Chapter 7 Stabilisers

About the structure of the Pauli group, which is the group generated by tensor products of the Pauli matrices, including the identity. It has nice algebraic properties which are useful in many areas of quantum information science, in particular quantum error correction and classical simulations of some types of quantum computation. We will discuss how certain subgroups of the Pauli group, and in particular stabilisers and normalisers of these subgroups, slice the Pauli group into interesting cosets that have a group structure of their own. We will also look at the Clifford group, which is a set of unitary operators that preserve the Pauli group under conjugation and describes the “easy” part of quantum computation.

N.B. This section is sort of an odd-one-out, since we won’t need any of this formalism until Sections 13 and 14. However, if you’re reading this book in order, then you might find this a nice detour halfway through, and it gives a taste of things to come.

We have already seen the (single-qubit) Pauli matrices, along with a brief look into their algebraic structure, in Section 3.3.

\mathbf{1}= \begin{bmatrix}1&0\\0&1\end{bmatrix} \qquad X = \begin{bmatrix}0&1\\1&0\end{bmatrix} \qquad Y = \begin{bmatrix}0&-i\\i&0\end{bmatrix} \qquad Z = \begin{bmatrix}1&0\\0&-1\end{bmatrix}

Recall that these matrices span the entire space of (2\times2) complex matrices, square to the identity (and thus can only have eigenvalues in the set \{+1,-1\}), and are both Hermitian and unitary. As such, they can represent both observables and unitary evolutions. Any two given Pauli matrices either commute or anticommute.

As one final reminder, we often refer to the Pauli matrices as “matrices”, but they are defined as operators by the commutations relations, without reference to any particular basis. That is, the Pauli operators X, Y, and Z are defined exactly by the relations \begin{gathered} X^2 = Y^2 = Z^2 = \mathbf{1} \\XY=iZ \qquad YZ=iX \qquad ZX=iY \\YX=-iZ \qquad ZY=-iX \qquad XZ=-iY. \end{gathered}