Quantum Error-Correcting Codes

Daiki Ueno, Ryutaroh Matsumoto

In the previous research by Grassl, Huber and Winter, they proved a theorem which can make entanglement-assisted quantum error-correcting codes (EAQECC) from general quantum error-correcting codes (QECC). In this paper, we prove that the shortened EAQECC is a stabilizer code if the original EAQECC is a stabilizer code.

Ian F. Blake

Giuliano G. La Guardia

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.

Shigeyori Nagahashi, Tsuyoshi Sasaki Usuda, Ichi Takumi

In [6], the 2-EPPs using quantum error correcting codes (QECCs) and binary search were proposed and shown to have a high purification capability. This protocol maintains a balance between error-detection and correction capabilities by using QECCs to detect errors and binary search to correct a single error. However, when we construct a protocol from codes whose code-length is long, sharing entanglement efficiently is difficult because of an incapability to perform error correction despite an error-detection capability. In this paper, we propose a method to improve the capability of error correction by using binary search more than once. We also show the performance of the protocol, which consists of a (16, 6, 4) stabilizer code, and show that this method is useful in improving performance.

Dan Cristian Marinescu, Gabriela M. Marinescu