A Family of Quantum Stabilizer Codes Based on the Weyl Commutation Relations over a Finite Field

Abstract

Using the Weyl commutation relations over a finite field \({\mathbb{F}_q}\) we introduce a family of error-correcting quantum stabilizer codes based on a class of symmetric matrices over \({\mathbb{F}_q}\) satisfying certain natural conditions. When q = 2 the existence of a rich class of such symmetric matrices is demonstrated by a simple probabilistic argument depending on the Chernoff bound for i.i.d. symmetric Bernoulli trials. If, in addition, these symmetric matrices are assumed to be circulant it is possible to obtain concrete examples by a computer program. The quantum codes thus obtained admit elegant encoding circuits.