Quantum Computation and Quantum Information 10th Anniversary Edition

This chapter explains how to do quantum information processing reliably in the presence of noise. The chapter covers three broad topics: the basic theory of quantum error-correcting codes, fault-tolerant quantum computation, and the threshold theorem. We begin by developing the basic theory of quantum error-correcting codes, which protect quantum information against noise. These codes work by encoding quantum states in a special way that make them resilient against the effects of noise, and then decoding when it is wished to recover the original state. Section 10.1 explains the basic ideas of classical error-correction, and some of the conceptual challenges that must be overcome to make quantum error-correction possible. Section 10.2 explains a simple example of a quantum error-correcting code, which we then generalize into a theory of quantum error-correcting codes in Section 10.3. Section 10.4 explains some ideas from the classical theory of linear codes, and how they give rise to an interesting class of quantum codes known as Calderbank–Shor–Steane (CSS) codes. Section 10.5 concludes our introductory survey of quantum error-correcting codes with a discussion of stabilizer codes, a richly structured class of codes with a close connection to classical error-correcting codes.