Abstract
We prove that the known formulae for computing the optimal number of maximally entangled pairs required for entanglement-assisted quantum error-correcting codes (EAQECCs) over the binary field hold for codes over arbitrary finite fields as well. We also give a Gilbert–Varshamov bound for EAQECCs and constructions of EAQECCs coming from punctured self-orthogonal linear codes which are valid for any finite field.
Highlights
The Shor’s proposal of using quantum error correction for reducing decoherence in quantum computation [24] and his polynomial-time algorithms for prime factorization and discrete logarithms on quantum computers [25] clearly illustrate the feasibility and importance of quantum computation and quantum error correction.Most of the quantum error-correcting codes (QECCs) come from classical codes
QECCs of length n over Fq can be derived from classical self-orthogonal codes with respect to the Hermitian inner product included in Fqn2 and from codes in Fqn which are self-orthogonal with respect to the Euclidean inner product
With this new formalism, entanglementassisted quantum stabilizer codes can be constructed from any classical linear code giving rise to entanglement-assisted quantum error-correcting codes (EAQECCs)
Summary
The Shor’s proposal of using quantum error correction for reducing decoherence in quantum computation [24] and his polynomial-time algorithms for prime factorization and discrete logarithms on quantum computers [25] clearly illustrate the feasibility and importance of quantum computation and quantum error correction. Brun et al [3] proposed to share entanglement between encoder and decoder to simplify the theory of quantum error correction and increase the communication capacity With this new formalism, entanglementassisted quantum stabilizer codes can be constructed from any classical linear code giving rise to entanglement-assisted quantum error-correcting codes (EAQECCs). This paper is devoted to prove formulae for the minimum required number c of pairs of maximally entangled quantum states, corresponding to EAQECCs codes obtained from linear codes C over any finite field, by using symplectic forms, or Hermitian or Euclidean inner products. Since fewer qudits should be transmitted through a noisy channel, they perform better Constructions of this type have been considered in the binary case for giving a coding scheme with imperfect ebits [15].
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