# Quantum Error Correction via Noise Guessing Decoding

**Authors:** Diogo Cruz, Francisco A. Monteiro, Bruno C. Coutinho  
**arXiv:** [2208.02744](https://arxiv.org/abs/2208.02744)  
**Date:** 2022-08-04  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2208.02744v1/

## Key points

- Error correction is critical in quantum technologies for both quantum communications and computation
- Quantum error correction codes (QECCs) are designed to mitigate errors due to the fragility of quantum systems
- Practical QECCs have rigid code lengths and rates, limiting their adaptability
- A recent paper proposes a novel approach for constructing and decoding QECCs that can achieve maximum performance for any chosen code length and sufficiently high code rate
- The method is based on a classical decoding strategy called GRAND, which has been shown to perform near capacity for classical random linear codes (RLCs)
- The authors extend these concepts to quantum random linear codes (QRLCs), which were previously not feasible to decode
- By combining QRLCs with a newly proposed quantum GRAND algorithm, versatile QECCs can be decoded that can adapt on-the-fly to changing conditions
- The paper presents an innovative approach towards constructing and decoding QECCs with greater flexibility and adaptability than existing stabilizer codes.

## Summary

In the field of quantum technologies, error correction is a critical aspect that plays a central role in both quantum communications and quantum computation. Quantum error correction codes (QECCs) are designed to mitigate errors that arise due to the inherent fragility of quantum systems. However, practical QECCs such as stabilizer codes have rigid code lengths and rates, which limit their adaptability to changing requirements. A recent paper proposes a novel approach for constructing and decoding QECCs that can achieve maximum performance in the finite blocklength regime for any chosen code length and sufficiently high code rate. The method is based on a recently proposed classical decoding strategy called GRAND (guessing random additive noise decoding), which has been shown to perform near capacity for classical random linear codes (RLCs). By leveraging noise statistics, GRAND provides an efficient universal decoder for classical codes when there is a simple code membership test. The authors extend these concepts to quantum random linear codes (QRLCs), which were known to be possible to construct but whose decoding was not yet feasible. By combining QRLCs with a newly proposed quantum GRAND algorithm, this paper shows that it is possible to decode versatile QECCs that can adapt on-the-fly to changing conditions. The authors first assess the minimum number of gates required in the coding circuit needed to reach QRLCs' asymptotic performance and then propose a quantum GRAND algorithm that utilizes quantum noise statistics not only for building an adaptive code membership test but also for efficiently implementing syndrome decoding. Overall, this paper presents an innovative approach towards constructing and decoding QECCs with greater flexibility and adaptability than existing stabilizer codes. This could pave the way towards more efficient and reliable quantum communication and computation systems in the future.