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Paired quantum Fourier transform with log2N Hadamard gates

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Abstract

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.

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References

  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)

  2. 2.

    Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information, 2nd edn. Cambridge University Press, Cambridge (2001)

  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)

  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)

  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)

  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)

  7. 7.

    Sang, J., Wang, S., Li, Q.: A novel quantum representation of color digital images. Quantum Inf. Process. 16(2), 1–14 (2017)

  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)

  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)

  10. 10.

    Coppersmith D.: An approximate Fourier transform useful in quantum factoring. Technical, Report RC19642, IBM (1994)

  11. 11.

    Chan, I.C., Ho, K.L.: Split vector-radix fast Fourier transform. IEEE Trans. Signal Process. 40(8), 2029–2040 (1992)

  12. 12.

    Cheung D.: Using generalized quantum Fourier transforms in quantum phase estimation algorithms, Thesis. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.572.9698&rep=rep1&type=pdf

  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)

  14. 14.

    Barenco, A., Ekert, A., Suominen, K.-A., Törmä, P.: Approximate quantum Fourier transform and decoherence. Phys. Rev. A 54, 139–146 (1996)

  15. 15.

    Yoran, N.N., Short, A.: Efficient classical simulation of the approximate quantum Fourier transform. Phys. Rev. A 76, 042321 (2007)

  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)

  17. 17.

    Karafyllidis, I.G.: Visualization of the quantum Fourier transform using a quantum computer simulator. Quantum Inf. Process. 2(4), 271–288 (2003)

  18. 18.

    Muthukrishnan, A., Stroud Jr., C.: Quantum fast fourier transform using multilevel atoms. J. Mod. Optics 49, 2115–2127 (2002)

  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)

  20. 20.

    Zilic, Z., Radecka, K.: Scaling and better approximating quantum fourier transform by higher radices. IEEE Trans. Comput. 56(2), 202–207 (2007)

  21. 21.

    Grigoryan, A.M.: New algorithms for calculating discrete Fourier transforms. USSR Comput. Math. Math. Phys. 26(5), 84–88 (1986)

  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)

  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)

  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)

  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)

  26. 26.

    Grigoryan, A.M., Agaian, S.S.: Practical Quaternion and Octonion Imaging with MATLAB. SPIE Press, Bellingham (2009)

  27. 27.

    Browne, D.E.: Efficient classical simulation of the semi-classical quantum Fourier transform. New J. Phys. 9, 146 (2007)

  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)

  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)

  30. 30.

    Perez, L.R., Garcia-Escartin, J.C.: Quantum arithmetic with the quantum Fourier transform. Quantum Inf. Process. 16, 14 (2017)

  31. 31.

    Maynard, C.E., Pius, E.: A quantum multiply-accumulator. Quantum Inf. Process. 13(5), 1127–1138 (2014)

  32. 32.

    Grigoryan, A.M.: Two classes of elliptic discrete Fourier transforms: properties and examples. J. Math. Imaging Vis. 0235(39), 210–229 (2011)

  33. 33.

    Grigoryan, A.M., Agaian, S.S.: Tensor transform-based quaternion Fourier transform algorithm. Inf. Sci. 320, 62–74 (2015). https://doi.org/10.1016/j.ins.2015.05.018

  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)

  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)

  36. 36.

    Grigoryan, A.M., Agaian, S.S.: Multidimensional Discrete Unitary Transforms: Representation, Partitioning, and Algorithms. Marcel Dekker, New York (2003)

  37. 37.

    Agaian, S.S.: Hadamard Matrices and Their Applications, Lecture Notes in Mathematics, vol. 1168. Springer, New York (1985)

  38. 38.

    Agaian, S.S., Sarukhanyan, H.G., Egiazarian, K.O., Astola, J.: Hadamard Transforms. SPIE Press, Bellingham (2011)

  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)

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Correspondence to Artyom M. Grigoryan.

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Grigoryan, A.M., Agaian, S.S. Paired quantum Fourier transform with log2N Hadamard gates. Quantum Inf Process 18, 217 (2019) doi:10.1007/s11128-019-2322-6

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Keywords

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