Catalytic quantum error correction

TL;DR

This paper introduces entanglement-assisted quantum error correcting codes (EAQEC), expanding the stabilizer formalism to include pre-shared entanglement, enabling the use of classical codes like LDPC for quantum error correction.

Contribution

It generalizes stabilizer codes to EAQEC, allowing the use of any classical code, including capacity-approaching LDPC codes, for quantum error correction.

Findings

EAQEC codes do not require dual-containing condition.

Any classical quaternary code can be transformed into an EAQEC code.

EAQEC codes can attain the hashing bound with LDPC codes.

Abstract

We develop the theory of entanglement-assisted quantum error correcting (EAQEC) codes, a generalization of the stabilizer formalism to the setting in which the sender and receiver have access to pre-shared entanglement. Conventional stabilizer codes are equivalent to dual-containing symplectic codes. In contrast, EAQEC codes do not require the dual-containing condition, which greatly simplifies their construction. We show how any quaternary classical code can be made into a EAQEC code. In particular, efficient modern codes, like LDPC codes, which attain the Shannon capacity, can be made into EAQEC codes attaining the hashing bound. In a quantum computation setting, EAQEC codes give rise to catalytic quantum codes which maintain a region of inherited noiseless qubits. We also give an alternative construction of EAQEC codes by making classical entanglement assisted codes coherent.