Chapter 8 - Quantum Error Correction

Abstract

The concept of this chapter is to gradually introduce readers to the quantum error correction coding principles, starting from an intuitive description to a rigorous mathematical description. The chapter starts with Pauli operators, basic definitions, and representation of quantum errors. Although the Pauli operators were introduced in Chapter 2, they are put here in the context of quantum errors and quantum error correction. Next, basic quantum codes, such as three-qubit flip code, three-qubit phase-flip code, and Shor's nine-qubit code, are presented. The projection measurements are used to determine the error syndrome and perform corresponding error correction action. Furthermore, the stabilizer formalism and the stabilizer group are introduced. The basic stabilizer codes are described as well. The whole of the next chapter is devoted to stabilizer codes; here only the basic concepts are introduced. An important class of codes, the class of Calderbank–Shor–Steane (CSS) codes, is described next. The connection between classical and quantum codes is then established, and two classes of CSS codes, dual-containing and quantum codes derived from classical codes over GF(4), are described. The concept of quantum error correction is then formally introduced, followed by the necessary and sufficient conditions for quantum code to correct a given set of errors. Then, the minimum distance of a quantum code is defined and used to relate it to the error correction capability of a quantum code. The CSS codes are then revisited by using this mathematical framework. The next section is devoted to important quantum coding bounds, such as the Hamming quantum bound, quantum Gilbert–Varshamov bound, and quantum Singleton bound (also known as the Knill–Laflamme bound). Next, the concept of operator-sum representation is introduced, and used to provide physical interpretation and describe the measurement of the environment. Finally, several important quantum channel models are introduced, such as depolarizing channel, amplitude-damping channel, and generalized amplitude-damping channel. After the summary section, the set of problems for self-study is provided, which enables readers to better understand the underlying concepts of quantum error correction.