# General structures of reversible and quantum gates

## TL;DR

This paper characterizes the general structure of single-qubit gates, provides criteria for identifying non-decomposable 2-qubit gates, and discusses their applications in optimizing reversible and quantum circuits.

## Contribution

It introduces a systematic method to identify non-decomposable 2-qubit gates and explores their role in universal quantum gate libraries.

## Key findings

- Derived conditions for testing 2-qubit gate decomposability
- Developed a systematic procedure for identifying non-decomposable 2-qubit gates
- Discussed applications in quantum circuit optimization

## Abstract

The most general structure (in matrix form) of a single-qubit gate is presented. Subsequently, used that to obtain a set of conditions for testing (a) whether a given 2-qubit gate is genuinely a 2-qubit gate, i.e., not decomposable into two single qubit gates and (b) whether a given single qubit gate is self-inverse? Relevance of the results reported here is discussed in the context of optimization of reversible and quantum circuits, especially for the optimization of quantum cost. A systematic procedure is developed for the identification of the non-decomposable 2-qubit gates. Such a non-decomposable 2-qubit gate along with all possible single qubit gates form a universal quantum gate library. Further, some possible applications of the present work are also discussed.