Paired quantum Fourier transform with log2N Hadamard gates

  • Artyom M. GrigoryanEmail author
  • Sos S. Agaian


The quantum Fourier transform (QFT) is perhaps the furthermost central building block in creation quantum algorithms. In this work, we present a new approach to compute the standard quantum Fourier transform of the length \( N = 2^{r} , \;r > 1 \), which also is called the r-qubit discrete Fourier transform. The presented algorithm is based on the paired transform developed by authors. It is shown that the signal-flow graphs of the paired algorithms could be used for calculating the quantum Fourier and Hadamard transform with the minimum number of stages. The calculation of all components of the transforms is performed by the Hadamard gates and matrices of rotations and all simple NOT gates. The new presentation allows for implementing the QFT (a) by using only the r Hadamard gates and (b) organizing parallel computation in r stages. Also, the circuits for the length-2r fast Hadamard transform are described. Several mathematical illustrative examples of the order the \( N = 4,\;8 \), and 16 cases are illustrated. Finally, the QFT for inputs being two, three and four qubits is described in detail.


Quantum Fourier transform Quantum computing Quantum Hadamard transform Paired transform Fast Fourier transform 



  1. 1.
    Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484–1509 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information, 2nd edn. Cambridge University Press, Cambridge (2001)zbMATHGoogle Scholar
  3. 3.
    Young R.C.D., Birch P.M., Chatwin C.R.: A simplification of the Shor quantum factorization algorithm employing a quantum Hadamard transform. In: Proceedings of SPIE 10649, Pattern Recognition and Tracking XXIX, 1064903, p. 11. Orlando, Florida, USA (2018)Google Scholar
  4. 4.
    Gong, L.H., He, X.T., Tan, R.C., Zhou, Z.H.: Single channel quantum color image encryption algorithm based on HSI model and quantum Fourier transform”. Int. J. Theor. Phys. 57, 59–73 (2018)zbMATHCrossRefGoogle Scholar
  5. 5.
    Yan, F., Iliyasu, A.M., Le, P.Q.: Quantum image processing: a review of advances in its security technologies. Int. J. Quantum Inf. 15(3), 1730001 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Yan, F., Iliyasu, A.M., Venegas-Andraca, S.E.: A survey of quantum image representations. Quantum Inf. Process. 15(1), 1–35 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Sang, J., Wang, S., Li, Q.: A novel quantum representation of color digital images. Quantum Inf. Process. 16(2), 1–14 (2017)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Zhang, W.W., Gao, F., Liu, B., et al.: A watermark strategy for quantum images based on quantum Fourier transform. Quantum Inf. Process. 12(2), 793–803 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Yang, Y.G., Jia, X., Xu, P., et al.: Analysis and improvement of the watermark strategy for quantum images based on quantum Fourier transform. Quantum Inf. Process. 12(8), 2765–2769 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Coppersmith D.: An approximate Fourier transform useful in quantum factoring. Technical, Report RC19642, IBM (1994)Google Scholar
  11. 11.
    Chan, I.C., Ho, K.L.: Split vector-radix fast Fourier transform. IEEE Trans. Signal Process. 40(8), 2029–2040 (1992)ADSzbMATHCrossRefGoogle Scholar
  12. 12.
    Cheung D.: Using generalized quantum Fourier transforms in quantum phase estimation algorithms, Thesis.
  13. 13.
    Marquezinoa, F.L., Portugala, R., Sasse, F.D.: Obtaining the quantum Fourier transform from the classical FFT with QR decomposition. J. Comput. Appl. Math. 235(1), 74–81 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Barenco, A., Ekert, A., Suominen, K.-A., Törmä, P.: Approximate quantum Fourier transform and decoherence. Phys. Rev. A 54, 139–146 (1996)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Yoran, N.N., Short, A.: Efficient classical simulation of the approximate quantum Fourier transform. Phys. Rev. A 76, 042321 (2007)ADSCrossRefGoogle Scholar
  16. 16.
    Cleve R., Watrous J.: Fast parallel circuits for the quantum Fourier transform. In: Proceedings of IEEE 41st Annual Symposium on Foundations of Computer Science, pp. 526–536, Redondo Beach, CA, USA (2000)Google Scholar
  17. 17.
    Karafyllidis, I.G.: Visualization of the quantum Fourier transform using a quantum computer simulator. Quantum Inf. Process. 2(4), 271–288 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Muthukrishnan, A., Stroud Jr., C.: Quantum fast fourier transform using multilevel atoms. J. Mod. Optics 49, 2115–2127 (2002)ADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Heo, J., Kang, M.S., Hong, C.H., et al.: Discrete quantum Fourier transform using weak cross-Kerr nonlinearity and displacement operator and photon-number-resolving measurement under the decoherence effect. Quantum Inf. Process. 15(12), 4955–4971 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Zilic, Z., Radecka, K.: Scaling and better approximating quantum fourier transform by higher radices. IEEE Trans. Comput. 56(2), 202–207 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Grigoryan, A.M.: New algorithms for calculating discrete Fourier transforms. USSR Comput. Math. Math. Phys. 26(5), 84–88 (1986)zbMATHCrossRefGoogle Scholar
  22. 22.
    Grigoryan, A.M.: An algorithm of computation of the one-dimensional discrete Fourier transform. Izvestiya VUZov SSSR, Radioelectronica 31(5), 47–52 (1988)Google Scholar
  23. 23.
    Grigoryan, A.M.: 2-D and 1-D multi-paired transforms: frequency-time type wavelets. IEEE Trans. Signal Process. 49(2), 344–353 (2001)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Grigoryan, A.M., Grigoryan, M.M.: Brief Notes in Advanced DSP: Fourier Analysis with MATLAB. CRC Press Taylor and Francis Group, Boca Raton (2009)zbMATHGoogle Scholar
  25. 25.
    Grigoryan, A.M., Agaian, S.S.: Split manageable efficient algorithm for Fourier and Hadamard transforms. IEEE Trans. Signal Process. 48(1), 172–183 (2000)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Grigoryan, A.M., Agaian, S.S.: Practical Quaternion and Octonion Imaging with MATLAB. SPIE Press, Bellingham (2009)Google Scholar
  27. 27.
    Browne, D.E.: Efficient classical simulation of the semi-classical quantum Fourier transform. New J. Phys. 9, 146 (2007)ADSCrossRefGoogle Scholar
  28. 28.
    Li, H.S., Fan, P., Xia, H., Song, S., He, X.: The quantum Fourier transform based on quantum vision representation. Quantum Inf. Process. 17, 333 (2018)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Agaian S.S., Klappenecker A.: Quantum computing and a unified approach to fast unitary transforms. In: Proceedings of SPIE 4667, Image Processing: Algorithms and Systems, p. 11 (2002)Google Scholar
  30. 30.
    Perez, L.R., Garcia-Escartin, J.C.: Quantum arithmetic with the quantum Fourier transform. Quantum Inf. Process. 16, 14 (2017)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Maynard, C.E., Pius, E.: A quantum multiply-accumulator. Quantum Inf. Process. 13(5), 1127–1138 (2014)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Grigoryan, A.M.: Two classes of elliptic discrete Fourier transforms: properties and examples. J. Math. Imaging Vis. 0235(39), 210–229 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Grigoryan, A.M., Agaian, S.S.: Tensor transform-based quaternion Fourier transform algorithm. Inf. Sci. 320, 62–74 (2015). MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Grigoryan A.M., S.S. Agaian S.S.: 2-D Octonion discrete Fourier transform: fast algorithms. In: Proceedings of IS&T International Symposium, Electronic Imaging: Algorithms and Systems XV, Burlingame, CA (2017)Google Scholar
  35. 35.
    Grigoryan A.M., Agaian S.S.: 2-D left-side quaternion discrete Fourier transform fast algorithms. In: Proceedings of IS&T International Symposium, 2016 Electronic Imaging: Algorithms and Systems XIV, San Francisco, California (2016)Google Scholar
  36. 36.
    Grigoryan, A.M., Agaian, S.S.: Multidimensional Discrete Unitary Transforms: Representation, Partitioning, and Algorithms. Marcel Dekker, New York (2003)zbMATHCrossRefGoogle Scholar
  37. 37.
    Agaian, S.S.: Hadamard Matrices and Their Applications, Lecture Notes in Mathematics, vol. 1168. Springer, New York (1985)CrossRefGoogle Scholar
  38. 38.
    Agaian, S.S., Sarukhanyan, H.G., Egiazarian, K.O., Astola, J.: Hadamard Transforms. SPIE Press, Bellingham (2011)zbMATHCrossRefGoogle Scholar
  39. 39.
    Grigoryan, A.M.: An algorithm of computation of the one-dimensional discrete Hadamard transform. Izvestiya VUZov SSSR Radioelectron. USSR Kiev 34(8), 100–103 (1991)Google Scholar

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Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringUniversity of Texas at San AntonioSan AntonioUSA
  2. 2.Computer Science DepartmentThe College of Staten Island New YorkNew York CityUSA

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