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.
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.
Quantum parallel processing techniques are capable of solving certain complex problems at a substantially lower complexity than their classical counterparts. From the perspective of telecommunications, this quantum-domain parallel processing provides a plausible solution for achieving full-search based multi-stream detection, which is vital for future gigabit-wireless systems. The peculiar laws of quantum mechanics have also spurred interest in the absolutely secure quantum-based communication systems. Unfortunately, quantum decoherence imposes a hitherto insurmountable impairment on the practical implementation of quantum computation as well as on quantum communication systems, which may be overcome with the aid of efficient error correction codes. In this thesis, we design error correction codes for the quantum domain, which is an intricate journey from the realm of classical channel coding theory to that of the Quantum Error Correction Codes (QECCs). Since quantum-based communication systems are capable of supporting the transmission of both classical and quantum information, we initially focus our attention on the code design for entanglementassisted classical communication over the quantum depolarizing channel. We conceive an Extrinsic Information Transfer (EXIT) chart aided near-capacity classical-quantum code design, which invokes a classical Irregular Convolutional Code (IRCC) and a Unity Rate Code (URC) in conjunction with our proposed soft-decision aided SuperDense Code (SD). Hence, it is referred to as an ‘IRCC-URCSD’ arrangement. The proposed scheme is intrinsically amalgamated both with 2-qubit as well as 3-qubit SD coding protocols and it is benchmarked against the corresponding entanglement-assisted classical capacity. Since the IRCC-URC-SD scheme is a bit-based design, it incurs a capacity loss. As a further advance, we design a symbol based concatenated code design, referred to as a symbol-based ‘CC-URC-SD’, which relies on a single-component classical Convolutional Code (CC). Additionally, for the sake of reducing the associated decoding complexity, we also investigate the impact of the constraint length of the convolutional code on the achievable performance. Our initial designs, namely IRCC-URC-SD and CC-URC-SD, exploit redundancy in the classical domain. By contrast, QECCs relying on the quantum-domain redundancy are indispensable for conceiving a quantum communication system supporting the transmission of quantum information and also for quantum computing. Therefore, we next provide insights into the transformation from the family of classical codes to the class of quantum codes known as ‘Quantum Stabilizer Codes’ (QSC), which invoke the classical syndrome decoding. Particularly, we detail the underlying quantum-to classical isomorphism, which facilitates the design of meritorious families of QECCs from the known classical codes. We further study the syndrome decoding techniques operating over classical channels, which may be exploited for decoding QSCs. In this context, we conceive a syndrome-based block decoding approach for the classical Turbo Trellis Coded Modulation (TTCM), whose performance is investigated for transmission over an Additive White Gaussian Noise (AWGN) channel as well as over an uncorrelated Rayleigh fading channel. Pursuing our objective of designing efficient QECCs, we next consider the construction of Hashingbound-approaching concatenated quantum codes. In this quest, we appropriately adapt the conventional non-binary EXIT charts for Quantum Turbo Codes (QTCs) by exploiting the intrinsic quantumto- classical isomorphism. We further demonstrate the explicit benefit of our EXIT-chart technique for achieving a Hashing-bound-approaching code design. We also propose a generically applicable structure for Quantum Irregular Convolutional Codes (QIRCCs), which can be dynamically adapted to a specific application scenario with the aid of the EXIT charts. More explicitly, we provide a detailed design example by constructing a 10-subcode QIRCC and use it as an outer code in a concatenated quantum code structure for evaluating its performance. Working further in the direction of iterative code structures, we survey Quantum Low Density Parity Check (QLPDC) codes from the perspective of code design as well as in terms of their decoding algorithms. Furthermore, we propose a radically new class of high-rate row-circulant Quasi-Cyclic QLDPC (QC-QLDPC) codes, which can be constructed from arbitrary row-circulant classical QC LDPC matrices. We also conceive a modified non-binary decoding algorithm for homogeneous Calderbank-Shor-Steane (CSS)-type QLDPC codes, which is capable of alleviating the problems imposed by the unavoidable length-4 cycles. Our modified decoder outperforms the state-of-the-art decoders in terms of their Word Error Rate (WER) performance, despite imposing a reduced decoding complexity. Finally, we intricately amalgamate our modified decoder with the classic Uniformly-ReWeighted Belief Propagation (URW-BP) for the sake of achieving further performance improvement.
High detection complexity is the main impediment in future gigabit-wireless systems. However, a quantum-based detector is capable of simultaneously detecting hundreds of user signals by virtue of its inherent parallel nature. This, in turn, requires near-capacity quantum error correction codes for protecting the constituent qubits of the quantum detector against undesirable environmental decoherence. In this quest, we appropriately adapt the conventional nonbinary EXtrinsic Information Transfer (EXIT) charts for quantum turbo codes (QTCs) by exploiting the intrinsic quantum-to-classical isomorphism. The EXIT chart analysis not only allows us to dispense with the time-consuming Monte Carlo simulations but facilitates the design of near-capacity codes without resorting to the analysis of their distance spectra as well. We have demonstrated that our EXIT chart predictions are in line with the Monte Carlo simulation results. We have also optimized the entanglement-assisted QTC using EXIT charts, which outperforms the existing distance-spectra-based QTCs. More explicitly, the performance of our optimized QTC is as close as 0.3 dB to the corresponding hashing bound.
In this contribution, the Hashing bound of entanglement-assisted quantum channels is investigated in the context of quantum devices built from a range of popular materials, such as trapped ion and relying on solid state nuclear magnetic resonance, which can be modeled as a so-called asymmetric channel. Then, quantum error correction codes (QECC) are designed based on extrinsic information transfer charts for improving performance when employing these quantum devices. The results are also verified by simulations. Our QECC schemes are capable of operating close to the corresponding Hashing bound.
An expert discussion of the potential evolution of quantum codes In From Classical to Quantum Coding, a team of distinguished researchers deliver a seamless book on the subject of quantum error correction codes (QECC) designed for mitigating the environment-induced decoherence imposed on quantum computing and communications. Commencing from first principles, Part I is dedicated to readers familiar with classical coding and wishing to move into quantum coding. Part II focuses on near-term quantum codes requiring a modest to moderate number of qubits. Finally, Part III of the book offers an outlook on the classical to quantum evolution of QECCs, to advanced codes that rely on numerous qubits as quantum technology matures. The book incorporates several advanced topics, including the universal decoding of arbitrary linear codes, iterative short turbo block codes, turbo convolutional codes, and the family of low-density parity check codes. The powerful design tool of extrinsic information transfer charts plays a central role in the associated near-hashing-bound designs. Readers will also find: An easy-reading introduction to quantum information processing and quantum coding An evolutionary portrayal of the classical to quantum coding paradigm Practical discussions of near-term quantum topological error correction codes and how they protect quantum gates from decoherence Detailed treatments of syndrome-based decoding of diverse quantum turbo codes and quantum low-density parity check codes From Classical to Quantum Coding will benefit doctoral students, and industrial and academic researchers wishing to expand their expertise from the classical to the quantum field of signal processing, computing and communications.
Quantum-based transmission is an attractive solution conceived for achieving absolute security. In this quest, we have conceived an EXtrinsic Information Transfer (EXIT) chart aided channel code design for symbol-based entanglement-assisted classical communication over quantum depolarizing channels. Our proposed concatenated code design incorporates a Convolutional Code (CC), a symbol-based Unity Rate Code (URC) and a soft-decision aided 2-qubit Superdense Code (2SD), which is hence referred to as a CC-URC-2SD arrangement. We have optimized our design with the aid of non-binary EXIT charts. Our proposed design operates within 1 dB of the achievable capacity, providing attractive performance gains over its bit-based counterpart. Quantitatively, the bit-based scheme requires 60% more iterations than our symbol-based scheme for the sake of achieving perfect decoding convergence. Furthermore, we demonstrate that the decoding complexity can be reduced by using memory-2 and memory-3 convolutional codes, while still outperforming the bit-based approach.
Quantum stabilizer codes (QSCs) suffer from a low quantum coding rate since they have to recover the quantum bits (qubits) in the face of both bit-flip and phase-flip errors. In this treatise, we conceive a low-complexity concatenated quantum turbo code (QTC) design exhibiting a high quantum coding rate. The high quantum coding rate is achieved by combining the quantum-domain version of short-block codes (SBCs) also known as single parity check (SPC) codes as the outer codes and quantum unity-rate codes (QURCs) as the inner codes. Despite its design simplicity, the proposed QTC yields a near-hashing-bound error correction performance. For instance, compared to the best half-rate QTC known in the literature, namely the QIrCC-QURC scheme, which operates at the distance of D = 0.037 from the quantum hashing bound, our novel QSBC-QURC scheme can operate at the distance of D = 0.029. In addition, it is worth mentioning that this is the first instantiation of QTCs capable of adjusting the quantum encoders according to the quantum coding rate required for mitigating the Pauli errors, given the different depolarizing probabilities of the quantum channel.