Artificial Intelligence Review

, Volume 20, Issue 3–4, pp 361–417 | Cite as

Evolutionary Approach to Quantum and Reversible Circuits Synthesis

  • Martin Lukac
  • Marek Perkowski
  • Hilton Goi
  • Mikhail Pivtoraiko
  • Chung Hyo Yu
  • Kyusik Chung
  • Hyunkoo Jeech
  • Byung-Guk Kim
  • Yong-Duk Kim


The paper discusses theevolutionary computation approach to theproblem of optimal synthesis of Quantum andReversible Logic circuits. Our approach usesstandard Genetic Algorithm (GA) and itsrelative power as compared to previousapproaches comes from the encoding and theformulation of the cost and fitness functionsfor quantum circuits synthesis. We analyze newoperators and their role in synthesis andoptimization processes. Cost and fitnessfunctions for Reversible Circuit synthesis areintroduced as well as local optimizingtransformations. It is also shown that ourapproach can be used alternatively forsynthesis of either reversible or quantumcircuits without a major change in thealgorithm. Results are illustrated onsynthesized Margolus, Toffoli, Fredkin andother gates and Entanglement Circuits. This isfor the first time that several variants ofthese gates have been automatically synthesizedfrom quantum primitives.

genetic algorithm minimizing transformation Quantum CAD Quantum Logic Synthesis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Al-Rabadi, A., Casperson, L., Perkowski, M. & Song, X. (2002). Canonical Representation for Two-Valued Quantum Computing. Proc. Fifth Intern. Workshop on Boolean Problems, 23–32. Freiberg, Sachsen, Germany, September 19–20.Google Scholar
  2. Al-Rabadi, A. (2002). Novel Methods for Reversible Logic Synthesis and Their Application to Quantum Computing. Ph.D. Thesis, Portland State University, Portland, Oregon, USA, October 24.Google Scholar
  3. Barenco, A. et al. (1995). Elementary Gates for Quantum Computation. Physical Review A 52: 3457–3467.Google Scholar
  4. Bennett, C. (1973). Logically Reversible Computation. I.B.M. J. Res. Dev. (17): 525–632.Google Scholar
  5. Cirac, J. I. & Zoller, P. (1995). Quantum Computation with Cold Trapped Ions. Physical Review Letters (15 May) 74: Issue20, 4091–4094.Google Scholar
  6. Dill, K. & Perkowski, M. (2001). Baldwinian Learning utilizing Genetic and Heuristic for Logic Synthesis and Minimization of Incompletely Specified Data with Generalized Reed-Muller (AND-EXOR) Forms. Journal of System Architecture 47: Issue6, 477–489.Google Scholar
  7. Dirac, P. A. M. (1930). The Principles of Quantum Mechanics, 1st edn. Oxford University Press.Google Scholar
  8. DiVincenzo, D. P. (1995). Quantum Computation. Science 270: 255–256.Google Scholar
  9. Dueck, G. W. & Maslov, D. (2003). Garbage in Reversible Designs of Multiple-Output Functions. Proc. RM: 162–170.Google Scholar
  10. Einstein, A., Podolsky, B. & Rosen, N. (1935). Physical Review (47): 777.Google Scholar
  11. Ekert, A. & Jozsa, R. (1996). Quantum Computation and Shor's Factoring Algorithm. Review of Modern Physics (July) 68: Issue3, 733–753.Google Scholar
  12. Feynman, R. (1996). Feynman Lectures on Computation. Addison Wesley.Google Scholar
  13. Fredkin, E. & Toffoli, T. (1982). Conservative Logic. Int. J. of Theoretical Physics (21): 219–253.Google Scholar
  14. Ge, Y. Z., Watson, L. T. & Collins, E. G. (1998). Genetic Algorithms for Optimization on a Quantum Computer. In Unconventional Models of Computation, 218–227. London: Springer Verlag.Google Scholar
  15. Goldberg, D. E. (1989). Genetic Algorithms in Search, Optimization, and Machine Learning. Addison Wesley.Google Scholar
  16. Graham, A. (1981). Kronecker Products and Matrix Calculus with Applications. Chichester, UK: Ellis Horwood Limited.Google Scholar
  17. Hirvensalo, M. (2001). Quantum Computing. Springer Verlag.Google Scholar
  18. Iwama, K., Kambayashi, Y. & Yamashita, S. (2002). Transformation Rules for Designing CNOT-based Quantum Circuits. In Proc. DAC, 419–424. New Orleans, Louisiana.Google Scholar
  19. Kerntopf, P. (2001). Maximally Efficient Binary and Multi-Valued Reversible Gates. In Proceedings of ULSI Workshop, 55–58. Warsaw, Poland, May.Google Scholar
  20. Kerntopf, P. (2000). A Comparison of Logical Efficiency of Reversible and Conventional Gates. Proc. of 3rd Logic Design and Learning Symposium (LDL). Portland, Oregon.Google Scholar
  21. Khan, M. H. A., Perkowski, M. & Kerntopf, P. (2003). Multi-Output Galois Field Sum of Products Synthesis with New Quantum Cascades. In Proceedings of 33rd International Symposium on Multiple-Valued Logic, ISMVL 2003, 146–153. Meiji University, Tokyo, Japan, 16–19 May.Google Scholar
  22. Khlopotine, A., Perkowski, M. & Kerntopf, P. (2002). Reversible Logic Synthesis by Gate Composition. Proceedings of IWLS: 261–266.Google Scholar
  23. Kim, J., Lee, J-S. & Lee, S. (2000). Implementation of the Refined Deutsch-Jozsa algorithm on a Three-Bit NMR Quantum Computer. Physical Review A 62: 022312.Google Scholar
  24. Kim, J., Lee, J-S. & Lee, S. (2000). Implementing Unitary Operators in Quantum Computation. Physical Review A: 032312.Google Scholar
  25. Klay, M. (1988). Einstein-Podolsky-Rosen Experiments: The Structure of the Sample Space I, II. Foundations of Physics Letters 1: 205–232.Google Scholar
  26. Koza, J. (1992). Genetic Programming. On the Programming of Computers by Means of Natural Selection. The MIT Press.Google Scholar
  27. Lee, J-S., Chung, Y., Kim, J. & Lee, S. (1999). A Practical Method of Constructing Quantum Combinational Logic Circuits. arXiv:quant-ph/9911053v1 (12 November).Google Scholar
  28. Lomont, Ch. (2003). Quantum Circuit Identities. arXiv:quant-ph/0307111v1 (16 July).Google Scholar
  29. Lukac, M., Pivtoraiko, M., Mishchenko, A. & Perkowski, M. (2002). Automated Synthesis of Generalized Reversible Cascades Using Genetic Algorithms. In Proc. Fifth Intern. Workshop on Boolean Problems, 33–45 Freiberg, Sachsen, Germany, September 19–20.Google Scholar
  30. Lukac, M. & Perkowski, M. (2002). Evolving Quantum Circuits Using Genetic Algorithms. In Proc. of 5th NASA/DOD Workshop on Evolvable Hardware, 177–185.Google Scholar
  31. Lukac, M., Lee, S. & Perkowski, M. (2003). Low Cost NMR Realizations of Ternary and Mixed Quantum Gates and Circuits. In preparation.Google Scholar
  32. Lukac, M., Lee, S. & Perkowski, M. (2003). Inexpensive NMR Realizations of Quantum Gates. In preparation.Google Scholar
  33. Miller, D. M. (2002). Spectral and Two-Place Decomposition Techniques in Reversible Logic. Proc. Midwest Symposium on Circuits and Systems, on CD-ROM, August.Google Scholar
  34. Miller, D. M. & Dueck, G. W. (2003). Spectral Techniques for Reversible Logic Synthesis. Proc. RM: 56–62.Google Scholar
  35. Mishchenko, A. & Perkowski, M. (2002). Logic Synthesis of Reversible Wave Cascades. In Proc. IEEE/ACM International Workshop on Logic Synthesis, 197–202. June.Google Scholar
  36. Monroe, C., Leibfried, D., King, B. E., Meekhof, D. M., Itano, W. M. & Wineland, D. J. (1997). Simplified Quantum Logic with trapped Ions. Physical Review A (April) 55: Issue4, 2489–2491.Google Scholar
  37. Monroe, C., Meekhof, D. M., King, B. E. & Wineland, D. J. (1996). A “Schroedinger Cat” Superposition State of an Atom. Science (May) 272: 1131–1136.Google Scholar
  38. Nielsen, M. A. & Chuang, I. L. (2000). Quantum Computation and Quantum Information. Cambridge University Press.Google Scholar
  39. Negotevic, G., Perkowski, M., Lukac, M. & Buller, A. (2002). Evolving Quantum Circuits and an FPGA Based Quantum Computing Emulator. In Proc. Fifth Intern. Workshop on Boolean Problems, 15–22. Freiberg, Sachsen, Germany, September 19–20.Google Scholar
  40. Peres, A. (1985). Reversible Logic and Quantum Computers. Physical Review A 32: 3266–3276.Google Scholar
  41. Perkowski, M., Kerntopf, P., Buller, A., Chrzanowska-Jeske, M., Mishchenko, A., Song, X., Al-Rabadi, A., Jozwiak, L., Coppola, A. & Massey, B. (2001). Regular Realization of symmetric Functions Using Reversible Logic. In Proceedings of EUROMICRO Symposium on Digital Systems Design, 245–252.Google Scholar
  42. Perkowski, M., Jozwiak, L., Kerntopf, P., Mishchenko, A., Al-Rabadi, A., Coppola, A., Buller, A., Song, X., Khan, M. M. H. A., Yanushkevich, S., Shmerko, V. & Chrzanowska-Jeske, M. (2001). A General Decomposition for Reversible Logic. Proceedings of RM: 119–138.Google Scholar
  43. Perkowski, M., Al-Rabadi, A. & Kerntopf, P. (2002). Multiple-Valued Quantum Logic Synthesis. Proc. of 2002 International Symposium on New Paradigm VLSI Computing, 41–47. Sendai, Japan, December 12–14.Google Scholar
  44. Price, M. D., Somaroo, S. S., Tseng, C. H., Core, J. C., Fahmy, A. H., Havel, T. F. & Cory, D. (1999). Construction and Implementation of NMR Quantum Logic Gates for Two Spin Systems. Journal of Magnetic Resonance 140: 371–378.Google Scholar
  45. Price, M. D., Somaroo, S. S., Dunlop, A. E., Havel, T. F. & Cory, D. G. (1999). Generalized Methods for the Development of Quantum Logic Gates for an NMR Quantum Information Processor. Physical Review A (Octover) 60(4): 2777–2780.Google Scholar
  46. Rubinstein, B. I. P. (2001). Evolving Quantum Circuits Using Genetic Programming. In Proceedings of the 2001 Congress on Evolutionary Computation (CEC2001), 144–151.Google Scholar
  47. Shende, V. V., Prasad, A. K., Markov, I. K. & Hayes, J. P. (2002). Reversible Logic Circuit Synthesis. Proc. 11th IEEE/ACM Intern. Workshop on Logic Synthesis (IWLS), 125–130.Google Scholar
  48. Smolin, J. & DiVincenzo, D. P. (1996). Five Two-Qubit Gates are Sufficient to Implement the Quantum Fredkin Gate. Physical Review A (April) 53(4), 2855–2856.Google Scholar
  49. Spector, L., Barnum, H., Bernstein, H. J. & Swamy, N. (1999). Finding a Better-than-Classical Quantum AND/OR Algorithm Using Genetic Programming. In Proc. 1999 Congress on Evolutionary Computation, Vol. 3, 2239–2246. Washington DC, 6–9 July; IEEE, Piscataway, NJ.Google Scholar
  50. Van Der Sypen, L. M. K., Steffen, M., Breyta, G., Yannoni, C. S., Sherwood, M. H. & Chuang, I. L. (2001). Experimental Realization of Shor's Quantum Factoring Algorithm Using Nuclear Magnetic Resonance. Nature (20/27 December) 414: 883–887.Google Scholar
  51. Vieri, C., Ammer, M. J., Frank, M., Margolus, N. & Knight, T. A Fully Reversible Asymptotically Zero Energy Microprocessor. MIT Artificial Intelligence Laboratory, Cambridge, MA 02139, USAGoogle Scholar
  52. von Neumann, J. (1950). Mathematical Foundations of Quantum Mechanics. Princeton University Press.Google Scholar
  53. Wheeler, J. A. & Zurek, W. H. (1983). Quantum Theory and Measurement. Princeton University Press.Google Scholar
  54. Williams, C. W. & Gray, A. G. (1999). Automated Design of Quantum Circuits. ETC Quantum Computing and Quantum Communication, QCQC '98, 113–125. Palm Springs, California: Springer-Verlag, February 17–20.Google Scholar
  55. Williams, C. P. & Clearwater, S. H. (1998)., Explorations in Quantum Computing. New York Inc.: Springer-Verlag.Google Scholar
  56. Scholar
  57. Yabuki, T. & Iba, H. (2000). Genetic Algorithms and Quantum Circuit Design, Evolving a Simpler Teleportation Circuit. In Late Breaking Papers at the 2000 Genetic and Evolutionary Computation Conference, 421–425.Google Scholar
  58. Yang, G., Hung, W. N. N., Song, X. & Perkowski, M. (2003). Majority-Based Reversible Logic Gate. In Proceedings of 6th International Symposium on Representations and Methodology of Future Computing Technology, 191–200. Trier, Germany, March 10–11.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Martin Lukac
    • 1
  • Marek Perkowski
    • 1
  • Hilton Goi
    • 1
  • Mikhail Pivtoraiko
    • 2
  • Chung Hyo Yu
    • 1
  • Kyusik Chung
    • 1
  • Hyunkoo Jeech
    • 1
  • Byung-Guk Kim
    • 1
  • Yong-Duk Kim
    • 1
  1. 1.Department of Electrical Engineering and Computer ScienceKorea Advanced Institute of Science and TechnologyYuseong-gu, DaejeonKorea
  2. 2.Department of Electrical EngineeringPortland State UniversityPortlandUSA

Personalised recommendations