# The magic of universal quantum computing with permutations

## TL;DR

This paper explores how permutation gates contribute to universal quantum computing by analyzing their eigenstates, Wigner functions, and contextuality, providing a classification of magic states in low dimensions.

## Contribution

It offers a new perspective on the role of permutation gates in quantum computing and classifies magic states in dimensions up to 9.

## Key findings

- Permutation gates are key to universal quantum computation.
- Eigenstates and Wigner functions reveal the 'magic' of permutation gates.
- A classification of magic states in low dimensions is provided.

## Abstract

The role of permutation gates for universal quantum computing is investigated. The \lq magic' of computation is clarified in the permutation gates, their eigenstates, the Wootters discrete Wigner function and state-dependent contextuality (following many contributions on this subject). A first classification of main types of resulting magic states in low dimensions $d \le 9$ is performed.