A4 Refereed article in a conference publication
Strongly Universal Reversible Gate Sets
Authors: Boykett T, Kari J, Salo V
Editors: Simon Devitt, Ivan Lanese
Conference name: International Conference on Reversible Computation
Publisher: SPRINGER INT PUBLISHING AG, GEWERBESTRASSE 11, CHAM, CH-6330, SWITZERLAND
Publication year: 2016
Book title : Reversible Computation: 8th International Conference, RC 2016, Bologna, Italy, July 7-8, 2016, Proceedings
Journal name in source: REVERSIBLE COMPUTATION, RC 2016
Journal acronym: LECT NOTES COMPUT SC
Series title: Lecture Notes in Computer Science
Volume: 9720
First page : 239
Last page: 254
Number of pages: 16
ISBN: 978-3-319-40577-3
eISBN: 978-3-319-40578-0
ISSN: 0302-9743
DOI: https://doi.org/10.1007/978-3-319-40578-0_18(external)
Web address : http://link.springer.com/chapter/10.1007/978-3-319-40578-0_18(external)
Self-archived copy’s web address: https://arxiv.org/abs/1602.04967(external)
It is well-known that the Toffoli gate and the negation gate together yield a universal gate set, in the sense that every permutation of {0, 1}(n) can be implemented as a composition of these gates. Since every bit operation that does not use all of the bits performs an even permutation, we need to use at least one auxiliary bit to perform every permutation, and it is known that one bit is indeed enough. Without auxiliary bits, all even permutations can be implemented. We generalize these results to non-binary logic: For any finite set A, a finite gate set can generate all even permutations of A(n) for all n, without any auxiliary symbols. This directly implies the previously published result that a finite gate set can generate all permutations of A(n) when the cardinality of A is odd, and that one auxiliary symbol is necessary and sufficient to obtain all permutations when the cardinality of A is even. We also consider the conservative case, that is, those permutations of A(n) that preserve the weight of the input word. The weight is the vector that records how many times each symbol occurs in the word. It turns out that no finite conservative gate set can, for all n, implement all conservative even permutations of A(n) without auxiliary bits. But we provide a finite gate set that can implement all those conservative permutations that are even within each weight class of A(n).
