Strongly Universal Reversible Gate Sets

TL;DR

This paper generalizes universal reversible gate sets to non-binary logic, characterizing when all permutations or even permutations can be implemented without auxiliary bits, and explores conservative permutations within weight classes.

Contribution

It extends the theory of universal reversible gates to non-binary alphabets and analyzes the capabilities of finite gate sets for permutations and conservative permutations.

Findings

Finite gate sets can generate all permutations of A^n for odd |A| without auxiliary bits.

Only even permutations are possible for large n when |A| is even, but all even permutations are achievable.

Finite conservative gate sets cannot implement all conservative even permutations without auxiliary bits.

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: If $A$ is a finite set of odd cardinality then a finite gate set can generate all permutations of $A^n$ for all $n$, without any auxiliary symbols. If the cardinality of $A$ is even then, by the same argument as above, only even permutations of $A^n$ can be implemented for large $n$, and we show that indeed all even permutations can be obtained from a finite universal gate set. 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 $A^n$.