Abstract. The hereby paper presents an overview of the current interpretation of decoherence in quantum theory, at the same time firmly stating that decoherence can be successfully fought against by means of the quantum error-correcting codes developed in the recent years. While decoherence itself is but a natural transition from quantum to classical, explaining the so disputed quantum origins of the classical world, its effects have to be combated or severely diminished if revolutionary techniques such as quantum communication or quantum computation are to be ever implemented. We will argue and try to prove by reference to existing leading research papers that quantum error-correction codes are a viable solution for the decoherence problem in particular settings. As a clear-cut example we tackle the five-qubit quantum circuit described by Shor, DiVincenzo and Terhal. Full generalization is in our view a simple problem of time as the theory describing it exists already and is continuously revised by top scientists. In this sense the general QECC framework developed by Knill and Laflamme is thoroughly analyzed and commented upon. 1. Scene-Setting: Decoherence as Transition Concept between Quantum and Classical If we were to consider decoherence in its full generality
Abstract. The hereby paper presents an overview of the current interpretation of decoherence in quantum theory, at the same time firmly stating that decoherence can be successfully fought against by means of the quantum error-correcting codes developed in the recent years. While decoherence itself is but a natural transition from quantum to classical, explaining the so disputed quantum origins of the classical world, its effects have to be combated or severely diminished if revolutionary techniques such as quantum communication or quantum computation are to be ever implemented. We will argue and try to prove by reference to existing leading research papers that quantum error-correction codes are a viable solution for the decoherence problem in particular settings. As a clear-cut example we tackle the five-qubit quantum circuit described by Shor, DiVincenzo and Terhal. Full generalization is in our view a simple problem of time as the theory describing it exists already and is continuously revised by top scientists. In this sense the general QECC framework developed by Knill and Laflamme is thoroughly analyzed and commented upon. 1. Scene-Setting: Decoherence as Transition Concept between Quantum and Classical If we were to consider decoherence in its full generality
In Chapter 4 we introduced different quantum gates and quantum circuits and in Chapter 5 we described several quantum algorithms. So far the analysis of the algorithms and the circuits has been made with the assumption that the required unitary operations can be implemented exactly without any error. This is an abstraction. In an actual physical system that implements a quantum algorithm or a quantum circuit there are several sources of errors. The situation is similar in classical computation and to circumvent this problem error correcting codes are used in classical digital computing. Since qubits are more delicate and more susceptible to errors we need to introduce quantum error correcting codes. It is very important, as without such codes it will not be possible to build a scalable quantum computer. Such quantum error correcting codes are introduced in this chapter. However, we will not limit ourselves to the discussion of error correcting codes alone. We will also discuss more general aspects of errors and error correction, like decoherence, decoherence free subspace and fault-tolerance.
In the current Noisy Intermediate Scale Quantum (NISQ) era of quantum computing, qubit technologies are prone to imperfections, giving rise to various errors such as gate errors, decoherence/dephasing, measurement errors, leakage, and crosstalk. These errors present challenges in achieving error-free computation within NISQ devices. A proposed solution to this issue is Quantum Error Correction (QEC), which aims to rectify the corrupted qubit state through a three-step process: (i) detection: identifying the presence of an error, (ii) decoding: pinpointing the location(s) of the affected qubit(s), and (iii) correction: restoring the faulty qubits to their original states. QEC is an expanding field of research that encompasses intricate concepts. In this paper, we aim to provide a comprehensive review of the historical context, current state, and future prospects of Quantum Error Correction, tailored to cater to computer scientists with limited familiarity with quantum physics and its associated mathematical concepts. In this work, we, (a) explain the foundational principles of QEC and explore existing Quantum Error Correction Codes (QECC) designed to correct errors in qubits, (b) explore the practicality of these QECCs concerning implementation and error correction quality, and (c) highlight the challenges associated with implementing QEC within the context of the current landscape of NISQ computers.
In the current Noisy Intermediate Scale Quantum (NISQ) era of quantum computing, qubit technologies are prone to imperfections, giving rise to various errors such as gate errors, decoherence/dephasing, measurement errors, leakage, and crosstalk. These errors present challenges in achieving error-free computation within NISQ devices. A proposed solution to this issue is Quantum Error Correction (QEC), which aims to rectify the corrupted qubit state through a three-step process: (i) detection: identifying the presence of an error, (ii) decoding: pinpointing the location(s) of the affected qubit(s), and (iii) correction: restoring the faulty qubits to their original states. QEC is an expanding field of research that encompasses intricate concepts. In this paper, we aim to provide a comprehensive review of the historical context, current state, and future prospects of Quantum Error Correction, tailored to cater to computer scientists with limited familiarity with quantum physics and its associated mathematical concepts. In this work, we, (a) explain the foundational principles of QEC and explore existing Quantum Error Correction Codes (QECC) designed to correct errors in qubits, (b) explore the practicality of these QECCs with regard to implementation and error correction quality, and (c) highlight the challenges associated with implementing QEC within the context of the current landscape of NISQ computers.
Research in the field of quantum algorithms and quantum error correction is progressing at an astounding rate. There are many good papers on both subjects, but reading even a few of these may seem a daunting task to the newcomer. The aim of this paper is to give a leisurely introduction to the basic theory of quantum error correcting codes without appealing to even the most basic notions in physics. Thus the article is not a substitute for important papers such as [12] or [7] but rather an advertisement for them. I would be pleased if, in addition, some readers view this as a useful companion article if and when they go on to read more substantial literature on the subject of quantum error correction. I present nothing new here. Rather, I give an elementary account of the important theorems and proofs which appear in these fundamental works using only undergraduate algebra and a bit of classical coding theory. In particular, I give a full proof of the Knill/Laflamme theorem as well as an elementary treatment of stabilizer codes. The goal is to make the literature dealing with this exciting new area more accessible to discrete mathematicians.
Quantum information is prone to suffer from errors caused by the so-called decoherence, which describes the loss in coherence of quantum states associated to their interactions with the surrounding environment. This decoherence phenomenon is present in every quantum information task, be it transmission, processing or even storage of quantum information. Consequently, the protection of quantum information via quantum error correction codes (QECC) is of paramount importance to construct fully operational quantum computers. Understanding environmental decoherence processes and the way they are modeled is fundamental in order to construct effective error correction methods capable of protecting quantum information. Moreover, quantum channel models that are efficiently implementable and manageable on classical computers are required in order to design and simulate such error correction schemes. In this article, we present a survey of decoherence models, reviewing the manner in which these models can be approximated into quantum Pauli channel models, which can be efficiently implemented on classical computers. We also explain how certain families of quantum error correction codes can be entirely simulated in the classical domain, without the explicit need of a quantum computer. A quantum error correction code for the approximated channel is also a correctable code for the original channel, and its performance can be obtained by Monte Carlo simulations on a classical computer.
Quantum technologies have shown immeasurable potential to effectively solve several information processing tasks such as prime number factorization, unstructured database search or complex macromolecule simulation. As a result of such capability to solve certain problems that are not classically tractable, quantum machines have the potential revolutionize the modern world via applications such as drug design, process optimization, unbreakable communications or machine learning. However, quantum information is prone to suffer from errors caused by the so-called decoherence, which describes the loss in coherence of quantum states associated to their interactions with the surrounding environment. This decoherence phenomenon is present in every quantum information task, be it transmission, processing or even storage of quantum information. Consequently, the protection of quantum information via quantum error correction codes (QECC) is of paramount importance to construct fully operational quantum computers. Understanding environmental decoherence processes and the way they are modeled is fundamental in order to construct effective error correction methods capable of protecting quantum information. In this thesis, the nature of decoherence is studied and mathematically modelled; and QECCs are designed and optimized so that they exhibit better error correction capabilities.