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arXiv:2205.13732

Explicit method to make shortened stabilizer EAQECC from stabilizer QECC

Daiki Ueno, Ryutaroh Matsumoto

Published May 27, 2022• 2 citations
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Abstract

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.
DOI
10.48550/arxiv.2205.13732

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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.
Entanglement-assisted quantum error-correcting codes (EAQECCs) make use of pre-existing 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 (OQECCs) 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.
The entanglement-assisted (EA) formalism is a generalization of the standard stabilizer formalism, and it can transform arbitrary quaternary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using of shared entanglement between the sender and the receiver. Using the special structure of linear EAQECCs, we derive an EA-Plotkin bound for linear EAQECCs, which strengthens the previous known EA-Plotkin bound. This linear EA-Plotkin bound is tighter then the EA-Singleton bound, and matches the EA-Hamming bound and the EA-linear programming bound in some cases. We also construct three families of EAQECCs with good parameters. Some of these EAQECCs saturate this linear EA-Plotkin bound and the others are near optimal according to this bound; almost all of these linear EAQECCs are degenerate codes.
The theory of entanglement-assisted quantum error-correcting codes (EAQECCs) is a generalization of the standard stabilizer quantum error-correcting codes, which can be possibly constructed from any classical codes by relaxing the duality condition and utilizing preshared entanglement between the sender and receiver. In this paper, a new family of EAQECCs is constructed from negacyclic codes of length n=q2+1an=\frac{q^2+1}{a}n=aq2+1​, where qqq is an odd prime power, a=m2+12a=\frac{m^2+1}{2}a=2m2+1​ and mmm is an odd integer. Some new entanglement-assisted quantum maximum distance separable (EAQMDS) codes are obtained in the sense that their parameters are not covered by the previously known ones.
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