Nonbinary Stabilizer Codes Over Finite Fields

By Avanti Ketkar, Andreas Klappenecker, Santosh Kumar, Pradeep Kiran Sarvepalli. IEEE Transactions on Information Theory, 2005

597 citations recorded by the source corpus.

One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. To address this difficulty, many good quantum error-correcting codes have been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over Fq in terms of classical codes over Fq 2 is provided that generalizes the well-known notion of additive codes over F4 of the binary case. This paper also derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum Bose-Chaudhuri-Hocquenghem (BCH) codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper

Subfields: Artificial Intelligence, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/cf5213547e71ab230197f750b04a949a59ef1b6c

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Citations in the corpus

On Quantum and Classical BCH Codes (2006, background): “Proof. Part b) is a special case of a) which is commonly referred to as the CSS construction and part c) is the Hermitian code construction, see, for instance, [ 13 ] for the proofs.”
Quantum channels and memory effects (2012, background): “For non binary codes the reader may refer to (Ashikhmin and Knill, 2001; Ketkar et al., 2006; Knill, 1996).”
Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer (2020, background): “In analogy to the qubit and qudit stabilizer formalisms [51–53], CV quantum computation also possesses an efficiently simulable sub-theory, Gaussian computation [44].”