Group Theory Based Synthesis of Binary Reversible Circuits

  • Guowu Yang
  • Xiaoyu Song
  • William N. N. Hung
  • Fei Xie
  • Marek A. Perkowski
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3959)


This paper presents an important result addressing a fundamental question in synthesizing binary reversible logic circuits for quantum computation. We show that any even-reversible-circuit of n (n>3) qubits can be realized using NOT gate and Toffoli gate (‘2’-Controlled-Not gate), where the numbers of Toffoli and NOT gates required in the realization are bounded by \((n + \lfloor \frac{n}{3} \rfloor)(3 \times 2^{2n-3}-2^{n+2})\) and \(4n(n + \lfloor \frac{n}{3} \rfloor)2^n\), respectively. A provable constructive synthesis algorithm is derived. The time complexity of the algorithm is \(\frac{10}{3}n^2 \cdot 2^n\). Our algorithm is exponentially lower than breadth-first search based synthesis algorithms with respect to space and time complexities.


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  1. 1.
    Nielsen, M.A., Isaac, L.: Chuang: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)Google Scholar
  2. 2.
    Iwama, K., Kambayashi, Y., Yamashita, S.: Transformation rules for designing CNOT-based quantum circuits. In: Proc. DAC, New Orleans, Louisiana 28.4 (2002)Google Scholar
  3. 3.
    Landauer, R.: Irreversibility and heat generation in the computational process. IBM Journal of Research and Development 5, 183–191 (1961)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bennett, C.: Logical reversibility of computation. IBM Journal of Research and Development 17, 525–532 (1973)zbMATHCrossRefGoogle Scholar
  5. 5.
    Fredkin, E., Toffoli, T.: Conservative logic. Int. Journal of Theoretical Physics. 21, 219–253 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Deutsch, D.: Quantum computational networks. Royal Society of London Series A 425, 73–90 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Khlopotine, A., Perkowski, M., Kerntopf, P.: Reversible logic synthesis by gate composition. In: Proc. IEEE/ACM Int. Workshop on Logic Synthesis, pp. 261–266 (2002)Google Scholar
  8. 8.
    Toffoli, T.: Bicontinuous extensions of invertible combinatorial functions. Mathematical Systems Theory 14, 13–23 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Yang, G., Hung, W.N.N., Song, X., Perkowski, M.: Majority-Based Reversible Logic Gates. Theoretical Computer Science 334, 274–295 (2005)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Yang, G., Song, X., Hung, W.N.N., Perkowski, M.: Fast Synthesis of Exact Minimal Reversible Circuits using Group Theory. In: ACM/IEEE Asia and South Pacific Design Automation Conference (ASP-DAC), pp. 1002–1005 (2005)Google Scholar
  11. 11.
    Dixon, J.D., Mortimer, B.: Permutation Groups. Springer, New York (1996)zbMATHGoogle Scholar
  12. 12.
    De. Vos, A.: Reversible computing. Qutantum Electronics 23, 1–49 (1999)CrossRefGoogle Scholar
  13. 13.
    Storme, L., De. Vos, A., Jacobs, G.: Group theoretical aspects of reversible logic gates. Journal of Universal Computer Science 5, 307–321 (1999)zbMATHGoogle Scholar
  14. 14.
    Michael Miller, D., Maslov, D., Dueck, G.W.: A Transformation Based Algorithm for Reversible Logic Synthesis. In: Proc. DAC, pp. 318–323 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Guowu Yang
    • 1
  • Xiaoyu Song
    • 2
  • William N. N. Hung
    • 2
  • Fei Xie
    • 1
  • Marek A. Perkowski
    • 2
  1. 1.Dept. of Computer SciencePortland State UniversityPortlandUSA
  2. 2.Dept. of Electrical and Computer EngineeringPortland State UniversityPortlandUSA

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