# Near-Hashing-Bound Multiple-Rate Quantum Turbo Short-Block Codes

## TL;DR

This paper introduces a high-rate, low-complexity quantum turbo code that approaches the quantum hashing bound, improving error correction performance and adaptability for different quantum channel conditions.

## Contribution

It presents a novel concatenated quantum turbo code combining short-block codes and quantum unity-rate codes, achieving near-hashing-bound performance with adjustable encoding rates.

## Key findings

- Operates at a distance of 0.029 from the quantum hashing bound
- Outperforms the best known half-rate quantum turbo code
- First quantum turbo code with adjustable encoding rate

## Abstract

Quantum stabilizer codes (QSCs) suffer from a low quantum coding rate, since they have to recover the quantum bits (qubits) in the face of both bit-flip and phase-flip errors. In this treatise, we conceive a low-complexity concatenated quantum turbo code (QTC) design exhibiting a high quantum coding rate. The high quantum coding rate is achieved by combining the quantum-domain version of short-block codes (SBCs) also known as single parity check (SPC) codes as the outer codes and quantum unity-rate codes (QURCs) as the inner codes. Despite its design simplicity, the proposed QTC yields a near-hashing-bound error correction performance. For instance, compared to the best half-rate QTC known in the literature, namely the QIrCC-QURC scheme, which operates at the distance of $D = 0.037$ from the quantum hashing bound, our novel QSBC-QURC scheme can operate at the distance of $D = 0.029$. It is worth also mentioning that this is the first instantiation of QTCs capable of adjusting the quantum encoders according to the quantum coding rate required for mitigating the Pauli errors given the different depolarizing probabilities of the quantum channel.