A quantum computer is a device capable of performing computational tasks that depend on characteristically quantum mechanical effects, in particular coherent quantum superposition. Such devices can efficiently perform classes of computation (e.g. factorisation) which are believed to be intractable on any classical computer. This makes it highly desirable to construct such devices. In this paper, we address the last remaining theoretical obstacle to such a construction, namely the problem of stability or error correction. This problem is more substantial in quantum computation than in classical computation because of the delicate nature of the interference phenomena on which quantum computation depends. We present a new, purely quantum mechanical method of error correction, which has no classical analogue, but can serve to stabilise coherent quantum computations. Like the classical methods, it utilises redundancy, but it does not depend on measuring intermediate results of the computation.>
A quantum computer is a device capable of performing computational tasks that depend on characteristically quantum mechanical effects, in particular coherent quantum superposition. Such devices can efficiently perform classes of computation (e.g. factorisation) which are believed to be intractable on any classical computer. This makes it highly desirable to construct such devices. In this paper, we address the last remaining theoretical obstacle to such a construction, namely the problem of stability or error correction. This problem is more substantial in quantum computation than in classical computation because of the delicate nature of the interference phenomena on which quantum computation depends. We present a new, purely quantum mechanical method of error correction, which has no classical analogue, but can serve to stabilise coherent quantum computations. Like the classical methods, it utilises redundancy, but it does not depend on measuring intermediate results of the computation.>
Recent theoretical results confirm that quantum theory provides the possibility of new ways of performing efficient calculations. The most striking example is the factoring problem. It has recently been shown that computers that exploit quantum features could factor large composite integers. This task is believed to be out of reach of classical computers as soon as the number of digits in the number to factor exceeds a certain limit. The additional power of quantum computers comes from the possibility of employing a superposition of states, of following many distinct computation paths and of producing a final output that depends on the interference of all of them. This ‘quantum parallelism’ outstrips by far any parallelism that can be thought of in classical computation and is responsible for the ‘exponential’ speed-up of computation. Experimentally, however, it will be extremely difficult to ‘decouple’ a quantum computer from its environment. Noise fluctuations due to the outside world, no matter how little, are sufficient to drastically reduce the performance of these new computing devices. To control the nefarious effects of this environmental noise, one needs to implement efficient error-correcting techniques.
Quantum computers herald the arrival of a new era in which previously intractable computational problems will be solved efficiently. However, quantum technology is held down by decoherence, a phenomenon that is omnipresent in the quantum paradigm and that renders quantum information useless when left unchecked. The science of quantum error correction, a discipline that seeks to combine and protect quantum information from the effects of decoherence using structures known as codes, has arisen to meet this challenge. Stabilizer codes, a particular subclass of quantum codes, have enabled fast progress in the field of quantum error correction by allowing parallels to be drawn with the widely studied field of classical error correction. This has resulted in the construction of the quantum counterparts of well-known capacity-approaching classical codes like sparse codes and quantum turbo codes. However, quantum codes obtained in this manner do not entirely evoke the stupendous error correcting abilities of their classical counterparts. This occurs because classical strategies ignore important differences between the quantum and classical paradigms, an issue that needs to be addressed if quantum error correction is to succeed in its battle with decoherence. In this dissertation we study a phenomenon exclusive to the quantum paradigm, known as degeneracy, and its effects on the performance of sparse quantum codes. Furthermore, we also analyze and present methods to improve the performance of a specific family of sparse quantum codes in various different scenarios.
Quantum computation exploits superposition, entanglement, interference, and measurement to process information.Research has moved from theoretical algorithms to functional devices where the performance is not only limited by qubit physics, but also by the control, compilation, error mitigation, and error correction.This review considers quantum computing on various levels of abstraction, including physical qubits, quantum algorithms, system architecture, fault-tolerance, and emerging applications.Study-specific evidence regarding scale, operational quality, connectivity, control and suppression of errors is compared across the different types of superconducting circuits, trapped ions, semiconductor spin qubits, neutral atoms and photonic processors.The review of the fundamental algorithms, noisy intermediate-scale quantum computing, variational and hybrid quantum-classical approaches, quantum simulation, quantum machine learning, and security implications is also discussed.Measurable progress has been demonstrated in published experiments for logical-qubit operation, below-threshold surface-code memory, programmable photonic sampling and platformspecific scaling.However, the reported metrics are not directly interchangeable, and practical advantage remains task-and baseline-dependent.Further progress will need to be accompanied by coordinated developments in hardware reliability, error correction, classical control, estimates of algorithmic resources, and clear classical method comparisons.
A mathematical model of computation is an idealized abstraction. We design algorithms and perform analysis on the assumption that the mathematical operations we specify will be carried out exactly, and without error. Physical devices that implement an abstract model of computation are imperfect and of limited precision. For example, when a digital circuit is implemented on a physical circuit board, unwanted electrical noise in the environment may cause components to behave differently than expected, and may cause voltage levels (bit-values) to change. These sources of error must be controlled or compensated for, or else the resulting loss of efficiency may reduce the power of the information-processing device. If individual steps in a computation succeed with probability p, then a computation involving t sequential steps will have a success probability that decreases exponentially as pt. Although it may be impossible to eliminate the sources of errors, we can devise schemes to allow us to recover from errors using a reasonable amount of additional resources. Many classical digital computing devices use error-correcting codes to perform detection of and recovery from errors. The theory of error-correcting codes is itself a mathematical abstraction, but it is one that explicitly accounts for errors introduced by the imperfection and imprecision of realistic devices. This theory has proven extremely effective in allowing engineers to build computing devices that are resilient against errors. Quantum computers are more susceptible to errors than classical digital computers, because quantum mechanical systems are more delicate and more difficult to control. If large-scale quantum computers are to be possible, a theory of quantum error correction is needed. The discovery of quantum error correction has given researchers confidence that realistic large-scale quantum computing devices can be built despite the presence of errors. We begin by considering fundamental concepts for error correction in a classical setting. We will focus on three of these concepts: (a) the characterization of the error model, (b) the introduction of redundancy through encoding, and (c) an error recovery procedure. We will later see that these concepts generalize quite naturally for quantum error correction.
Quantum computing is necessary to be described as a powerful tool is the new generation of computational power that works based on the principles of quantum mechanics, allowing the solving of problems that are intractable for classical computers. Due to the utilization of such phenomena as superposition, quantum entanglement and quantum interference, quantum computers are said to be able to solve problems in such areas as cryptography, artificial intelligence, optimization problems and material sciences, among others. This paper attempts to examine the basic concepts of quantum computing while focusing on their different aspects and their potential impacts. This study its experience and potential in its current form and in future applications through examples, illustrations, and theoretical discussion. Specific problems, including error correction, hardware scalability, and requirements for resources to address, make the paper a comprehensive overview of the quantum computing potential and weaknesses. Moving towards practical examples, the discussion covers all aspects of the book, providing readers with a holistic understanding of how quantum computing is going to change technological paradigms in the forthcoming decades.
The field of quantum computing holds immense promise for solving complex problems that are currently intractable for classical computers. However, the practical realization of quantum computers faces a critical challenge: the inherent fragility of qubits, the fundamental units of quantum information. Quantum systems are susceptible to various types of errors and decoherence due to their interaction with the surrounding environment, which can lead to a rapid degradation of quantum information. This chapter delves into the pivotal aspect of quantum error correction techniques as a means to mitigate the adverse effects of errors and decoherence in quantum computing systems. Begins by elucidating the theoretical foundations of quantum error correction, exploring the principles that underlie the encoding of quantum information into larger, redundant codes that are resilient to errors. A comprehensive overview of prominent quantum error correction codes, such as the stabilizer codes and topological codes, is provided, along with a discussion of their respective strengths and limitations.