A Flexible Representation and Invertible Transformations for Images on Quantum Computers

  • Phuc Q. Le
  • Abdullahi M. Iliyasu
  • Fangyan Dong
  • Kaoru Hirota
Part of the Studies in Computational Intelligence book series (SCI, volume 372)


A flexible representation for quantum images (FRQI) is proposed to provide a representation for images on quantum computers which captures information about colors and their corresponding positions in the images. A constructive polynomial preparation for the FRQI state from an initial state, an algorithm for quantum image compression (QIC), and invertible processing operations for quantum images are combined to build the whole process for quantum image processing based on FRQI. The simulation experiments on FRQI include storage and retrieval of images and detecting a line from binary images by applying quantum Fourier transform as a processing operation. The compression ratios of QIC between groups of same color positions range from 68.75% to 90.63% on single digit images and 6.67% to 31.62% on the Lena image. The FRQI provides a foundation not only to express images but also to explore theoretical and practical aspects of image processing on quantum computers.


Compression Ratio Binary String Quantum Circuit Quantum Image Boolean Expression 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Barenco, A., Bennett, C.H., Cleve, R., DiVincenzo, D.P., Margolus, N., Shor, P., Sleator, T., Smolin, J.A., Weinfurter, H.: Elementary gates for quantum computation. Phys. Rev. A 52, 3457 (1995)CrossRefGoogle Scholar
  2. 2.
    Beach, G., Lomont, C., Cohen, C.: Quantum image processing (quip). In: Proc. of Applied Imagery Pattern Recognition Workshop, pp. 39–44 (2003)Google Scholar
  3. 3.
    Brayton, R.K., Sangiovanni-Vincentelli, A., McMullen, C., Hachtel, G.: Logic minimization algorithms for VLSI synthesis. Kluwer Academic Publishers, Dordrecht (1984)CrossRefzbMATHGoogle Scholar
  4. 4.
    Caraiman, S., Manta, V.I.: New applications of quantum algorithms to computer graphics: the quantum random sample consensus algorithm. In: Proc. of the 6th ACM Conference on Computing Frontier, pp. 81–88 (2009)Google Scholar
  5. 5.
    Curtis, D., Meyer, D.A.: Towards quantum template matching. In: Proc. of the SPIE, vol. 5161, pp. 134–141 (2004)Google Scholar
  6. 6.
    Feynman, R.P.: Simulating physics with computers. International Journal of Theoretical Physics 21(6/7), 467–488 (1982)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Fijany, A., Williams, C. P.: Quantum wavelet transform: fast algorithm and complete circuits. arXiv:quant-ph/9809004 (1998)Google Scholar
  8. 8.
    Grover, L.: A fast quantum mechanical algorithm for database search. In: Proc. of the 28th Ann. ACM Symp. on the Theory of Computing (STOC 1996), pp. 212–219 (1996)Google Scholar
  9. 9.
    Klappenecker, A., Rötteler, M.: Discrete cosine transforms on quantum computers. In: Proc. of the 2nd Inter. Symp. on Image and Signal Processing and Analysis, pp. 464–468 (2001)Google Scholar
  10. 10.
    Latorre, J. I.: Image compression and entanglement. arXiv:quant-ph/0510031 (2005)Google Scholar
  11. 11.
    Lomont, C.: Quantum convolution and quantum correlation algorithms are physically impossible. arXiv:quant-ph/0309070 (2003)Google Scholar
  12. 12.
    Lomont, C.: Quantum circuit identities. arXiv:quant-ph/0307111 (2003)Google Scholar
  13. 13.
    Maslov, D., Dueck, G.W., Miller, D.M., Camille, N.: Quantum circuit simplification and level compaction. IEEE Trans. on Computer-Aided Design of Integrated Circuits and Systems 27(3), 436–444 (2008)CrossRefGoogle Scholar
  14. 14.
    Nielsen, M., Chuang, I.: Quantum computation and quantum information. Cambridge University Press, New York (2000)zbMATHGoogle Scholar
  15. 15.
    Shor, P.W.: Algorithms for quantum computation: discrete logarithms and factoring. In: Proc. 35th Ann. Symp. Foundations of Computer Science, pp. 124–134. IEEE Computer Soc. Press, Los Almitos (1994)CrossRefGoogle Scholar
  16. 16.
    Tseng, C.C., Hwang, T.M.: Quantum circuit design of 8×8 discrete cosine transforms using its fast computation flow graph. In: ISCAS 2005, vol. I, pp. 828–831 (2005)Google Scholar
  17. 17.
    Venegas-Andraca, S. E., Ball, J. L.: Storing Images in engtangled quantum systems. arXiv:quant-ph/0402085 (2003)Google Scholar
  18. 18.
    Venegas-Andraca, S.E., Bose, S.: Storing, processing and retrieving an image using quantum mechanics. In: Proc. of the SPIE Conf. Quantum Information and Computation, pp. 137–147 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Phuc Q. Le
    • 1
  • Abdullahi M. Iliyasu
    • 1
  • Fangyan Dong
    • 1
  • Kaoru Hirota
    • 1
  1. 1.Department of Computational Intelligence and Systems Science, Interdisciplinary Graduate School of Science and EngineeringTokyo Institute of TechnologyMidori-kuJapan

Personalised recommendations