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Efficient Constructions for Simulating Multi Controlled Quantum Gates

Part of the Lecture Notes in Computer Science book series (LNCS,volume 13353)


Multi Controlled Gates, with Multi Controlled Toffoli as primary example are a building block for a lot of complex quantum algorithms in the domains of discrete arithmetic, cryptography, machine learning, and image processing. However, these gates cannot be physically implemented in quantum hardware and therefore they need to be decomposed into many smaller elementary gates. In this work we analyse previously proposed circuit constructions for MCT gates and describe 6 new methods for generating MCT circuits with efficient costs, less restrictions, and improved applicability.


  • Multi-controlled Toffoli gate
  • Quantum circuit
  • Ancilla qubit
  • Efficient quantum algorithms

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  1. Baker, J.M., Duckering, C., Hoover, A., Chong, F.T.: Decomposing quantum generalized Toffoli with an arbitrary number of ancilla (2019)

    Google Scholar 

  2. Barenco, A., et al.: Elementary gates for quantum computation. Phys. Rev. A 52(5), 3457–3467 (1995).

  3. Cross, A.W., Bishop, L.S., Sheldon, S., Nation, P.D., Gambetta, J.M.: Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019).

    CrossRef  Google Scholar 

  4. Dirac, P.A.M.: The Principles of Quantum Mechanics. International Series of Monographs on Physics. Clarendon Press (1981)

    Google Scholar 

  5. DiVincenzo, D.P.: The physical implementation of quantum computation. Fortsch. Phys. 48(9–11), 771–783 (2000)

    Google Scholar 

  6. Gidney, C.: Using quantum gates instead of ancilla bits (2015).

  7. Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing (STOC 1996), pp. 212–219. Association for Computing Machinery, New York (1996).

  8. He, Y., Luo, M.-X., Zhang, E., Wang, H.-K., Wang, X.-F.: Decompositions of n-qubit Toffoli gates with linear circuit complexity. Int. J. Theoret. Phys. 56(7), 2350–2361 (2017).

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. Héctor Abraham, E.A.: Qiskit: an open-source framework for quantum computing (2019).

  10. Kay, A.: Quantikz (2018).

  11. Krovi, H., Magniez, F., Ozols, M., Roland, J.: Quantum walks can find a marked element on any graph. Algorithmica 74(2), 851–907 (2015).

  12. Maslov, D.: Advantages of using relative-phase Toffoli gates with an application to multiple control Toffoli optimization. Phys. Rev. A 93(2) (2016).

  13. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press (2010).

  14. Saeedi, M., Pedram, M.: Linear-depth quantum circuits for \(n\)-qubit Toffoli gates with no ancilla. Phys. Rev. A 87, 062318 (2013).

  15. Shor, P.: Algorithms for quantum computation: discrete logarithms and factoring. In: Proceedings 35th Annual Symposium on Foundations of Computer Science, pp. 124–134 (1994).

  16. Toffoli, T.: Reversible computing. In: de Bakker, J., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 632–644. Springer, Heidelberg (1980).

    CrossRef  Google Scholar 

  17. Vandersypen, L.M.K., Steffen, M., Breyta, G., Yannoni, C.S., Sherwood, M.H., Chuang, I.L.: Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance. Nature 414(6866), 883–887 (2001).

  18. Yanofsky, N.S., Mannucci, M.A.: Quantum Computing for Computer Scientists. Cambridge University Press, Cambridge (2008).

  19. Yao, X.W., Wang, H., et al.: Quantum image processing and its application to edge detection: theory and experiment. Phys. Rev. X 7(3) (2017).

  20. Zu, H., Dai, W., de Waele, A.: Development of dilution refrigerators-a review. Cryogenics 121, 103390 (2022).

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Correspondence to Stefan Balauca .

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Balauca, S., Arusoaie, A. (2022). Efficient Constructions for Simulating Multi Controlled Quantum Gates. In: Groen, D., de Mulatier, C., Paszynski, M., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P.M.A. (eds) Computational Science – ICCS 2022. ICCS 2022. Lecture Notes in Computer Science, vol 13353. Springer, Cham.

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  • Print ISBN: 978-3-031-08759-2

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