Quantum Error Correction and Its Application

Abstract

Quantum error correction is a technology to protect quantum state from noise and decoherence. This technology is essentially based on group symmetry. In particular, a stabilizer code and a CSS (Calderbank-Shor-Smolin) code (typical quantum error corrections) are essentially based on the discrete Heisenberg representation. Since the quantum error correction is complicated, we firstly explain error correction in the classical case in Sect. 5.1.1. Section 5.2 shows the general formulation of quantum error correction. In Sect. 5.3, combining this knowledge and Discrete Heisenberg representation, which is summarized in Sect. 3.3.1, we explain stabilizer code, which is a typical example of quantum error correction. We also discuss CSS (Calderbank-Shor-Smolin) code, which is a special case of a stabilizer code. For simplicity, our analysis in Sect. 5.3 assumes a Pauli channel. In Sect. 5.4, we introduce Clifford Code, which is a generalization of stabilizer code as a general framework of quantum error correction with group symmetry. In Sect. 5.5, we apply quantum error correction to entanglement distillation. Also, we discuss our error correction when the noisy channel is a general channel beyond a Pauli channel. In Sect. 5.6, we apply our discussion of stabilizer code to quantum secure communication. In Sect. 5.7, we discuss quantum cryptography (quantum key distribution) based on the framework of this chapter.