Grand Unification of continuous-variable codes

TL;DR

This paper provides a unified theoretical framework for continuous-variable quantum error correction codes, including GKP and cat codes, revealing their common properties and interrelations to advance fault-tolerant quantum computing.

Contribution

It introduces a general description and mapping rules for continuous-variable codes, unifying various code families under a common theoretical structure.

Findings

All CV codes' properties derive from the GKP code.

Explicit construction and properties of rotation-symmetric codes are re-derived.

New properties of CV codes are identified through the unified framework.

Abstract

Quantum error correction codes in continuous variables (also called CV codes, or single-mode bosonic codes) have recently been identified to be a technologically viable option for building fault-tolerant quantum computers. The best-known examples are the GKP code and the cat-code, both of which were shown to have some advantageous properties over any discrete-variable, or qubit codes. It was recently shown that the cat-code, as well as other kinds of CV codes, belong to a set of codes with common properties called rotation-symmetric codes. We expand this result by giving a general description of sets of codes with common properties, and rules by which they can be mapped into one another, effectively creating a unified description of continuous-variable codes. We prove that the properties of all of these sets of codes can be obtained from the properties of the GKP code. We also show explicitly how this construction works in the case of rotation-symmetric codes, re-deriving known properties and finding new ones.