Chapter 13 - Fault-tolerant quantum error correction

Abstract

This chapter is devoted to fault-tolerant quantum error correction concepts. The chapter starts with an introduction to fault-tolerance basics and traversal operations. It continues with fault-tolerant quantum information processing concepts and procedures. The universal set of fault-tolerant quantum gates is introduced, followed by fault-tolerant measurement, fault-tolerant state preparation, and fault-tolerant encoding state preparation, using Steane code as an example. The focus then moves to a rigorous description of fault-tolerant quantum error correction. After a short review of some basic concepts from stabilizer codes, fault-tolerant syndrome extraction is described, followed by a description of fault-tolerant encoding operations. Next described is the application of quantum gates on a quantum register through a measurement protocol. When applied transversally, this method enables fault-tolerant error correction based on arbitrary quantum stabilizer code. Fault-tolerant stabilizer codes are described next. Finally, the [5,1,3] fault-tolerant stabilizer code is described as an example.