Quantum Data-Syndrome Codes

• Published in: IEEE journal on selected areas in communications Vol. 38; no. 3; pp. 449 - 462
• Main Authors: Ashikhmin, Alexei, Lai, Ching-Yi, Brun, Todd A.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.03.2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Atmospheric measurements; Circuit reliability; Codes; Data-syndrome (DS) codes; Disorders; Error analysis; Error correction; Generators; Golay codes; Linear programming; linear programming bounds; macWilliams identities; Measurement uncertainty; Particle measurements; quantum codes; quantum syndrome errors; Qubit; Qubits (quantum computing); stabilizer codes; Upper bounds
• ISSN: 0733-8716; 1558-0008
• DOI: 10.1109/JSAC.2020.2968997

Performing active quantum error correction to protect fragile quantum states highly depends on the correctness of measured error syndromes. To obtain reliable error syndromes using imperfect physical circuits, we propose syndrome measurement (SM) and quantum data-syndrome (DS) codes. SM codes protect syndrome with linearly dependent redundant stabilizer measurements. DS codes generalize this idea for simultaneous correction of 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 the minimum distance of 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. Finally, we propose a family of CSS-type quantum DS codes based on classical cyclic codes, which include the Steane code and the quantum Golay code.