Introduction

In the previous lesson, we took a first look at quantum error correction, focusing specifically on the 9-qubit Shor code. In this lesson, we'll introduce the stabilizer formalism, which is a mathematical framework through which a broad class of quantum error correcting codes, known as stabilizer codes, can be specified and analyzed. This includes the 9-qubit Shor code along with many other examples, including codes that seem likely to be well-suited to real-world quantum devices. Not every quantum error correcting code is a stabilizer code, but many are, including every example that we'll see in this course.

The lesson begins with a short discussion of Pauli matrices, and tensor products of Pauli matrices more generally, which can represent not only operations on qubits, but also measurements of qubits — in which case they're typically referred to as observables. We'll then go back and take a second look at the repetition code and see how it can be described in terms of Pauli matrix observables. This will both inform and lead into a general discussion of stabilizer codes, including several examples, basic properties of stabilizer codes, and how the fundamental tasks of encoding, detecting errors, and correcting those errors can be performed.

Lesson video

In the following video, John Watrous steps you through the content in this lesson on stabilizer formalism. Alternatively, you can open the YouTube video for this lesson in a separate window. Download the slides for this lesson.