Abstract
Recently, a reversible garbage-free 2k ±1 constant-multiplier circuit was presented by Axelsen and Thomsen. This was the first construction of a garbage-free, reversible circuit for multiplication with non-trivial constants. At the time, the strength, that is, the range of constants obtainable by cascading these circuits, was unknown.
In this paper, we show that there exist infinitely many constants we cannot multiply by using cascades of 2k±1-multipliers; in fact, there exist infinitely many primes we cannot multiply by. Using these results, we further provide an algorithm for determining whether one can multiply by a given constant using a cascade of 2k ±1-multipliers, and for generating the minimal cascade of 2k ±1-multipliers for an obtainable constant, giving a complete characterization of the problem. A table of minimal cascades for multiplying by small constants is provided for convenience.
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
Axelsen, H.B., Thomsen, M.K.: Garbage-free reversible integer multiplication with constants of the form 2k±2l±1. In: Glück, R., Yokoyama, T. (eds.) RC 2012. LNCS, vol. 7581, pp. 171–182. Springer, Heidelberg (2013)
Cuccaro, S.A., Draper, T.G., Kutin, S.A., Moulton, D.P.: A New Quantum Ripple-carry Addition Circuit arXiv:quant-ph/0410184 (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)
Fredkin, E., Toffoli, T.: Conservative Logic. International Journal of Theoretical Physics 21(3-4), 219–253 (1982)
Mogensen, T.Æ.: Garbage-Free Reversible Constant Multipliers for Arbitrary Integers. In: Dueck, G.W., Miller, D.M. (eds.) RC 2013. LNCS, vol. 7948, pp. 70–83. Springer, Heidelberg (2013)
Parvardi, A.H.: Lifting The Exponent, LTE (2011), http://www.artofproblemsolving.com/Resources/Papers/LTE.pdf
Thomsen, M.K., Axelsen, H.B.: Parallelization of Reversible Ripple-carry Adders. Parallel Processing Letters 19(1), 205–222 (2009)
Toffoli, T.: Reversible Computing. In: de Bakker, J.W., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 632–644. Springer, Heidelberg (1980)
Van Rentergem, Y., De Vos, A.: Optimal Design of a Reversible Full Adder. International Journal of Unconventional Computing 1(4), 339–355 (2005)
Vedral, V., Barenco, A., Ekert, A.: Quantum Networks for Elementary Arithmetic Operations. Physical Review A 54(1), 147–153 (1996)
Zsigmondy, K.: Zur Theorie der Potenzreste. Monatshefte für Mathematik und Physik 3(1), 265–284 (1892)
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
Rotenberg, E., Cranch, J., Thomsen, M.K., Axelsen, H.B. (2013). Strength of the Reversible, Garbage-Free 2k ±1 Multiplier. In: Dueck, G.W., Miller, D.M. (eds) Reversible Computation. RC 2013. Lecture Notes in Computer Science, vol 7948. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38986-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-38986-3_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38985-6
Online ISBN: 978-3-642-38986-3
eBook Packages: Computer ScienceComputer Science (R0)