The Exercise

A stabilizer code $C_1$ encodes one qubit in $n_1$ qubits, and a second code $C_2$ encodes one qubit in $n_2$. In their concatenation a qubit is first encoded by $C_2$ into $n_2$ qubits, and each of these is then encoded by $C_1$ into a block of $n_1$, so that the logical qubit is carried by $n_1 n_2$ qubits in $n_2$ blocks. The exercise is to show that the concatenated code again encodes one qubit.^✦

The library states this for any two such codes, each given with its generators and a pair of logical operators.

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)^✦ The $X$ bit $x(\sigma)$ of a Pauli matrix $\sigma$ is $1$ for $X$ and $Y$ and $0$ for $I$ and $Z$^✦; its $Z$ bit $z(\sigma)$ is $1$ for $Y$ and $Z$ and $0$ for $I$ and $X$.^✦ The two bits form the binary representation of $\sigma$, and up to a phase $\sigma = X^{x(\sigma)} Z^{z(\sigma)}$. With phases dropped, the product of two Pauli matrices is again one of the four: each squares to $I$, and the product of two different ones among $X$, $Y$, $Z$ is the third.^✦

A Pauli string on $n$ qubits is the operator $\sigma_{p_1} \otimes \cdots \otimes \sigma_{p_n}$ on the $n$ qubits given by a choice of one Pauli matrix $\sigma_{p_k}$ for each qubit $k$, with no phase in front. Two Pauli strings are multiplied qubit by qubit, with phases dropped as above.^✦

Stabilizer Codes

The subspace $V_S$ stabilized by a family $S = (g_1, \dots, g_m)$ of Pauli strings on $n$ qubits is the subspace of the state space of the $n$ qubits formed by the vectors that every $g_i$ leaves invariant, the intersection of their $+1$ eigenspaces: $$V_S = \{\, |\psi\rangle : g_i |\psi\rangle = |\psi\rangle \text{ for } i = 1, \dots, m \,\}.$$(2)^✦

An $[n, 1]$ stabilizer code $C$ on $n$ qubits is a family of $n - 1$ Pauli strings $g_1, \dots, g_{n-1}$ on the $n$ qubits, its generators, together with two Pauli strings $\bar X$ and $\bar Z$, its logical operators, such that $$g_i g_j = g_j g_i, \qquad \bar X g_i = g_i \bar X, \qquad \bar Z g_i = g_i \bar Z, \qquad \bar X \bar Z = -\bar Z \bar X$$(3)^✦ for all $i$ and $j$, and such that the generators are independent: the rows of their check matrix, which hold the $X$ and $Z$ bits of the factors of each generator, are linearly independent modulo $2$.

The Concatenated Code

Let $C_1$ be an $[n_1, 1]$ code with generators $g_1, \dots, g_{n_1 - 1}$ and logical operators $\bar X$ and $\bar Z$, and let $C_2$ be an $[n_2, 1]$ code with generators $g'_1, \dots, g'_{n_2 - 1}$. The encoded version of a Pauli matrix $\sigma$ under $C_1$ is the Pauli string on $n_1$ qubits $$\bar\sigma = \bar X^{x(\sigma)} \bar Z^{z(\sigma)},$$(4)^✦ with the phase dropped, so that $\bar I = I^{\otimes n_1}$ and $\bar Y$ is the product of $\bar X$ and $\bar Z$.

The qubit $(b, q)$ of the concatenated code is the $q$th qubit of block $b$, for $1 \le b \le n_2$ and $1 \le q \le n_1$; the qubits are numbered block by block, the qubit $q$ of block $b$ receiving the number $(b-1)\,n_1 + q$, so that they run from $1$ to $n_1 n_2$. The generators of the concatenated code of $C_1$ and $C_2$ are labelled by the index set $$\mathcal I = \bigl(\{1, \dots, n_2\} \times \{1, \dots, n_1 - 1\}\bigr) \sqcup \{1, \dots, n_2 - 1\}, \qquad |\mathcal I| = n_2 (n_1 - 1) + n_2 - 1 = n_1 n_2 - 1,$$(5) For $1 \le b \le n_2$ and $1 \le a \le n_1 - 1$ the generator $G_{(b, a)}$ places $g_a$ on block $b$, and for $1 \le j \le n_2 - 1$, with $g'_j = \sigma_{p_1} \otimes \cdots \otimes \sigma_{p_{n_2}}$, the generators are $$G_{(b, a)} = I^{\otimes n_1} \otimes \cdots \otimes \underbrace{g_a}_{\text{block } b} \otimes \cdots \otimes I^{\otimes n_1}, \qquad G_j = \overline{\sigma_{p_1}} \otimes \cdots \otimes \overline{\sigma_{p_{n_2}}}.$$(6) With the qubits numbered as above, each $G_c$ with $c \in \mathcal I$, that is $G_{(b, a)}$ or $G_j$, is a Pauli string on $n_1 n_2$ qubits.

An enumeration $\iota$ of the index set $\mathcal I$ is a bijection $$\iota : \mathcal I \to \{1, \dots, n_1 n_2 - 1\},$$(7)^✦ and numbered by one, the generators of the concatenated code are $$h_r = G_{\iota^{-1}(r)}, \qquad r = 1, \dots, n_1 n_2 - 1.$$(8)^✦

The Claim

What Is Checked

The claim is stated over the stabilized subspace (2) of the generators $h_r$ of (8), which are built from two $[n, 1]$ stabilizer codes through the encoded Paulis (4) and the enumeration (7). About these constructions the library proves the count of generators in (5).