The Exercise

Let $S$ be a subgroup of the Pauli group $G_n$ on $n$ qubits, the group of operators $i^s \sigma_{a_1} \otimes \cdots \otimes \sigma_{a_n}$ defined below. The exercise asks to show that $S$ lies in its own normalizer $N(S)$^✦, the elements of $G_n$ that conjugate $S$ onto itself: if $E$ and $g$ are elements of $S$, the conjugate $E g E^\dagger$ is again an element of $S$.^✦

Take for $S$ the group generated by $X \otimes X$ and $Z \otimes Z$, where $X$ and $Z$ are two of the Pauli matrices, written out in (1) below.

The Pauli Group

The Pauli matrices are the four complex $2 \times 2$ matrices $\sigma_0, \sigma_1, \sigma_2, \sigma_3$, with rows and columns indexed by $\{0, 1\}$: $\sigma_0 = I$ is the identity, and $\sigma_1 = X$, $\sigma_2 = Y$, $\sigma_3 = Z$ are $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(1)

For a choice $a$ of one Pauli matrix for each of $n$ qubits, $a_k \in \{0, 1, 2, 3\}$ for qubit $k$, the Pauli string $\sigma_{a_1} \otimes \cdots \otimes \sigma_{a_n}$ is the operator on the $n$ qubits whose matrix, with rows and columns labelled by bit strings $x, y \in \{0, 1\}^n$, has the entries $$(\sigma_{a_1} \otimes \cdots \otimes \sigma_{a_n})_{xy} = \prod_{k=1}^{n} (\sigma_{a_k})_{x_k y_k}.$$(2)^✦

For $s$ an integer modulo $4$, the phase $i^s$ is the imaginary unit raised to the representative of $s$ in $\{0, 1, 2, 3\}$.^✦

The Pauli group $G_n$ on $n$ qubits consists of the operators $i^s \sigma_{a_1} \otimes \cdots \otimes \sigma_{a_n}$.^✦ An element $p$ of $G_n$ is fixed by its phase power $s \in \mathbb{Z}_4$ and its Pauli string $a$^✦; the matrix of $p$, a complex $2^n \times 2^n$ matrix with rows and columns labelled by $\{0, 1\}^n$, is the Pauli string times the phase, $$p = i^s \, \sigma_{a_1} \otimes \cdots \otimes \sigma_{a_n}.$$(3)^✦

The Claim

For a subgroup $S$ of $G_n$, the normalizer $N(S)$ of $S$ in $G_n$ is the subgroup of the elements $E$ of $G_n$ for which $E g E^{-1}$ lies in $S$ exactly when $g$ does.

What Is Checked

The claim is stated over the Pauli group $G_n$, the matrices (3) of its elements, and the normalizer of a subgroup within $G_n$. The library proves both parts of the theorem for every number of qubits $n$ and every subgroup $S$ of $G_n$.