The Exercise

The Steane code encodes one qubit in seven. Its logical states $|0_L\rangle$ and $|1_L\rangle$ are equal superpositions of computational basis states of the seven qubits, labelled by the codewords of a classical code built from the $[7,4,3]$ Hamming code. Its stabilizer has six generators: three are tensor products of Pauli $X$ operators and identities, the other three of Pauli $Z$ operators and identities. The exercise is to check that every generator stabilizes both logical states: applied to $|0_L\rangle$ or to $|1_L\rangle$, it returns the same state.

A Register of Qubits

A quantum system $S$ is described by its state space, a finite dimensional complex Hilbert space.^✦ A unitary $U$ of $S$ is an operator on the state space of $S$ with $U^\dagger U = U U^\dagger = 1$.^✦

The register of $n$ qubits is the quantum system whose state space is the space of complex functions on the bit vectors $w \in \mathbb{Z}_2^n$.^✦ The computational basis of the register is the orthonormal basis of the states $|w\rangle$, one for each bit vector $w$, with $|w\rangle$ the function equal to $1$ at $w$ and $0$ elsewhere.^✦ Bit vectors add componentwise modulo $2$. The Steane code lives on the register of seven qubits.

For a bit vector $v \in \mathbb{Z}_2^n$, the bit flip string $X^v = X^{v_1} \otimes \cdots \otimes X^{v_n}$ is the unitary of the register that flips the qubits where $v$ has a $1$, adding $v$ to the label of each basis state: $$X^v |w\rangle = |w + v\rangle .$$(1)^✦ For a bit vector $u \in \mathbb{Z}_2^n$, the phase flip string $Z^u = Z^{u_1} \otimes \cdots \otimes Z^{u_n}$ is the unitary of the register that is diagonal in the computational basis, with $$Z^u |w\rangle = (-1)^{u \cdot w} |w\rangle ,$$(2)^✦ where $u \cdot w = \sum_j u_j w_j$ modulo $2$.

The Generators

The parity check matrix $H$ is the $3 \times 7$ matrix over $\mathbb{Z}_2$ whose $j$th column is the binary numeral of $j$, most significant bit on top; it is the parity check matrix of the $[7,4,3]$ Hamming code, defined in the section on the logical states: $$H = \begin{pmatrix} 0&0&0&1&1&1&1 \\ 0&1&1&0&0&1&1 \\ 1&0&1&0&1&0&1 \end{pmatrix} .$$(3)^✦ Its rows $h_1, h_2, h_3$ are bit vectors in $\mathbb{Z}_2^7$.

The $X$ type generators $g_1, g_2, g_3$ of the Steane code are the bit flip strings of the rows of $H$, $$g_i = X^{h_i}, \qquad i = 1, 2, 3,$$(4)^✦ and the $Z$ type generators $g_4, g_5, g_6$ are the phase flip strings of the same rows, $$g_{i+3} = Z^{h_i}, \qquad i = 1, 2, 3.$$(5)^✦ Together, the $X$ type first, they are the six stabilizer generators of the Steane code.^✦ Writing $IIIXXXX$ for $I \otimes I \otimes I \otimes X \otimes X \otimes X \otimes X$, and so on, they read $$\begin{aligned} g_1 &= IIIXXXX, & g_2 &= IXXIIXX, & g_3 &= XIXIXIX, \\ g_4 &= IIIZZZZ, & g_5 &= IZZIIZZ, & g_6 &= ZIZIZIZ. \end{aligned}$$(6)^✦

The Logical States

The Hamming code $C_1$ is the binary linear code of length seven whose codewords are the bit vectors $x$ with $Hx = 0$. The dual Hamming code $C_2 = C_1^\perp$ is the binary linear code of length seven made of the bit vectors $y$ with $x \cdot y = 0$ for every codeword $x$ of $C_1$.^✦

A pure state $|\psi\rangle$ of a quantum system $S$ is a unit vector in the state space of $S$, determined physically up to a global phase.^✦

For a binary linear code $C$ of length $n$ and a bit vector $x \in \mathbb{Z}_2^n$, the coset state $|x + C\rangle$ is the pure state of the register of $n$ qubits that superposes with equal amplitudes the basis states $|x + y\rangle$, one for each codeword $y$ of $C$: $$|x + C\rangle = \frac{1}{\sqrt{|C|}} \sum_{y \in C} |x + y\rangle ,$$(7)^✦ where $|C|$ is the number of codewords of $C$.

The logical states $|0_L\rangle$ and $|1_L\rangle$ of the Steane code are the coset states of $C_2$ at the zero vector and at the vector of seven ones: $$|0_L\rangle = |0000000 + C_2\rangle ,$$(8)^✦ $$|1_L\rangle = |1111111 + C_2\rangle .$$(9)^✦

The Claim

A pure state $|\psi\rangle$ of a quantum system $S$ is an eigenstate of a unitary $U$ of $S$ with eigenvalue $\mu \in \mathbb{C}$ when $U|\psi\rangle = \mu\,|\psi\rangle$.^✦

What Is Checked

The claim is stated over the six generators (6), unitaries of the register of seven qubits obtained from the rows of $H$ by (4) and (5), and over the logical states (8) and (9), coset states of the dual Hamming code $C_2$.

The library proves that the bit flip and phase flip strings are unitaries. It proves that each coset state $|x + C\rangle$ has unit norm, the basis states $|x + y\rangle$ being distinct for distinct codewords $y$ and each weighted by $1/\sqrt{|C|}$.