The Exercise

A stabilizer on $n$ qubits is a group $S$ of Pauli operators given by generators: $S = \langle g_1, \dots, g_l \rangle$ is the group of all products of the $g_j$ . The exercise asks to show that $-I$ is not an element of $S$ if and only if every generator satisfies $g_j^2 = I$ and $g_j \neq -I$ .

The library states one direction for every set of generators, and the other under extra conditions on the generators.

The Pauli Group

The Pauli group $G_n$ on $n$ qubits consists of the operators $i^s\, P_1 \otimes \cdots \otimes P_n, \qquad s \in \mathbb{Z}_4, \quad P_1, \dots, P_n \in \{I, X, Y, Z\},$ (1)^✦ each given by its phase power $s$ and its Pauli string $P_1 \otimes \cdots \otimes P_n$ . The element $-I$ of $G_n$ is the string $I \otimes \cdots \otimes I$ with phase power $s = 2$ , so that $i^2 I = -I$ .^✦

A Pauli $P$ on a single qubit has two check bits in $\mathbb{Z}_2$ , its $X$ bit $x_P$ , which marks an $X$ factor, and its $Z$ bit $z_P$ , which marks a $Z$ factor: $(x_I, z_I) = (0, 0), \quad (x_X, z_X) = (1, 0), \quad (x_Y, z_Y) = (1, 1), \quad (x_Z, z_Z) = (0, 1).$ (2)^✦ The check row $r(g)$ of a Pauli string $g = P_1 \otimes \cdots \otimes P_n$ is the vector of $\mathbb{Z}_2^{2n}$ formed by its $X$ part $x$ and its $Z$ part $z$ , which list the check bits of the Paulis on the $n$ qubits in turn: $r(g) = (x \,|\, z), \qquad x = (x_{P_1}, \dots, x_{P_n}), \quad z = (z_{P_1}, \dots, z_{P_n}).$ (3)^✦ The check row of an element of $G_n$ is that of its string, whatever the phase: $r(i^s g) = r(g)$ .^✦

The Claim

Theorem 1. Let $T$ be a set of elements of $G_n$ such that $-I$ is not in the group $\langle T \rangle$ generated by $T$ . Then $g^2 = I \quad \text{and} \quad g \neq -I \qquad \text{for every } g \in T.$ (4)^✦

On a single qubit, the operators $X$ and $-X$ square to $I$ , neither is $-I$ , and they commute, yet $X(-X) = -I$ .

Theorem 2. There is a set $T$ of elements of the Pauli group $G_1$ on one qubit such that $g^2 = I \quad \text{and} \quad g \neq -I \quad \text{for every } g \in T, \qquad -I \in \langle T \rangle.$ (5)^✦ There is even one whose elements commute pairwise: $gh = hg, \quad g^2 = I \quad \text{and} \quad g \neq -I \quad \text{for all } g, h \in T, \qquad -I \in \langle T \rangle.$ (6)^✦

The operators $X$ and $-X$ have the same check row $(1|0)$ . The pair $X$ and $Z$ has the independent check rows $(1|0)$ and $(0|1)$ , but $X$ and $Z$ anticommute, and $(XZ)^2 = -I$ . Commutation and independent check rows together exclude both pairs.

Theorem 3. Let $g_1, \dots, g_l$ be elements of $G_n$ that commute pairwise and whose check rows $r(g_1), \dots, r(g_l)$ are linearly independent over $\mathbb{Z}_2$ . Then $-I \notin \langle g_1, \dots, g_l \rangle \iff g_j^2 = I \text{ and } g_j \neq -I \text{ for } j = 1, \dots, l.$ (7)^✦

What Is Checked

The theorems are stated over the Pauli group $G_n$ of (1) with its element $-I$ , over the group that a set of its elements generates, and, for the equivalence, over the check rows (3) of the generators, built from the check bits (2).