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.
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.
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.
Noise is one of the central obstacles to building useful quantum computers, and quantum error correction (QEC) provides the framework for protecting quantum information against it. Unlike classical error correction, QEC must preserve fragile quantum states without copying them, measuring them directly, or destroying the information they encode. Driven by rapid progress in both theory and experiment, this challenge has grown into one of the most active areas of quantum information science. This tutorial gives a guided introduction to modern QEC, developing the core concepts of codes, syndromes, stabilizers, decoding, and fault tolerance before connecting them to major code families and current research directions. We cover both established constructions and newer developments, including topological and subsystem codes, bosonic and qudit codes, dynamical codes, and quantum low-density parity-check (qLDPC) codes. The emphasis is on building operational understanding: explaining not only what the main objects are, but how they are used in code design, error diagnosis, decoding, and fault-tolerant computation. The tutorial is intended for newcomers seeking a first path through QEC, as well as researchers looking for a coherent reference for the concepts, code families, and tools that arise in current work.
Quantum computing has emerged as one of the most transformative technological developments of the twenty-first century. By exploiting the fundamental principles of quantum mechanics—namely superposition, entanglement, and measurement—quantum computers offer the potential to address problems that are beyond the practical reach of classical computing systems. As the field advances from theoretical exploration to real-world implementation, the demand for clear, structured, and comprehensive educational resources continues to grow. This book presents a systematic and accessible introduction to quantum computing, beginning with the essential concepts of classical information and computation and progressively advancing toward modern quantum algorithms and communication protocols. The initial chapters establish a strong foundation in classical bits, logic gates, Boolean algebra, reversible logic, and computational complexity, providing the conceptual bridge necessary to understand quantum computation. The discussion then moves to the quantum bit (qubit), examining its behavior through superposition, measurement, Bloch sphere representation, and physical realization. Special emphasis is placed on linear algebra as the mathematical framework underpinning quantum computing. Through this approach, readers are guided to develop an intuitive yet rigorous understanding of quantum states, quantum gates, and circuit-based computation. The text further extends these ideas to multi-qubit systems, quantum entanglement, universal gate sets, and quantum error correction, integrating theoretical principles with practical insights. To support experiential learning, the book incorporates hands-on exposure to leading quantum programming platforms, including IBM Quantum, OpenQASM, and Qiskit. These sections connect theory with practice, enabling readers to design, simulate, and execute quantum circuits on both simulators and real quantum hardware. Subsequent chapters explore key quantum communication protocols such as superdense coding, quantum teleportation, and quantum key distribution, followed by an in-depth treatment of foundational quantum algorithms including Deutsch–Jozsa, Grover’s search, Simon’s algorithm, and Shor’s factoring algorithm. This text is intended for undergraduate and postgraduate students, research scholars, and industry professionals seeking a coherent and structured pathway into quantum computing. No prior knowledge of quantum mechanics is assumed, although familiarity with linear algebra and classical computation will enhance comprehension. Each chapter is carefully organized to promote conceptual clarity, supported by illustrative examples, circuit diagrams, and concise summaries. Beyond explaining the principles and mechanisms of quantum computation, this book aims to inspire readers to engage with the future of the field—whether through academic research, industrial application, or interdisciplinary innovation. As quantum technologies continue to evolve, it is hoped that this text will serve as a dependable guide and a meaningful entry point into the quantum era.
Quantum computing has emerged as one of the most transformative technological developments of the twenty-first century. By exploiting the fundamental principles of quantum mechanics—namely superposition, entanglement, and measurement—quantum computers offer the potential to address problems that are beyond the practical reach of classical computing systems. As the field advances from theoretical exploration to real-world implementation, the demand for clear, structured, and comprehensive educational resources continues to grow. This book presents a systematic and accessible introduction to quantum computing, beginning with the essential concepts of classical information and computation and progressively advancing toward modern quantum algorithms and communication protocols. The initial chapters establish a strong foundation in classical bits, logic gates, Boolean algebra, reversible logic, and computational complexity, providing the conceptual bridge necessary to understand quantum computation. The discussion then moves to the quantum bit (qubit), examining its behavior through superposition, measurement, Bloch sphere representation, and physical realization. Special emphasis is placed on linear algebra as the mathematical framework underpinning quantum computing. Through this approach, readers are guided to develop an intuitive yet rigorous understanding of quantum states, quantum gates, and circuit-based computation. The text further extends these ideas to multi-qubit systems, quantum entanglement, universal gate sets, and quantum error correction, integrating theoretical principles with practical insights. To support experiential learning, the book incorporates hands-on exposure to leading quantum programming platforms, including IBM Quantum, OpenQASM, and Qiskit. These sections connect theory with practice, enabling readers to design, simulate, and execute quantum circuits on both simulators and real quantum hardware. Subsequent chapters explore key quantum communication protocols such as superdense coding, quantum teleportation, and quantum key distribution, followed by an in-depth treatment of foundational quantum algorithms including Deutsch–Jozsa, Grover’s search, Simon’s algorithm, and Shor’s factoring algorithm. This text is intended for undergraduate and postgraduate students, research scholars, and industry professionals seeking a coherent and structured pathway into quantum computing. No prior knowledge of quantum mechanics is assumed, although familiarity with linear algebra and classical computation will enhance comprehension. Each chapter is carefully organized to promote conceptual clarity, supported by illustrative examples, circuit diagrams, and concise summaries. Beyond explaining the principles and mechanisms of quantum computation, this book aims to inspire readers to engage with the future of the field—whether through academic research, industrial application, or interdisciplinary innovation. As quantum technologies continue to evolve, it is hoped that this text will serve as a dependable guide and a meaningful entry point into the quantum era.
Quantum Computing The twenty-first century has witnessed remarkable advancements in computing technologies, transforming every aspect of modern society. As conventional computing systems approach their physical and computational limits, the search for new paradigms capable of solving increasingly complex problems has led to the emergence of Quantum Computing. By harnessing the principles of quantum mechanics, quantum computing represents a revolutionary approach that has the potential to redefine computation, communication, and information processing on an unprecedented scale. Quantum Computing is designed to provide readers with a comprehensive understanding of the concepts, principles, technologies, and applications that form the foundation of this rapidly evolving field. This book aims to bridge the gap between theoretical quantum mechanics and practical computational applications, enabling students, researchers, academicians, and professionals to explore the transformative potential of quantum technologies. The chapters in this book cover a broad spectrum of topics, including the fundamentals of quantum mechanics, quantum bits (qubits), quantum gates, superposition, entanglement, quantum circuits, quantum algorithms, quantum cryptography, quantum communication, and quantum error correction. The book also explores the architecture of quantum computers and examines the challenges involved in building scalable and fault-tolerant quantum systems. As industries increasingly seek innovative solutions to complex computational problems, quantum computing is emerging as a promising technology with applications in cryptography, artificial intelligence, machine learning, optimization, financial modeling, healthcare, pharmaceutical research, materials science, logistics, and climate modeling. The ability of quantum computers to process vast amounts of information simultaneously offers opportunities that were previously beyond the reach of classical computing systems. Special attention has been given to contemporary developments in quantum technologies, including quantum supremacy, hybrid quantum-classical computing, cloud-based quantum platforms, and the growing role of quantum computing in scientific discovery and industrial innovation. The book also discusses ethical, security, and societal implications associated with the advancement of quantum technologies. The primary objective of this book is to equip readers with both foundational knowledge and practical insights into the field of quantum computing. Through a balanced presentation of theory, applications, case studies, and emerging research trends, the book seeks to inspire curiosity, innovation, and further exploration in this transformative domain. We extend our sincere gratitude to the authors, reviewers, researchers, academicians, and industry experts whose valuable contributions have enriched the quality and relevance of this publication. Their dedication and expertise have helped create a resource that reflects the latest developments and future possibilities of quantum computing. We hope that Quantum Computing will serve as a valuable guide for students, educators, researchers, and technology professionals seeking to understand the next frontier of computational science. May this book inspire new ideas, foster innovation, and contribute to the advancement of knowledge in one of the most exciting technological revolutions of our time. Mr. M. Rajeshkumar Dr. R. Mariappan Mr. E.Janakiraman,
The employment of quantum error correction codes (QECCs) within quantum computers potentially offers a reliability improvement for both quantum computation and communications tasks. However, the laws of quantum mechanics prevent us from directly invoking the mature family of classical error correction codes in the quantum domain. In order to circumvent the associated problems, the notion of quantum stabilizer codes (QSCs) was proposed in conjunction with syndrome-based decoding in the quantum regime. However, most of the powerful QSC schemes require long codewords for achieving a high performance, which potentially imposes additional challenges for implementation concerning their implementation, since their decoding may require longer than the quantum circuit's coherence time. Hence at the time of writing, QSCs exhibiting short to moderate codeword lengths are preferable. We commence by describing the pivotal problem encountered by classical error-correction codes, which also emerges when designing the QSCs, namely the intrinsic trade-off between the minimum distance versus coding rate. The complete formulation of this particular trade-off does not exist, but several lower and upper bounds can be found in the literature. It has been shown that a substantial gap can be observed between the upper and lower bound of the minimum distance, given the codeword length and the quantum coding rate. Hence, we propose an appealingly simple and invertible analytical approximation, for characterizing the trade-off between the quantum coding rate and the minimum distance of QSCs as well as their corresponding quantum bit error rate (QBER) performance upper-bound. For example, for a half-rate QSC having a codeword length of n = 128, the minimum distance is bounded by 11 < d < 22, while our approximation yields a minimum distance of d = 17 for the above-mentioned code. Next, we link this parametric study of the minimum distance versus quantum coding rate to the popular QSCs, namely to the family of quantum topological error correction codes (QTECCs). In order to construct the classical-to-quantum isomorphism, we conceive and investigate the family of classical topological error correction codes (TECCs), assuming that the bits of a codeword can be arranged in a lattice structure. We then present the classical-to-quantum isomorphism to pave the way for constructing their dual pairs in the quantum domain, which are the QTECCs. Finally, we characterize the performance of QTECCs in the face of the quantum depolarizing channel in terms of both their QBER and fidelity. Specifically, we demonstrate that for quantum coding rate rQ ~ 0, the threshold probability of the QBER below which the colour, rotated-surface, surface, and toric codes become capable of improving the uncoded QBER are given by 1.8 ×10-2, 1.3 ×10-2, 6.3 ×10-2 and 6.8 ×10-2, respectively. Furthermore, we also demonstrate that we can achieve beneficial fidelity improvements above the minimum fidelity of 0.94, 0.97 and 0.99 by employing the rQ = 1/7 colour code, the rQ = 1/9 rotated-surface code, and the rQ = 1/13 surface code, respectively. However, QSCs require additional quantum gates for their employment. Incorporating more quantum gates for performing error correction potentially introduces further sources of quantum decoherence into quantum computers. In this scenario, the primary challenge is to find the sufficient condition required by each of the quantum gates for beneficially employing QECCs in order to yield reliability improvements, given that the quantum gates utilized by the QECCs also introduce quantum decoherence. In this treatise, we approach this problem by firstly presenting the general framework of protecting quantum gates by the amalgamation of the transversal configuration of quantum gates and QSCs, which can be viewed as syndrome-based QECCs. Secondly, we provide examples of the advocated framework by invoking QTECCs for protecting both transversal Hadamard gates and controlled (CNOT) gates. Both our simulation and analytical results explicitly show that by utilizing QTECCs, the fidelity of the quantum gates can be beneficially improved, provided that quantum gates satisfying a certain minimum depolarization fidelity threshold (Fth) are available. For instance, for protecting transversal Hadamard gates, the minimum fidelity values required for each of the gates in order to attain fidelity improvements are 99.74%, 99.73%, 99.87%, and 99.86%, when they are protected by colour, rotated-surface, surface, and toric codes, respectively. Unfortunately, these specific Fth values can only be obtained for a very large number of physical qubits (n → ∞), when the quantum coding rate of the QTECCs approaches zero (rQ → 0). Finally, in order to conceive QSCs exhibiting a high quantum coding rate, we modify the construction of QTECCs for conceiving a low-complexity concatenated quantum turbocode (QTC). The above-mentioned high quantum coding rate is obtained 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 quantum irregular convolutional codes (QIrCCs) combined with the 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 lower distance of D = 0:029. It is also 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 imposed by the time-variant depolarizing probabilities of the quantum channel.