Finite generating sets for reversible gate sets under general conservation laws

TL;DR

This paper generalizes the concept of finite generating sets for reversible gates under conservation laws from binary to non-binary logic, analyzing the necessity of auxiliary bits for generating permutations and their conservative variants.

Contribution

It extends known results to non-binary alphabets and characterizes the minimal auxiliary resources needed for generating all permutations and conservative permutations.

Findings

Finite gate sets can generate all even permutations without auxiliary bits for non-binary alphabets.

One auxiliary symbol is necessary and sufficient to generate all permutations when the alphabet size is even.

No finite conservative gate set can generate all conservative even permutations without auxiliary bits, but some can generate all within each weight class.

Abstract

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 or, more generally, the image of the word under a fixed monoid homomorphism from $A^*$ to a commutative monoid. 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 $A^n$.