The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure

Abstract

Abstract-Powerful Quantum Error Correction Codes (QECCs) are required for stabilizing and protecting fragile qubits against the undesirable effects of quantum decoherence. Similar to classical codes, hashing bound approaching QECCs may be designed by exploiting a concatenated code structure, which invokes iterative decoding. Therefore, in this paper we provide an extensive step-by-step tutorial for designing EXtrinsic Information Transfer (EXIT) chart aided concatenated quantum codes based on the underlying quantum-to-classical isomorphism. These design lessons are then exemplified in the context of our proposed Quantum Irregular Convolutional Code (QIRCC), which constitutes the outer component of a concatenated quantum code. The proposed QIRCC can be dynamically adapted to match any given inner code using EXIT charts, hence achieving a performance close to the hashing bound. It is demonstrated that our QIRCC-based optimized design is capable of operating within 0.4 dB of the noise limit.

Serially Concatenated Unity-Rate Codes Improve Quantum Codes Without Coding-Rate Reduction

“…13 N EAR-CAPACITY concatenated coding and detection 14 schemes [1], which rely on the low-complexity itera- tional Code (QIRCC) structure was proposed in [5].…”

“…1 are stabilizer codes [5], [10], which 130 belong to the family of Clifford operators [11]. A distinct 131 feature of the Clifford family is that they preserve the elements 132 of the Pauli group under conjugation.…”

“…135 Consequently, the stabilizer generators of the encoder V are 136 intrinsically commutative. Such an encoder V may be real-137 ized using the Hadamard (H), phase (S) and controlled-NOT 138 (C-NOT) gates, which are defined as [5]:…”

“…proposed QURC, we designed a 1/2 rate concatenated code 194 using the 10-subcode QIRCC of [5] as the outer component.…”

“…QURC merely acts 77 as a precoder invoked for supporting iterative decoding. The 78 precoded stream |ψ 2 is transmitted through a quantum depo-79 larizing channel, which is a commonly used channel model [2], 80 [3], [5]. Explicitly, a quantum depolarizing channel having a 81 depolarizing probability p inflicts an n-qubit Pauli error P 2 , 82 1558-2558 © 2016 IEEE.…”