Self-Inverse Functions and Palindromic Circuits

TL;DR

This paper explores self-inverse reversible functions and their realization as palindromic circuits, providing conditions for their implementation and methods to construct such circuits with minimal resources.

Contribution

It characterizes which self-inverse functions can be realized as palindromic circuits and offers alternative constructions requiring additional resources.

Findings

Identifies which self-inverse functions are realizable as palindromic circuits.

Provides methods to construct palindromic circuits for non-realizable functions.

Uses involutions in symmetric groups to analyze circuit realizations.

Abstract

We investigate the subclass of reversible functions that are self-inverse and relate them to reversible circuits that are equal to their reverse circuit, which are called palindromic circuits. We precisely determine which self-inverse functions can be realized as a palindromic circuit. For those functions that cannot be realized as a palindromic circuit, we find alternative palindromic representations that require an extra circuit line or quantum gates in their construction. Our analyses make use of involutions in the symmetric group $S_{2^n}$ which are isomorphic to self-inverse reversible function on $n$ variables.