Maximality of reversible gate sets

Tim Boykett

Home/Archive/arXiv

arXiv · 2002.02899

Maximality of reversible gate sets

Tim Boykett

View original source ↗Read source document ↗Export metadata ↓

Abstract

In order to better understand the structure of closed collections of reversible gates, we investigate the lattice of closed sets and the maximal members of this lattice. In this note, we find the maximal closed sets over a finite alphabet. We find that for odd sized alphabets, there are a finite number of maximal closed sets, while for the even case we have a countable infinity, almost all of which are related to an alternating permutations. We then extend to other forms of closure for reversible gates, ancilla and borrow closure. Here we find some structural results, including some examples of maximal closed sets.

Explore related subjects

math.GR cs.DM cs.LO math.CO

Keep this discovery

BibTeX RIS

Tim Boykett. 2020-02-06. Maximality of reversible gate sets. https://arxiv.org/abs/2002.02899

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Reversibility and its asymptotic counting in Picard group

We investigate reversible elements in the Picard modular group $\mathrm{PSL}(2,\mathbb{Z}[i])$. We show that reversibility coincides with strong reversibility for Kleinian groups, in particular for the Picard group. We classify reversible elements in the Picard group and characterize loxodromic reversible elements up to conjugacy. We prove that each such conjugacy class contains exactly eight special representatives. We also obtain asymptotic estimates for the number of reversible conjugacy classes with bounded trace.

math.GR↗

arXiv2026

Conjugator length in finitely generated groups

We describe all functions $\mathbb{N}\rightarrow \mathbb{N}$ that can be realized, up to the standard equivalence, as conjugator length functions of finitely generated groups. Furthermore, we show that any two increasing functions $f,g\colon \mathbb N\to \mathbb N$ can be simultaneously realized as conjugator length functions of finitely generated, commensurable (in particular, quasi-isometric) groups.

math.GR↗

arXiv2026

The spectrum of conjugator length functions

A recent program tries to find which functions appear as conjugator length functions. In this note, we show that any (computable) increasing function larger than $n$ appears as $\mathrm{Cl}_G$ for some finitely generated (recursively presented) group. On the other hand, we demonstrate that either $\mathrm{Cl}_G$ must be constant or $\mathrm{Cl}_G(n)\succ n$. Combining these, we obtain a complete description of which functions appear as conjugator length functions of finitely generated groups.

math.GR↗