# Quantum Data-Syndrome Codes

## TL;DR

This paper introduces quantum data-syndrome (DS) codes that correct errors in both data qubits and syndrome bits, providing theoretical bounds, properties, and constructions to enhance quantum error correction reliability.

## Contribution

It develops the theory of quantum DS codes, including bounds, properties, and new constructions based on classical codes, improving error correction in quantum computing.

## Key findings

- Quantum DS codes can correct both data and syndrome errors.
- Random DS codes achieve the Gilbert-Varshamov bound.
- Constructed DS codes include CSS-type codes like Steane and Golay codes.

## Abstract

Performing active quantum error correction to protect fragile quantum states highly depends on the correctness of error information--error syndromes. To obtain reliable error syndromes using imperfect physical circuits, we propose the idea of quantum data-syndrome (DS) codes that are capable of correcting both data qubits and syndrome bits errors. We study fundamental properties of quantum DS codes, including split weight enumerators, generalized MacWilliams identities, and linear programming bounds. In particular, we derive Singleton and Hamming-type upper bounds on degenerate quantum DS codes. Then we study random DS codes and show that random DS codes with a relatively small additional syndrome measurements achieve the Gilbert-Varshamov bound of stabilizer codes. Constructions of quantum DS codes are also discussed. A family of quantum DS codes is based on classical linear block codes, called syndrome measurement codes, so that syndrome bits are encoded in additional redundant stabilizer measurements. Another family of quantum DS codes is CSS-type quantum DS codes based on classical cyclic codes, and this includes the Steane code and the quantum Golay code.