In this article, we make a review of quantum error-correcting codes (QECCs) constructed from the concatenation of a phase flip code with a bit flip code. In particular, two examples are reviewed in details: the rate-1/9 quantum block code (QBC) proposed by Shor, and the rate 1/4 quantum convolutional code (QCC) proposed by Almeida and Palazzo. The purpose of this article is to introduce the study of QECCs to the community of telecommunication engineers.
In this article, we make a review of quantum error-correcting codes (QECCs) constructed from the concatenation of a phase flip code with a bit flip code. In particular, two examples are reviewed in details: the rate-1/9 quantum block code (QBC) proposed by Shor, and the rate 1/4 quantum convolutional code (QCC) proposed by Almeida and Palazzo. The purpose of this article is to introduce the study of QECCs to the community of telecommunication engineers.
Decoherence is one of the major challenges facing the field of quantum computation.The field of quantum error correction has developed to meet this challenge.A group-theoretical structure and associated class of quantum codes, the stabilizer codes, has proved particularly fruitful in producing codes and in understanding the structure of both specified codes and classes of codes.All stabilizer codes discovered so far are block codes.In this thesis, we will construct a class of concatenated quantum convolutional codes.We will introduce the concept of quantum convolutional memory and some simple techniques to produce good quantum convolutional codes from classes of classical convolutional codes.
This chapter deals with the subject of quantum error correction and the related codes (QECC), which can be applied to noisy quantum channels and quantum memories with the purpose of preserving or protecting the information integrity. I first describe the basics of quantum repetition codes , as applicable to bit-flip and phase-flip quantum channels. Then I consider the 9-qubit Shor code , which has the capability of diagnosing and correcting any combination of bit-flip and phase-flip errors, up to one error of each type. Furthermore, it is shown that the Shor code is, in fact, capable of fully restoring qubit integrity under a continuum of bit or phase errors, a property that has no counterpart in the classical world of error-correction codes. But the exploration of QECC does not stop here! We shall discover the elegant Calderbank–Shor–Steane (CSS) codes, which have the capability of correcting any number of errors t , both bit-flip and phase-flip. As an application of the CSS code, I then describe the 7-qubit Hadamard–Steane code , which can correct up to one error on single qubits. A corresponding quantum circuit, based on an original generator-matrix example, is presented. Quantum repetition code In Chapter 11, we saw that the simplest form of error-correction code (ECC) is the repetition code , based on the principle of majority logic . The background assumption is that in a given message sequence, or bit string, the probability of a bit error is sufficiently small for the majority of bits to be correctly transmitted through the channel.
Em um computador quântico, da mesma forma que em um computador clássico, a informação está sujeita a erros que precisam ser detectados e corrigidos, de onde surge a necessidade dos códigos quânticos de correção de erros. Neste trabalho estudamos os códigos CWS (Codeword Stabilized quantum codes) que generalizam os códigos estabilizadores. Primeiramente, descrevemos detalhadamente os códigos CWS sobre sistemas quânticos de mais de um nível (qudits) a partir de uma nova abordagem. Deixamos claro quais resultados valem em geral e quais valem apenas para qubits, qupits (sistemas com número primo de níveis) e quais valem no caso do código CWS ser baseado em um estado-grafo. Apresentamos também um novo resultado que relaciona códigos CWS com códigos estabilizadores generalizando os resultados presentes na literatura. Posteriormente, caracterizamos um tipo de operador de medida para códigos CWS para qubits. Criamos então um procedimento para buscar estes operadores, que nem sempre são suficientes para identificar o erro ocorrido, mas quando são, o fazem de forma eficiente.
We report new construction of nine-qubit error-correcting code, which introduces two new nine-qubit codes and one new three-qubit code. Because both the new two nine-qubit codes have the normal logical operators, as opposed to the nine-qubit Shor code, it results in different performance when the three codes are applied in concatenated quantum error-correction. On the other hand, one of the two nine-qubit codes has the same stabilizer generators as the nine-qubit Shor code, they are more suitable for the high-wight bit-flip noise, and the other code has the different stabilizer generators, which is more suitable for the high-wight phase-flip noise. This work is enlightening to the construction of quantum error-correcting codes, and adds more options for optimizing the performance of quantum error-correction.
Quantum systems operate using quantum bits (qubits), which are highly sensitive to external noises and disturbances and they are easily prone to errors. In the Noisy Intermediate-Scale Quantum (NISQ) era, these errors make difficult to build reliable, fault tolerant quantum circuits for real world deployment. This work provides an overview of need for Quantum Error Correction (QEC) in NISQ devices, as qubits commonly get affected by bit-flip, phase-flip, and combined errors that degrades system performance and examines Quantum Error Correction Code (QECC) techniques usage in communication, computation, and quantum multimedia applications such as image and video transmission. In addition, a simple three-qubit repetition circuit is introduced to demonstrate an error-mitigation approach. These findings, also show how QEC improves fidelity, reduces logical errors, and improves noise tolerance. The analysis also identifies challenges and outlines the future research direction aimed at achieving efficient and scalable fault tolerant quantum circuits.
Quantum systems operate using quantum bits (qubits), which are highly sensitive to external noises and disturbances and they are easily prone to errors. In the Noisy Intermediate-Scale Quantum (NISQ) era, these errors make difficult to build reliable, fault tolerant quantum circuits for real world deployment. This work provides an overview of need for Quantum Error Correction (QEC) in NISQ devices, as qubits commonly get affected by bit-flip, phase-flip, and combined errors that degrades system performance and examines Quantum Error Correction Code (QECC) techniques usage in communication, computation, and quantum multimedia applications such as image and video transmission. In addition, a simple three-qubit repetition circuit is introduced to demonstrate an error-mitigation approach. These findings, also show how QEC improves fidelity, reduces logical errors, and improves noise tolerance. The analysis also identifies challenges and outlines the future research direction aimed at achieving efficient and scalable fault tolerant quantum circuits.