# Multimode rotationally symmetric bosonic codes from group-theoretic construction

## TL;DR

This paper introduces a new family of multi-mode, rotationally symmetric bosonic codes based on group theory, enabling linear-optics implementation of logical gates and improved noise protection, including exact correction of correlated dephasing.

## Contribution

It presents a novel group-theoretic construction of multi-mode bosonic codes that preserve rotational symmetry and allow full Pauli group implementation via linear optics.

## Key findings

- Outperform single-mode and earlier multi-mode codes in dephasing protection
- Enable exact correction of correlated dephasing noise
- Show no trade-off between dephasing and photon loss protection

## Abstract

We introduce a new family of multi-mode, rotationally symmetric bosonic codes inspired by the group-theoretic framework of [Phys. Rev. Lett. 133, 240603 (2024)]. Such a construction inverts the traditional paradigm of code design by identifying codes from the requirement that a group of chosen logical gates should be implemented by means of physically simple logical operations, such as linear optics. Leveraging previously unexplored degrees of freedom within this framework, our construction preserves rotational symmetry across multiple modes, enabling linear-optics implementation of the full Pauli group. These codes exhibit improved protection against dephasing noise, outperforming both single-mode analogues and earlier multi-mode constructions. Notably, they allow exact correction of correlated dephasing and support qudit encoding in arbitrary dimensions. We analytically construct and numerically benchmark two-mode binomial codes instances, and demonstrate that, unlike single-mode rotationally symmetric bosonic codes, these exhibit no trade-off between protection against dephasing and photon loss.