Chapter 11 - Fault-Tolerant Quantum Error Correction and Fault-Tolerant Quantum Computing

Abstract

This chapter is devoted to one of the most important applications of quantum error correction, namely the protection of quantum information as it dynamically undergoes computation through so-called fault-tolerant quantum computing. The chapter starts with an introduction of fault-tolerance basics and transversal operations (Section 11.1). It continues with fault-tolerant quantum computation concepts and procedures (Section 11.2). In Section 11.2, the universal set of fault-tolerant quantum gates is introduced, followed by fault-tolerant measurement, fault-tolerant state preparation, and fault-tolerant-encoded state preparation by using Steane's code as an illustrative example. Section 11.3 is devoted to a rigorous description of fault-tolerant quantum error correction. This section starts with a short review of some basic concepts from stabilizer codes. In Subsection 11.3.1, fault-tolerant syndrome extraction is described, which is followed by a description of fault-tolerant-encoded operations in Subsection 11.3.2. Subsection 11.3.3 is concerned with the application of quantum gates on a quantum register by means of a measurement protocol. This method, when applied transversally, is used in Subsection 11.3.4 to enable fault-tolerant error correction based on arbitrary quantum stabilizer code. Fault-tolerant stabilizer codes are described in Subsection 11.3.4. In Subsection 11.3.5, the [5,1,3] fault-tolerant stabilizer code is described as an illustrative example. Section 11.4 is devoted to fault-tolerant computing, in particular to fault-tolerant implementation of the Toffoli gate. Furthermore, in Section 11.5, accuracy threshold theorem is formulated and proved. Finally, in Section 11.6 surface code-based large-scale quantum computing is discussed. After providing the summary remarks in Section 11.7, in the final section (Section 11.8) a set of problems is provided for interested readers to get a deeper understanding of fault-tolerance theory.