The Exercise

A Pauli string $g$ on $n$ qubits acts on each qubit with one of the matrices $I$, $X$, $Y$, $Z$. In the stabilizer formalism it is recorded as a row $r(g)$ of the check matrix: $2n$ bits, the first $n$ marking the qubits on which $g$ has an $X$ factor, the last $n$ those on which it has a $Z$ factor.^✦ The exercise asks to show that two Pauli strings $g$ and $h$ commute as operators if and only if $r(g) \Lambda r(h)^T = 0$, with arithmetic modulo two and $\Lambda$ the $2n \times 2n$ matrix that exchanges the two halves of a row.^✦

Pauli Strings

The Pauli matrices are the four complex $2 \times 2$ matrices $$\sigma_0 = I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad \sigma_1 = X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_2 = Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_3 = Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(1)^✦

A Pauli string $g$ on $n$ qubits is a choice of one index $g_k \in \{0, 1, 2, 3\}$ for each qubit $k$. The operator $P_g$ of a Pauli string $g$ on $n$ qubits is the tensor product of the chosen matrices, an operator on the $2^n$ dimensional state space of the qubits; in the computational basis, with rows and columns labelled by bit strings $i$ and $j$, each matrix element is a product over the qubits: $$P_g = \sigma_{g_1} \otimes \cdots \otimes \sigma_{g_n}, \qquad (P_g)_{ij} = \prod_{k=1}^{n} (\sigma_{g_k})_{i_k j_k}.$$(2)^✦

The Check Matrix

The check bits of a single qubit Pauli matrix $\sigma \in \{I, X, Y, Z\}$ are the pair of elements of $\mathbb{Z}_2$ consisting of its $X$ bit and its $Z$ bit, assigned as follows: $$I \mapsto (0, 0), \qquad X \mapsto (1, 0), \qquad Y \mapsto (1, 1), \qquad Z \mapsto (0, 1).$$(3)^✦

The $X$ part $x_g$ and the $Z$ part $z_g$ of a Pauli string $g$ are the vectors in $\mathbb{Z}_2^n$ whose entries on qubit $k$ are the $X$ bit and the $Z$ bit of $\sigma_{g_k}$.^✦ The check matrix row $r(g)$ of $g$ is the vector of $2n$ bits made of the two parts, the $X$ part first: $$r(g) = (x_g \,|\, z_g) = (x_{g,1}, \ldots, x_{g,n}, z_{g,1}, \ldots, z_{g,n}).$$(4)^✦

The symplectic matrix $\Lambda$ over $\mathbb{Z}_2$ exchanges the two halves of a vector of $2n$ bits, with $I_n$ the identity on one half: $$\Lambda = \begin{pmatrix} 0 & I_n \\ I_n & 0 \end{pmatrix}.$$(5)^✦

The twisted inner product $u \Lambda v^T$ of two vectors $u = (u_X \,|\, u_Z)$ and $v = (v_X \,|\, v_Z)$ in $\mathbb{Z}_2^{2n}$ is the element of $\mathbb{Z}_2$ obtained as the dot product of $u$ with $\Lambda v^T$: $$u \Lambda v^T = u_X \cdot v_Z + u_Z \cdot v_X = \sum_{k=1}^{n} \bigl( u_{X,k}\, v_{Z,k} + u_{Z,k}\, v_{X,k} \bigr).$$(6)^✦

The Claim

What Is Checked

The claim is stated over the operators $P_g$ of Pauli strings (2), built from the four matrices (1), and over the check matrix rows (4), read from the check bits (3) and paired by the twisted inner product (6) with the symplectic matrix (5).