# Quantum Pin Codes

## TL;DR

Quantum pin codes are a new class of quantum CSS codes that generalize existing codes and have properties useful for magic-state distillation, with structural features like gauge operators and orthogonal stabilizer spaces.

## Contribution

Introduction of quantum pin codes, a novel class of quantum CSS codes with unique structural properties and potential applications in quantum error correction and magic-state distillation.

## Key findings

- Quantum pin codes generalize color and Reed-Muller codes.
- They have gauge operators and an unfolding procedure.
- Their stabilizers form $ ext{ell}$-orthogonal spaces, aiding distillation protocols.

## Abstract

We introduce quantum pin codes: a class of quantum CSS codes. Quantum pin codes are a generalization of quantum color codes and Reed-Muller codes and share a lot of their structure and properties. Pin codes have gauge operators, an unfolding procedure and their stabilizers form so-called $\ell$-orthogonal spaces meaning that the joint overlap between any $\ell$ stabilizer elements is always even. This last feature makes them interesting for devising magic-state distillation protocols, for instance by using puncturing techniques. We study examples of these codes and their properties.