Codes for the Quantum Erasure Channel

TL;DR

This paper studies the quantum erasure channel, where the locations of errors are known, and shows that four qubits are both necessary and sufficient to encode one qubit and correct a single erasure, unlike five qubits for arbitrary errors. It provides a concrete four-qubit QEC code and proves minimality, then constructs a family of QEC codes from binary BCH codes (Quantum BCH codes) with efficient erasure decoding. The results highlight how knowledge of error positions enables more compact codes and offer practical pathways toward near-term demonstrations of quantum error correction.

Abstract

The quantum erasure channel (QEC) is considered. Codes for the QEC have to correct for erasures, i. e., arbitrary errors at known positions. We show that four qubits are necessary and sufficient to encode one qubit and correct one erasure, in contrast to five qubits for unknown positions. Moreover, a family of quantum codes for the QEC, the quantum BCH codes, that can be efficiently decoded is introduced.