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Calcolo

, Volume 45, Issue 3, pp 193–206 | Cite as

Synthesis of reversible circuits with minimal costs

  • Guowu Yang
  • Xiaoyu Song
  • William N. N. Hung
  • Marek A. Perkowski
  • Chang-Jun Seo
Article

Abstract

We present fast algorithms to synthesize exact minimal reversible circuits for various types of gate and cost. By reducing reversible logic synthesis problems to permutation group problems, we use the powerful algebraic software GAP to solve such problems. Our approach can minimize for arbitrary cost functions of gates. In addition, we show that Peres gates are a better choice than the standard Toffoli gates in libraries of universal reversible gates.

Keywords

Circuits networks switching theory application of Boolean algebra Boolean function discrete mathematics in relation to computer science combinatorics 

Mathematics Subject Classification (2000)

94C10 68R05 

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References

  1. 1.
    De Vos, A., Raa, B., Storme, L.: Generating the group of reversible logic gates. J. Phys. A 35, 7063–7078 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Dixon, J.D., Mortimer, B.: Permutation groups. New York: Springer 1996zbMATHGoogle Scholar
  3. 3.
    Gulde, S. et al.: Implementation of the Deutsch-Jozsa algorithm on an ion-trap quantum computer. Nature 421, 48–50 (2003)CrossRefGoogle Scholar
  4. 4.
    Hung, W.N.N., Song, X., Yang, G., Yang, J., Perkowski, M.: Quantum logic synthesis by symbolic reachability analysis. In: Malin, S. (ed.): Proceedings of the 41st conference on design automation. New York: ACM 2004, pp. 838–841Google Scholar
  5. 5.
    Hung, W.N.N., Song, X., Yang, G., Yang, J., Perkowski, M.: Optimal synthesis of multiple output Boolean functions using a set of quantum gates by symbolic reachability analysis. IEEE Trans. Computer-Aided Design 25, 1652–1663 (2006)CrossRefGoogle Scholar
  6. 6.
    Kargapolov, M.I., Merzljakov, Ju.I.: Fundamentals of the theory of groups. Berlin: Springer 1979zbMATHGoogle Scholar
  7. 7.
    Korf, R.E.: Artificial intelligence search algorithms. In: Atallah, M.J. (ed.): Algorithms and theory of computation handbook. Boca Raton, FL CRC 1999, pp. 36-1–36-20Google Scholar
  8. 8.
    Lee, S., Lee, J.-S., Lee, S.-J., Kim, T.: Costs of basic gates in quantum computation. Submitted to Journal of Physics A in 2004. Cited with author’s permission from private communication.Google Scholar
  9. 9.
    Miller, D.M., Maslov, D., Dueck, G.W.: A transformation based algorithm for reversible logic synthesis. In: Getreu, I. (ed.): Proceedings of the 40th conference on design automation. New York: ACM 2003, pp. 318–323Google Scholar
  10. 10.
    Perkowski, M., Lukac, M., Pivtoraiko, M., Kerntopf, P., Folgheraiter, M.: A hierarchical approach to computer aided design of quantum circuits. In: Proceedings of the 6th international symposium on representations and methodology of future computing technologies. Trier 2003, pp. 201–209Google Scholar
  11. 11.
    Schönert, M. et al.: GAP — groups, algorithms, and programming. Aachen: Lehrstuhl D für Mathematik, RheinischWestfälische Technische Hochschule 1995Google Scholar
  12. 12.
    Smolin, J.A., DiVincenzo, D.P.: Five two-bit quantum gates are sufficient to implement the quantum Fredkin gate. Phys. Rev. A53, 2855–2856 (1996)MathSciNetGoogle Scholar
  13. 13.
    Song, X., Yang, G., Perkowski, M., Wang, Y.: Algebraic characterization of reversible logic gates. Theory Comput. Syst. 39, 311–319 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Storme, L., De Vos, A., Jacobs, G.: Group theoretical aspects of reversible logic gates. J Universal Comput. Sci. 5, 307–321 (1999)zbMATHGoogle Scholar
  15. 15.
    Toffoli, T.: Reversible computing. Technical Report MIT/LCS/TM-151. Cambridge, MA: Laboratory for Computer Science, MIT 1980Google Scholar
  16. 16.
    Vandersypen, L.M.K., Chuang, I.L.: NMR techniques for quantum control and computation. Rev. Modern Phys. 76, 1037–1070 (2004)CrossRefGoogle Scholar
  17. 17.
    Yang, G., Hung, W.N.N., Song, X., Perkowski, M.: Majority-based reversible logic gates. Theoret. Comput. Sci. 334, 259–274 (2005)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Italia 2008

Authors and Affiliations

  • Guowu Yang
    • 1
  • Xiaoyu Song
    • 2
  • William N. N. Hung
    • 2
  • Marek A. Perkowski
    • 2
  • Chang-Jun Seo
    • 3
  1. 1.University of Electronic Science and Technology of ChinaChengduChina
  2. 2.Portland State UniversityPortlandUSA
  3. 3.Dept. of EIREInje UniversityInjeKorea

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