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

TL;DR

This paper presents a new entanglement-assisted quantum error-correcting code construction that saturates the classical Singleton bound with minimal entanglement, applicable across various code rates, and introduces an efficient encoding protocol.

Contribution

The authors develop a construction for EAQECCs that saturate the Singleton bound with less entanglement than existing methods, and show how any classical code can be transformed into an EAQECC with specific parameters.

Findings

Codes saturate the Singleton bound with minimal entanglement for certain rates.

Any classical [n,k,d]_q code can be transformed into an EAQECC with parameters [[n,k,d;2k]]_q.

Encoding complexity is linear in k for moderate systems, but increases for larger systems.

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.