Chapter 7 - Quantum error correction fundamentals

Abstract

The concept of this chapter is to gradually introduce the reader to quantum error correction coding principles, starting from intuitive descriptions and advancing to rigorous mathematical discussions. The chapter starts with Pauli operators, basic definitions, and the representation of quantum errors. Although Pauli operators were introduced in Chapter 3, they are discussed here in the context of quantum errors and quantum error correction. Next, basic quantum codes, such as three-qubit flip, three-qubit phase flip, and Shor's nine-qubit codes are presented. Projection measurements are used to determine error syndrome and perform corresponding error correction action. Stabilizer formalism and stabilizer groups are also introduced, and basic stabilizer codes are described. The entire next chapter is devoted to stabilizer codes; only the basic concepts are introduced here. An important class of codes, Calderbank–Shor–Steane (CSS) codes, is described next. The connection between classical and quantum codes is established, and two classes of CSS codes are described—dual-containing and quantum codes derived from classical codes over GF(4). The quantum error correction concept is formally introduced, followed by the necessary and sufficient conditions for quantum code to correct a given set of errors. Then, the minimum distance for a quantum code is defined and used to relate it to the error correction capability of a quantum code. CSS codes are then revisited under this mathematical framework. The section after that is devoted to important quantum coding bounds: Hamming quantum, quantum Gilbert–Varshamov, and quantum Singleton (also known as Knill–Laflamme). Next, the concept of operator-sum representation is introduced and used to provide a physical interpretation and describe measurement of the environment. Finally, several important quantum channel models are introduced, such as depolarizing, amplitude damping, and generalized amplitude damping.