# Analog quantum error correction with encoding a qubit into an oscillator

## TL;DR

This paper introduces a hybrid quantum error correction method that leverages both digital and analog information from GKP qubits, significantly enhancing error correction capabilities in continuous-variable quantum computing.

## Contribution

It presents the first approach to utilize both digital and analog data from a single GKP qubit for improved quantum error correction performance.

## Key findings

- Three-qubit bit-flip code corrects double errors
- Concatenated C4/C6 code reaches quantum capacity bound
- Analog information improves error correction efficiency

## Abstract

To implement fault-tolerant quantum computation with continuous variables, Gottesman-Kitaev-Preskill (GKP) qubits have been recognized as an important technological element. However, the analog outcome of GKP qubits, which includes beneficial information to improve the error tolerance, has been wasted, because the GKP qubits have been treated as only discrete variables. In this paper, we propose a hybrid quantum error correction approach that combines digital information with the analog information of the GKP qubits using the maximum-likelihood method. As an example, we demonstrate that the three-qubit bit-flip code can correct double errors, whereas the conventional method based on majority voting on the binary measurement outcome can correct only a single error. As another example, a concatenated code known as Knill's C4/C6 code can achieve the hashing bound for the quantum capacity of the Gaussian quantum channel. To the best of our knowledge, this approach is the first attempt to draw both digital and analog information from a single quantum state to improve quantum error correction performance.