Quantum error correction codes (QECCs) play a central role in both quantum communications and quantum computation. Practical quantum error correction codes, such as stabilizer codes, are generally structured to suit a specific use, and present rigid code lengths and code rates. This paper shows that it is possible to both construct and decode QECCs that can attain the maximum performance of the finite blocklength regime, for any chosen code length when the code rate is sufficiently high. A recently proposed strategy for decoding classical codes called GRAND (guessing random additive noise decoding) opened doors to efficiently decode classical random linear codes (RLCs) performing near the maximum rate of the finite blocklength regime. By using noise statistics, GRAND is a noise-centric efficient universal decoder for classical codes, provided that a simple code membership test exists. These conditions are particularly suitable for quantum systems, and therefore this work extends 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 and a newly proposed quantum GRAND, this work shows that it is possible to decode QECCs that are easy to adapt to changing conditions. Our work starts by assessing the minimum number of gates in the coding circuit needed to reach the QRLCs' asymptotic performance, and subsequently proposes a quantum GRAND algorithm that makes use of quantum noise statistics, not only to build an adaptive code membership test, but also to efficiently implement syndrome decoding.

Prof. Frank Kschischang (University of Toronto), Dr. Ioannis Chatzigeorgiou (Lancaster University), Dr. Bill Munro (NTT Basic Research Labs, Japan) and Prof. Kae Nemoto (National Institute of Informatics, Japan).

GRAND,ML decoding,quantum error correction codes,short codes,syndrome decoding

• Computer and Information Sciences - Natural Sciences
• Electrical Engineering, Electronic Engineering, Information Engineering - Engineering and Technology