Abstract
Quantum error correction codes (QECCs) can be constructed from the known classical coding paradigm by exploiting the inherent isomorphism between the classical and quantum regimes, while also addressing the challenges imposed by the strange laws of quantum physics. In this spirit, this paper provides deep insights into the duality of quantum and classical coding theory, hence aiming for bridging the gap between them. Explicitly, we survey the rich history of both classical as well as quantum codes. We then provide a comprehensive slow-paced tutorial for constructing stabilizer-based QECCs from arbitrary binary as well as quaternary codes, as exemplified by the dual-containing and non-dual-containing Calderbank–Shor–Steane (CSS) codes, non-CSS codes and entanglement-assisted codes. Finally, we apply our discussions to two popular code families, namely to the family of Bose–Chaudhuri–Hocquenghem as well as of convolutional codes and provide detailed design examples for both their classical as well as their quantum versions.
Highlights
I F COMPUTERS that you build are quantum, spies everywhere will all want ’em
Based on the duality of Quantum Stabilizer Codes (QSCs) and classical linear block codes established in Section V, we present the isomorphism between these two regimes, which in turn helps in constructing the quantum-domain versions of the known classical codes
Continuing further our discussions, we present the taxonomy of stabilizer codes with the aid of Fig. 28, which is based on the structure of the underlying equivalent classical Parity Check Matrix (PCM) H
Summary
I F COMPUTERS that you build are quantum, spies everywhere will all want ’em. Our codes will all fail, And they’ll read our e-mail, Till we get crypto that’s quantum, and daunt ’em. Similar to the classical error correction codes, QECCs redress the perturbations resulting from quantum impairments, enabling qubits to retain their coherent quantum states for longer durations with a high probability. This has been experimentally demonstrated in [35]–[37].
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