The Exercise

A stabilizer code on $n$ qubits with $n-k$ generators has a check matrix with one row for each generator, made of an $X$ part and a $Z$ part of $n$ columns each. Multiplying one generator by another adds the second row to the first modulo $2$ and leaves the code unchanged; renumbering the qubits permutes the columns of both parts alike. With these two moves, Gaussian elimination brings the check matrix to standard form, $$\left(\begin{array}{ccc|ccc} I & A_1 & A_2 & B & 0 & C \\ 0 & 0 & 0 & D & I & E \end{array}\right),$$(1) where the upper block has $r$ rows, $r$ the rank of the $X$ part, the lower block the other $n-k-r$, and the column blocks of each part have widths $r$, $n-k-r$ and $k$.

The exercise asks for the standard forms of the five qubit code and of Shor’s nine qubit code. The library writes down a standard form for each, the Shor one after a renumbering of the qubits, and states the blocks of its check matrix and that the code is unchanged.

Check Rows

The check bits of a single qubit Pauli $a$, one of $I$, $X$, $Y$ and $Z$, are the pair $(x_a, z_a)$ in $\mathbb{Z}_2 \times \mathbb{Z}_2$ of its $X$ bit $x_a$, which is $1$ when $a$ is $X$ or $Y$ and $0$ otherwise, and its $Z$ bit $z_a$, which is $1$ when $a$ is $Y$ or $Z$ and $0$ otherwise.^✦

The check row of a string $g = g_1 g_2 \cdots g_n$ of single qubit Paulis, $g_k$ acting on qubit $k$, is the row $r(g)$ of $2n$ bits whose $X$ part $x_g$ and $Z$ part $z_g$ hold the check bits of $g_1, \dots, g_n$: $$r(g) = (x_g \mid z_g), \qquad (x_g)_k = x_{g_k}, \quad (z_g)_k = z_{g_k}.$$(2)^✦

The Five Qubit Code

The five qubit code is the stabilizer code on five qubits with the four generators of Figure 10.12, $$g_1 = XZZXI, \quad g_2 = IXZZX, \quad g_3 = XIXZZ, \quad g_4 = ZXIXZ,$$(3)^✦ The standard form of the five qubit code has the generators $$g'_1 = YZIZY, \quad g'_2 = IXZZX, \quad g'_3 = ZZXIX, \quad g'_4 = ZIZYY,$$(4)^✦ on the same five qubits in the same order.

The Shor Code

Write $Z_j Z_k$ for $Z$ on qubits $j$ and $k$ and the identity on the rest, and likewise for products of $X$. Shor’s code is the stabilizer code on nine qubits with the eight generators $$\begin{gathered} s_1 = Z_1 Z_2, \quad s_2 = Z_2 Z_3, \quad s_3 = Z_4 Z_5, \\ s_4 = Z_5 Z_6, \quad s_5 = Z_7 Z_8, \quad s_6 = Z_8 Z_9, \\ s_7 = X_1 X_2 X_3 X_4 X_5 X_6, \quad s_8 = X_4 X_5 X_6 X_7 X_8 X_9, \end{gathered}$$(5)^✦

Its standard form needs the qubits renumbered: positions $1$ to $9$ hold the qubits $1, 7, 2, 3, 4, 5, 8, 9, 6$ in turn, and $\sigma(k)$ is the position of qubit $k$, so that $\sigma(7) = 2$ and $\sigma(6) = 9$.^✦ In the new numbering the standard form has the generators $$\begin{aligned} s'_1 &= XIXXXXIIX, & s'_2 &= IXIIXXXXX, \\ s'_3 &= ZIZIIIIII, & s'_4 &= ZIIZIIIII, \\ s'_5 &= IIIIZIIIZ, & s'_6 &= IIIIIZIIZ, \\ s'_7 &= IZIIIIZII, & s'_8 &= IZIIIIIZI, \end{aligned}$$(6)^✦ the $j$th letter acting at position $j$. Put back in the original numbering, $s'_i$ becomes the string $\tilde s_i$ with $$(\tilde s_i)_k = (s'_i)_{\sigma(k)}, \qquad k = 1, \dots, 9.$$(7)^✦

The Claim

What Is Checked

The claims are stated over the check rows (2) of fixed strings of Paulis, so each block is an equality between bits of $\mathbb{Z}_2$, and the sameness of two codes is the equality of the spaces their check rows span.