Abstract
Optimal synthesis of reversible circuits is a very hard task. For example, up to year 2009 this problem had not been solved even for 4-bit reversible functions, in spite of intensive research during previous decade. In 2010, a method and a tool of practical usage for finding optimal circuits for any 4-bit reversible specification were finally developed. Namely, with sophisticated optimizations it was possible to find gate count optimal circuits for any 4-bit reversible function built from multi-control Toffoli gates. Last year, we published an extension to the algorithm, which allows to reduce the quantum cost of the resulting circuits. In this paper we present another extension to this approach. Namely, we have extended the reversible gate library to mixed-polarity multi-control Toffoli gates (i.e. with both positive and negative controls). Our experimental results for the known reversible benchmarks show that using mixed-polarity Toffoli gates gives significant savings in gate count. The paper presents results of different computational experiments including optimal 4-bit circuits for the known reversible benchmarks with respect to both gate count and quantum cost criteria.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arabzadeh, M., Saeedi, M., Zamani, M.S.: Rule-Based Optimization of Reversible Circuits. In: Proc. Asia-South Pacific Design Automation Conference, pp. 849–854. IEEE (2010)
Ardestani, E.K., Zamani, M.S., Sedighi, M.: A Fast Transformation-Based Synthesis Algorithm for Reversible Circuits. In: Proc. EUROMICRO Conference on Digital System Design, pp. 803–806 (2008)
Barenco, A., Bennett, C.H., Cleve, R., DiVincenzo, D., Margolus, N., Shor, P., Sleator, T., Smolin, J., Weinfurter, H.: Elementary Gates for Quantum Computation. Phys. Rev. A 52, 3457–3467 (1995)
Golubitsky, O., Falconer, S.M., Maslov, D.: Synthesis of the Optimal 4-bit Reversible Circuits. In: Proc. Design Automation Conference, pp. 653–656. ACM (2010)
Golubitsky, O., Maslov, D.: A Study of Optimal 4-bit Reversible Toffoli Circuits and Their Synthesis. IEEE Trans. on Computers 61, 1341–1353 (2012)
Grosse, D., Wille, R., Dueck, G.W., Drechsler, R.: Exact Multiple Control Toffoli Network Synthesis with SAT Techniques. IEEE Trans. on CAD 28, 703–715 (2009)
Li, M., Zheng, Y., Hsiao, M.S., Huang, C.: Reversible Logic Synthesis Through Ant Colony Optimization. In: Proc. Design and Test in Europe Conference, pp. 307–310 (2010)
Markov, I.L., Saeedi, M.: Constant-Optimized Quantum Circuits for Modular Multiplication and Exponentiation. Quantum Information and Computation 12, 361–394 (2012)
Maslov, D.: Reversible Logic Synthesis Benchmarks Page, http://www.cs.uvic.ca/~dmaslov
Maslov, D., Dueck, G.W.: Improved Quantum Cost for n-bit Toffoli Gates. Electronics Letters 39, 1790–1791 (2003)
Miller, D.M., Wille, R., Sasanian, Z.: Elementary Quantum Gate Realizations for Multiple-Control Toffoli Gates. In: Proc. 41st IEEE International Symposium on Multiple-Valued Logic, pp. 288–293. IEEE (2011)
Moraga, C.: Mixed Polarity Reed Muller Expressions for Quantum Computing Circuits. In: Proc. 10th Reed-Muller Workshop, pp. 119–125 (2011)
Moraga, C.: Hybrid GF(2) – Boolean Expressions for Quantum Computing Circuits. In: De Vos, A., Wille, R. (eds.) RC 2011. LNCS, vol. 7165, pp. 54–63. Springer, Heidelberg (2012)
Moraga, C.: Using Negated Control Signals in Quantum Computing Circuits. Facta Universitatis (Niš), Ser.: Elec. Energ. 24, 423–435 (2011)
Saeedi, M., Markov, I.L.: Synthesis and Optimization of Reversible Circuits – A Survey. ACM Computing Surveys (accepted 2012), available at arXiv: 1110.2574v1 (2011)
Saeedi, M., Zamani, M.S., Sedighi, M.: Moving Forward: A Non-Search Based Synthesis Method toward Efficient CNOT-based Quantum Circuit Synthesis Algorithms. In: Proc. Asia-South Pacific Design Automation Conference, pp. 83–88. IEEE (2008)
Saeedi, M., Sedighi, M., Zamani, M.S.: A Novel Synthesis Algorithm for Reversible Circuits. In: Proc. International Conference on Computer Aided Design, pp. 65–68 (2007)
Sasanian, Z., Miller, D.M.: Mapping a Multiple-Control Toffoli Gate Cascade to an Elementary Quantum Gate Circuit. Journal of Multiple-Valued Logic and Soft Computing 18, 83–98 (2012)
Szyprowski, M., Kerntopf, P.: Estimating the Quality of Complexity Measures in Heuristics for Reversible Logic Synthesis. In: Proc. IEEE Congress on Computational Intelligence, Congress on Evolutionary Computation (CD), 8 p. IEEE (2010)
Szyprowski, M., Kerntopf, P.: Reducing Quantum Cost in Reversible Toffoli Circuits. In: Proc. 10th Reed-Muller Workshop, pp. 127–136 (2011), corrected version available at arXiv: 1105.5831v2
Szyprowski, M., Kerntopf, P.: An Approach to Quantum Cost Optimization in Reversible Circuits. In: Proc. 11th IEEE Conference on Nanotechnology, pp. 1521–1526. IEEE (2011)
Wille, R., Grosse, D., Teuber, L., Dueck, G.W., Drechsler, R.: RevLib: An Online Resourse for Reversible Functions and Reversible Circuits, http://www.revlib.org
Wille, R., Soeken, M., Przigoda, N., Drechsler, R.: Exact Synthesis of Toffoli Gate Circuits with Negative Control Lines. In: Proc. 42nd IEEE International Symposium on Multiple-Valued Logic, pp. 69–74. IEEE (2012)
Zheng, Y., Huang, C.: A Novel Toffoli Network Synthesis Algorithm for Reversible Logic. In: Proc. Asia-South Pacific Design Automation Conference, pp. 739–744. IEEE (2009)
Zhu, W., Guan, Z., Hang, Y.: Reversible Logic Synthesis of Networks of Positive/Negative Control Gates. In: Proc. 5th IEEE International Conference on Natural Computation, pp. 538–542. IEEE (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Szyprowski, M., Kerntopf, P. (2013). Optimal 4-bit Reversible Mixed-Polarity Toffoli Circuits. In: Glück, R., Yokoyama, T. (eds) Reversible Computation. RC 2012. Lecture Notes in Computer Science, vol 7581. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36315-3_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-36315-3_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36314-6
Online ISBN: 978-3-642-36315-3
eBook Packages: Computer ScienceComputer Science (R0)