Chapter 7 - Quantum Error Correction

Abstract

This chapter is devoted to the quantum error correction concepts, ranging from an intuitive description to rigorous mathematical framework. After the Pauli operators introduction, we describe basic quantum codes, such as three-qubit flip code, three-qubit phase flip code, Shor's nine-qubit code, stabilizer codes and CSS codes. We then formally introduce the quantum error correction, including the quantum error correction mapping, quantum error representation, stabilizer group definition, quantum-check matrix representation and quantum syndrome equation. We further provide the necessary and sufficient conditions for quantum error correction, discuss the distance properties and error correction capability, and revisit the CSS codes. Next we describe important quantum coding bounds including quantum Hamming bound, Gilbert–Varshamov bound and Singleton bound. In same section we also discuss the quantum weight enumerators and quantum MacWilliams identities. We further discuss the quantum superoperators and various quantum channels including the quantum depolarizing, amplitude damping and generalized amplitude damping channels. For deeper understanding of presented quantum error correction material, in final section we provide the set of homework problems.