457 (1946)
Mallat, S.G.: A theory for multiresolution signal decomposition—The wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 11(7), 674–693 (1989)
Grigoryan, A.M., Grigoryan, M.M.: Brief Notes in Advanced DSP: Fourier analysis with MATLAB. CRC Press, Boca Raton (2009)
Grigoryan, A.M., Du, N.: 2-D images in frequency-time representation: direction images and resolution map. J. Electron. Imaging 19(3), 033012 (2010)
Yan, F., Iliyasu, A.M., Venegas-Andraca, S.E.: A survey of quantum image representations. Quant. Inf. Process. 15(1), 1–35 (2015)
Zhang, Y., Lu, K., Gao, Y., Wang, M.: NEQR: a novel enhanced quantum representation of digital images. Quant. Inf. Process. 12(8), 2833–2860 (2013)
Grigoryan, A.M.: New algorithms for calculating discrete Fourier transforms. USSR Comput. Math. Math. Phys. 26(5), 84–88 (1986)
Grigoryan, A.M.: An algorithm of computation of the one-dimensional discrete Fourier transform. Izvestiya VUZov SSSR, Radioelectronica 31(5), 47–52 (1988)
Grigoryan, A.M., Agaian, S.S.: Paired quantum Fourier transform with log2N Hadamard gates. Quant. Inf. Process. 18, 217 (2019)
Li, H.S., Fan, P., Xia, H., Song, S., He, X.: The quantum Fourier transform based on quantum vision representation. Quant. Inf. Process. 17, 333 (2018)
Perez, L.R., Garcia-Escartin, J.C.: Quantum arithmetic with the quantum Fourier transform. Quant. Inf. Process. 16, 14 (2017)
Karafyllidis, I.G.: Visualization of the quantum Fourier transform using a quantum computer simulator. Quant. Inf. Process. 2(4), 271–288 (2003)
Grigoryan, A.M.: 2-D and 1-D multi-paired transforms: frequency-time type wavelets. IEEE Trans. Signal Process. 49(2), 344–353 (2001)
Grigoryan, A.M.: Fourier transform representation by frequency-time wavelets. IEEE Trans. Signal Process. 53(7), 2489–2497 (2005)
Grigoryan, A.M.: Representation of the Fourier transform by Fourier series. J. Math. Imaging Vis. 25(1), 87–105 (2006)
Grigoryan, A.M., Agaian, S.S.: Split manageable efficient algorithm for Fourier and Hadamard transforms. IEEE Trans. Signal Process. 48(1), 172–183 (2000)
Grigoryan, A.M., Agaian, S.S.: Multidimensional discrete unitary transforms: representation, partitioning, and algorithms. Marcel Dekker, New York (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Grigoryan, A.M. Resolution map in quantum computing: signal representation by periodic patterns. Quantum Inf Process 19, 177 (2020). https://doi.org/10.1007/s11128-020-02685-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11128-020-02685-7