Entanglement-assisted Quantum Error Correcting Code Saturating The Classical Singleton Bound

• URL: http://arxiv.org/abs/2410.04130v3
• Date: Sun, 13 Oct 2024 15:12:06 GMT
• Title: Entanglement-assisted Quantum Error Correcting Code Saturating The Classical Singleton Bound
• Authors: Soham Ghosh, Evagoras Stylianou, Holger Boche,
• Abstract summary: We introduce a construction for entanglement-assisted quantum error-correcting codes (EAQECCs) that saturates the classical Singleton bound with less shared entanglement than any known method for code rates below $ frackn = frac13 $. We demonstrate that any classical $[n,k,d]_q$ code can be transformed into an EAQECC with parameters $[n,k,d;2k]]_q$ using $2k$ pre-shared maximally entangled pairs.
• Score: 44.154181086513574
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We introduce a construction for entanglement-assisted quantum error-correcting codes (EAQECCs) that saturates the classical Singleton bound with less shared entanglement than any known method for code rates below $ \frac{k}{n} = \frac{1}{3} $. For higher rates, our EAQECC also meets the Singleton bound, although with increased entanglement requirements. Additionally, we demonstrate that any classical $[n,k,d]_q$ code can be transformed into an EAQECC with parameters $[[n,k,d;2k]]_q$ using $2k$ pre-shared maximally entangled pairs. The complexity of our encoding protocol for $k$-qudits with $q$ levels is $\mathcal{O}(k \log_{\frac{q}{q-1}}(k))$, excluding the complexity of encoding and decoding the classical MDS code. While this complexity remains linear in $k$ for systems of reasonable size, it increases significantly for larger-levelled systems, highlighting the need for further research into complexity reduction.

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