# Quantum spherical codes

## TL;DR

This paper introduces a new framework for quantum spherical codes, extending bosonic cat codes with improved performance and novel noise protection methods by leveraging geometric and design principles.

## Contribution

It presents a framework for quantum codes on spheres, introduces polytope-based cat codes with spherical designs, and recasts concatenated CSS codes as quantum spherical codes for enhanced noise protection.

## Key findings

- Multimode extensions of cat codes outperform previous constructions.
- Polytope-based cat codes utilize spherical designs for large separation and robustness.
- Recasting CSS code concatenations as quantum spherical codes reveals new protection strategies.

## Abstract

We introduce a framework for constructing quantum codes defined on spheres by recasting such codes as quantum analogues of the classical spherical codes. We apply this framework to bosonic coding, obtaining multimode extensions of the cat codes that can outperform previous constructions while requiring a similar type of overhead. Our polytope-based cat codes consist of sets of points with large separation that at the same time form averaging sets known as spherical designs. We also recast concatenations of CSS codes with cat codes as quantum spherical codes, revealing a new way to autonomously protect against dephasing noise.