International Journal of Theoretical Physics

, Volume 57, Issue 9, pp 2575–2584 | Cite as

Quantum Adder for Superposition States

  • Xiaowei Lu
  • Nan JiangEmail author
  • Hao Hu
  • Zhuoxiao Ji


Quantum superposition is one of the essential features that make quantum computation surpass classical computation in space complexity and time complexity. However, it is a double-edged sword. For example, it is troublesome to add all the numbers stored in a superposition state. The usual solution is taking out and adding the numbers one by one. If there are \(2^{n}\) numbers, the complexity of this scheme is \(O(2^{n})\) which is the same as the complexity of the classical scheme \(O(2^{n})\). Moreover, taking account to the current physical computing speed, quantum computers will have no advantage. In order to solve this problem, a new method for summing all numbers in a quantum superposition state is proposed in this paper, whose main idea is that circularly shifting the superposition state and summing the new one with the original superposition state. Our scheme can effectively reduce the time complexity to \(O(n)\).


Quantum adder Superposition state Quantum computation Quantum image processing 



This work is supported by the National Natural Science Foundation of China under Grants No. 61502016, and the Joint Open Fund of Information Engineering Team in Intelligent Logistics under Grants No. LDXX2017KF152.


  1. 1.
    Feynman, R.P.: Simulating physics with computers. Int. J. Theor. Phys. 21(6/7), 467–488 (2005)MathSciNetGoogle Scholar
  2. 2.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  3. 3.
    Shor, P.W.: Algorithms for Quantum Computation: Discrete Logarithms and Factoring. Proceedings of the 35th Annual Symposium on Foundations of Computer Science, Santa Fe, pp. 124–134 (1994)Google Scholar
  4. 4.
    Grover, L.K.: A fast quantum mechanical algorithm for database search. Proceedings of the 28Th Annual ACM Symposium on the Theory of Computing, pp. 212–219 (2011)Google Scholar
  5. 5.
    Vedral, V., Barenco, A., Ekert, A.L.: Quantum networks for elementary arithmetic operations. Phys. Rev. A 54(1), 147–153 (1996)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Amlan, C., Susmita, S.K.: Designing quantum adder circuits and evaluating their error performance. Int. Conf. Electron. Des. 4, 1–6 (2008)Google Scholar
  7. 7.
    Alvarez-Rodriguez, U., Sanz, M., Lamata, L., et al.: The forbidden quantum adder. Sci. Rep. 5, 11983 (2015)ADSCrossRefGoogle Scholar
  8. 8.
    Barbosa, G.: Quantum half-adder. Phys. Rev. A 73(5), 485–485 (2006)CrossRefGoogle Scholar
  9. 9.
    Montaser, R., Younes, A., Abdelaty, M.: New Design of Reversible Full Adder/Subtractor using R gate. arXiv:1708.00306
  10. 10.
    Shekoofeh, M., Mohammad, R.: A Novel \(4\times 4\) Universal Reversible Gate as a Cost Efficient Full Adder/Subtractor in Terms of Reversible and Quantum Metrics. Mod. Educ. Comput. Sci. 11, 28–34 (2015)Google Scholar
  11. 11.
    Chowdhury, A.K., Tan, D.Y.W., Yew, S.L.B.: Design of full adder/subtractor using irreversible IG-a gate. Design of full adder/subtractor using irreversible IG-a gate (2015)Google Scholar
  12. 12.
    Jiang, N., Wang, L.: Quantum image scaling using nearest neighbor interpolation. Quantum Inf. Process. 5, 1559–1571 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Jiang, N., Wang, J., Mu, Y.: Quantum image scaling up based on nearest-neighbor interpolation with integer scaling ratio. Quantum Inf. Process. 11, 4001–4026 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Nan, J., Yijie, D., Jian, W.: Quantum image matching. Quantum Inf. Process. 9, 3543–3572 (2016)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Wang, J.: QRDA Quantum representation of digital audio. Int. J. Theor. Phys. 3, 1622–1641 (2015)zbMATHGoogle Scholar
  16. 16.
    Wang, J., Wang, H., Song, Y.: Quantum endpoint detection based on QRDA. J. Theor. Phys. 10, 3257–3270 (2017)CrossRefzbMATHGoogle Scholar
  17. 17.
    Jiang, N., Wang, L.: Analysis and improvement of the quantum Arnold image scrambling. Quantum Inf. Process. 7, 1545–1551 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jiang, N., Wu, W., Wang, L.: The quantum realization of Arnold and Fibonacci image scrambling. Quantum Inf. Process. 5, 1223–1236 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Yan, F., Iliyasu, A.M., Venegas-Andraca, S.E.: A survey of quantum image representations. Quantum Inf. Process. 1, 1–35 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Song, X.H., Wang, S., Liu, S., El-Latif, A.A.A., Niu, X.M.: A dynamic watermarking scheme for quantum images using quantum wavelet transform. Quantum Inf. Process. 12, 3689–3706 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hillery, M, Buzek, V, Ziman, M.: Probabilistic implementation of universal quantum processors. Phys. Rev. A 2, 022301 (2012)MathSciNetGoogle Scholar
  22. 22.
    Jones, N.C., Whitfield, J.D., Mcmahon, P.L., Yung, M.H., Meter, R.V.: Faster quantum chemistry simulation on fault-tolerant quantum computers. J. Phys. 11, 115023 (2012)Google Scholar
  23. 23.
    Zhou, R.-G., Hu, W., Fan, P., Ian, H.: Quantum realization of the bilinear interpolation method for NEQR. Sci. Rep. 1, 2511 (2017)ADSCrossRefGoogle Scholar
  24. 24.
    Zhou, R.-G., Tan, C., Ian, H.: Global and Local Translation Designs of Quantum Image Based on FRQI. Int. J. Theor. Phys. 4, 1382–1398 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Wang, D, Liu, ZH, Zhu, W.M., et al.: Design of quantum comparator based on extended general toffoli gates with multiple targets. Comput. Sci. 39(9), 302–306 (2012)Google Scholar

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Authors and Affiliations

  1. 1.Faculty of Information TechnologyBeijing University of TechnologyBeijingChina
  2. 2.School of Information Science and TechnologyLinyi UniversityLinyiChina
  3. 3.Beijing Key Laboratory of Trusted ComputingBeijingChina
  4. 4.National Engineering Laboratory for Critical Technologies of Information Security Classified ProtectionBeijingChina

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