Additive Asymmetric Quantum Codes

Abstract

We present a general construction of asymmetric quantum codes based on additive codes under the trace Hermitian inner product. Various families of additive codes over $\F_{4}$ are used in the construction of many asymmetric quantum codes over $\F_{4}$.Comment: Accepted for publication March 2, 2011, IEEE Transactions on Information Theory, to appea

Asymmetric quantum codes: new codes from old

“…In [32], the Giuliano construction of nonadditive AQECC as well as constructions of asymptotically good AQECC derived from algebraic-geometry codes were presented. In [9], the Calderbank-Shor-Steane (CSS) construction [5,16,24] was extended to include codes endowed with the Hermitian and also trace Hermitian inner product. In [8], asymmetric quantum MDS codes derived from generalized Reed-Solomon (GRS) codes were constructed.…”

A Survey on Quantum Channel Capacities

“…By contrast, the materials considered at the time of writing for building quantum devices, including trapped ions [460] and solid state Nuclear Magnetic Resonance [519], exhibit asymmetric depolarization property defined as the ratio of the phase flip probability over the bit flip probability, where the grade of asymmetry is in the range spanning from α = 10 2 to α = 10 6 [333][334][335][336][337]. QECCs designed for the asymmetric depolarizing channel were termed as asymmetric QECCs in [338][339][340][341][342][343], where a limited range of α values was assumed and no entanglement assistance was addressed. In [344], a more general framework covering both symmetric and asymmetric depolarizing channels was proposed for Entanglement Assisted QECCs (EAQECCs).…”

Asymmetric Entanglement-Assisted Quantum Error-Correcting Codes and BCH Codes

“…The fact that dephasing usually happens much more often that relaxation [27] motivated the study and searching of asymmetric quantum error-correcting codes [14,15,16,30,31,32,33]. For this purpose, the most used procedure is the CSS construction because it easily allows us to get parameters d z and d x such that our previous stabilizer code detects phase-shift (respectively, qudit-flip) errors up to weight d z − 1 (respectively, d x − 1).…”