Abstract
Previously, Soeken and Thomsen presented six basic semantics-preserving rules for rewriting reversible logic circuits, defined using the well-known diagrammatic notation of Feynman. While this notation is both useful and intuitive for describing reversible circuits, its shortcomings in generality complicates the specification of more sophisticated and abstract rewriting rules.
In this paper, we introduce Ricercar, a general textual description language for reversible logic circuits designed explicitly to support rewriting.
Taking the not gate and the identity gate as primitives, this language allows circuits to be constructed using control gates, sequential composition, and ancillae, through a notion of ancilla scope. We show how the above-mentioned rewriting rules are defined in this language, and extend the rewriting system with five additional rules to introduce and modify ancilla scope. This treatment of ancillae addresses the limitations of the original rewriting system in rewriting circuits with ancillae in the general case.
To set Ricercar on a theoretical foundation, we also define a permutation semantics over symmetric groups and show how the operations over permutations as transposition relate to the semantics of the language.
M.K. Thomsen—This work was partly funded by the European Commission under the 7th Framework Programme.
M.K. Thomsen—A preliminary version of Ricercar was presented as work-in-progress at 6th Conference on Reversible Computation, 2014.
This is a preview of subscription content, log in via an institution to check access.
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
Abdessaied, N., Soeken, M., Thomsen, M.K., Drechsler, R.: Upper bounds for reversible circuits based on Young subgroups. Information Processing Letters 114(6), 282–286 (2014)
Bennett, C.H.: Time/Space Trade-Offs for reversible computation. SIAM Journal on Computing 18(4), 766–776 (1989)
Buhrman, H., Tromp, J., Vitányi, P.: Time and space bounds for reversible simulation. Journal of Physics A: Mathematical and General 34(35), 6821–6830 (2001)
Chan, T., Munro, J.I., Raman, V.: Selection and sorting in the “restore” model. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 995–1004 (2014)
Cuccaro, S.A., Draper, T.G., Kutin, S.A., Moulton, D.P.: A new quantum ripple-carry addition circuit. arXiv:quant-ph/0410184v1 (2005)
De Vos, A., Rentergem, Y.V.: Reversible computing: from mathematical group theory to electronical circuit experiment. In: Proceedings of the Second Conference on Computing Frontiers, 2005, Ischia, Italy, May 4–6, 2005, pp. 35–44 (2005)
Draper, T.G., Kutin, S.A., Rains, E.M., Svore, K.M.: A logarithmic-depth quantum carry-lookahead adder. arXiv:quant-ph/0406142 (2008)
Green, A.S., Lumsdaine, P.L.F., Ross, N.J., Selinger, P., Valiron, B.: An introduction to quantum programming in quipper. In: Dueck, G.W., Miller, D.M. (eds.) RC 2013. LNCS, vol. 7948, pp. 110–124. Springer, Heidelberg (2013)
Green, A.S., Lumsdaine, P.L., Ross, N.J., Selinger, P., Valiron, B.: Quipper: a scalable quantum programming language. In: Conference on Programming Language Design and Implementation, PLDI 2013, pp. 333–342. ACM (2013)
Jordan, S.P.: Strong equivalence of reversible circuits is \({\sf coNP}\)-complete. Quantum Information & Computation 14(15–16), 1302–1307 (2014)
Kaarsgaard, R.: Towards a propositional logic for reversible logic circuits. In: de Haan, R. (ed.) Proceedings of the ESSLLI 2014 Student Session, pp. 33–41 (2014). http://www.kr.tuwien.ac.at/drm/dehaan/stus2014/proceedings.pdf
Miller, D.M., Maslov, D., Dueck, G.W.: A transformation based algorithm for reversible logic synthesis. In: Design Automation Conference, DAC, pp. 318–323 (2003)
Shende, V.V., Prasad, A.K., Markov, I.L., Hayes, J.P.: Synthesis of reversible logic circuits. IEEE Trans. on CAD of Integrated Circuits and Systems 22(6), 710–722 (2003)
Soeken, M., Thomsen, M.K.: White dots do matter: rewriting reversible logic circuits. In: Dueck, G.W., Miller, D.M. (eds.) RC 2013. LNCS, vol. 7948, pp. 196–208. Springer, Heidelberg (2013)
Soeken, M., Thomsen, M.K., Dueck, G.W., Miller, D.M.: Self-inverse functions and palindromic circuits. arXiv 1502.05825 (2015)
Storme, L., De Vos, A., Jacobs, G.: Group theoretical aspects of reversible logic gates. J. UCS 5(5), 307–321 (1999)
Takahashi, Y., Kunihiro, N.: A fast quantum circuit for addition with few qubits. Quantum Info. Comput. 8(6), 636–649 (2008)
Thomsen, M.K.: Describing and optimising reversible logic using a functional language. In: Gill, A., Hage, J. (eds.) IFL 2011. LNCS, vol. 7257, pp. 148–163. Springer, Heidelberg (2012)
Thomsen, M.K., Axelsen, H.B.: Parallelization of reversible ripple-carry adders. Parallel Processing Letters 19(1), 205–222 (2009)
Vedral, V., Barenco, A., Ekert, A.: Quantum networks for elementary arithmetic operations. Physical Review A 54(1), 147–153 (1996)
Wille, R., Soeken, M., Miller, D.M., Drechsler, R.: Trading off circuit lines and gate costs in the synthesis of reversible logic. Integration, the VLSI Journal 47(2), 284–294 (2014)
Yokoyama, T., Axelsen, H.B., Glück, R.: Principles of a reversible programming language. In: Conference on Computing Frontiers, CF, pp. 43–54. ACM Press (2008)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Thomsen, M.K., Kaarsgaard, R., Soeken, M. (2015). Ricercar: A Language for Describing and Rewriting Reversible Circuits with Ancillae and Its Permutation Semantics. In: Krivine, J., Stefani, JB. (eds) Reversible Computation. RC 2015. Lecture Notes in Computer Science(), vol 9138. Springer, Cham. https://doi.org/10.1007/978-3-319-20860-2_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-20860-2_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-20859-6
Online ISBN: 978-3-319-20860-2
eBook Packages: Computer ScienceComputer Science (R0)