Quantum error-correcting codes from the quantum construction X

Jeonghwan Shin, Jun Heo, Todd A. Brun

In this paper, we introduce a unified framework to construct entanglement-assisted quantum error-correcting codes, including additive and nonadditive codes, based on the codeword stabilized framework on subsystems. The codeword stabilized (CWS) framework is a scheme to construct quantum error-correcting codes (QECCs) including both additive and nonadditive codes, and gives a method to construct a QECC from a classical error-correcting code in standard form. Entangled pairs of qubits (ebits) can be used to improve capacity of quantum error correction. In addition, it gives a method to overcome the dual-containing constraint. Operator quantum error correction (OQEC) gives a general framework to construct quantum error-correcting codes. We construct OQEC codes with ebits based on the CWS framework. This new scheme, entanglement-assisted operator codeword stabilized (EAOCWS) quantum codes, is the most general framework we know of to construct both additive and nonadditive codes from classical error-correcting codes. We describe the formalism of our scheme, demonstrate the construction with examples, and give several EAOCWS codes.

Akshay Degwekar, Kenza Guenda, Thomas Aaron Gulliver

Min-Hsiu Hsieh, Igor Devetak, Todd A. Brun

Entanglement-assisted quantum error-correcting codes (EAQECCs) make use of preexisting entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQECC from any classical linear code, unlike standard QECCs, which can only be constructed from dual-containing codes. Operator quantum error-correcting codes allow certain errors to be corrected (or prevented) passively, reducing the complexity of the correction procedure. We combine these two extensions of standard quantum error correction into a unified entanglement-assisted quantum error-correction formalism. This new scheme, which we call entanglement-assisted operator quantum error correction (EAOQEC), is the most general and powerful quantum error-correcting technique known, retaining the advantages of both entanglement-assistance and passive correction. We present the formalism, show the considerable freedom in constructing EAOQECCs from classical codes, and demonstrate the construction with examples.

Akshay Degwekar, Kenza Guenda, Thomas Aaron Gulliver

Yue RAN, Juan XU, Hanwu Chen +2 more

Since Calderbank et al. established CRSS construction, which constructs quantum error-correcting codes (QECC) from classical error-correcting codes, many QECC demonstrating good performance have been constructed from classical error-correcting codes; these codes are referred to as quantum stabilizer codes. Recent physical experiments indicate that the probabilities of occurrence of qubit- ip errors are much less than those of phase- ip errors in most quantum mechanical systems. Error-correcting codes thus constructed are referred to as asymmetric quantum codes. In this paper, series of asymmetric quantum stabilizer codes are constructed using nested Goppa codes and dual contained Goppa codes. In the construction of asymmetric quantum codes based on nested Goppa codes, the choice of Goppa codes is limited to construct codes with analysis methods. In normal circumstances, the construction is executed with the help of mathematical software Matlab, which is used to compute the minimum distance of Goppa codes. In the construction of asymmetric quantum codes based on dual contained Goppa codes, the error correction ability of the codes constructed in this paper is re ected in the correction of Z-type errors

Bei Zeng, Xie Chen, D. L. Zhou +1 more