The Exercise

A stabilizer code on $n$ qubits that encodes $k$ qubits can have its check matrix brought to standard form. The qubits then fall into three blocks, of $r$, $s = n - k - r$ and $k$ qubits, and the first $r$ generators carry a binary $r \times k$ matrix $A_2$ on the last block of their $X$ part. From $A_2$ the book reads off the encoded operators $\bar Z_1, \dots, \bar Z_k$, one logical $Z$ for each encoded qubit, each a tensor product of $Z$’s and identities. The exercise is to show that these $k$ operators are independent: none of them can be dropped without shrinking the group they generate.^✦

The Encoded Operators

The Pauli group $G_n$ on $n$ qubits is the group of operators $i^m P_1 \otimes \cdots \otimes P_n$, with the phase power $m$ an integer modulo $4$ and each $P_q$ one of $I$, $X$, $Y$, $Z$. The Pauli string of an element $i^m P_1 \otimes \cdots \otimes P_n$ of $G_n$ is the tensor product $P_1 \otimes \cdots \otimes P_n$ of its single-qubit factors, with the phase left out.

The pure $Z$ string $Z(w)$ of a string $w = (w_1, \dots, w_n)$ of $n$ bits is the Pauli string on $n$ qubits with $Z$ on each qubit $q$ where $w_q = 1$ and $I$ on the others: $$Z(w) = Z^{w_1} \otimes Z^{w_2} \otimes \cdots \otimes Z^{w_n}, \qquad Z^0 = I, \quad Z^1 = Z.$$(1)^✦

Fix a binary $r \times k$ matrix $A_2$ and a size $s$ of the middle block. The $Z$ support $w^{(j)}$ of the $j$th encoded operator, for $j = 1, \dots, k$, is the string of $r + s + k$ bits made of column $j$ of $A_2$, then $s$ zeros, then a single $1$ in place $j$ of the last block: $$w^{(j)} = \bigl( (A_2)_{1j}, \dots, (A_2)_{rj},\, \underbrace{0, \dots, 0}_{s},\, \delta_{1j}, \dots, \delta_{kj} \bigr).$$(2)^✦

The encoded operator $\bar Z_j$ is the element of $G_{r+s+k}$ with phase factor $1$ and Pauli string $Z\bigl(w^{(j)}\bigr)$: $$\bar Z_j = Z\bigl(w^{(j)}\bigr) = Z^{(A_2)_{1j}} \otimes \cdots \otimes Z^{(A_2)_{rj}} \otimes I^{\otimes (s + j - 1)} \otimes Z \otimes I^{\otimes (k - j)}.$$(3)^✦

The Claim

A family $g_1, \dots, g_l$ of elements of a group $G$ is independent when none of its members is redundant: for every $i$, the subgroup $\langle g_j : j \neq i \rangle$ generated by the other members is strictly smaller than the subgroup $\langle g_1, \dots, g_l \rangle$ generated by all of them.^✦

What Is Checked

The claim is stated over the Pauli group $G_{r+s+k}$, of which the encoded operators $\bar Z_j$ of (3) are elements, and over independence of a family of group elements as defined above.