The Exercise

The Steane code stores one qubit in seven. Its stabilizer is generated by six products of Pauli operators $g_1, \dots, g_6$, three built from $X$ and three from $Z$, and its logical $\bar Z$ applies $Z$ to every qubit. The logical zero $|0_L\rangle$ is an equal superposition of eight computational basis states of the seven qubits. The exercise is to show that the seven operators $g_1, \dots, g_6, \bar Z$ single this state out: it is an eigenstate of each of them with eigenvalue $+1$, and every state of the seven qubits with that property is $|0_L\rangle$ up to a global phase.

A Quantum System

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$.^✦ A pure state $|\psi\rangle$ of $S$ is a unit vector in the state space of $S$, determined physically up to a global phase^✦; the state $|\psi\rangle\langle\psi|$ it determines is the rank one projector onto that vector.^✦

A unitary $U$ of $S$, a closed system evolution of $S$, is an operator on the state space of $S$ with $U^\dagger U = U U^\dagger = 1$.^✦ A pure state $|\psi\rangle$ of $S$ is an eigenstate of $U$ with eigenvalue the complex number $\mu$ when $$U|\psi\rangle = \mu|\psi\rangle .$$(1)^✦

The register of $n$ qubits is the quantum system with one qubit for each of $n$ positions.^✦ Its state space is the space of complex functions on the bit vectors $w \in \mathbb{Z}_2^n$^✦, and its computational basis $\{|w\rangle\}$ is the orthonormal basis with one vector $|w\rangle$ for each bit vector, the function equal to $1$ at $w$ and to $0$ elsewhere. The Pauli operators of a single qubit are $X$, which exchanges its two basis states, and $Z$, which changes the sign of the second one. For a bit vector $v$, 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 $1$.^✦ For a bit vector $u$, 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.^✦ On the basis they act as $$X^v|w\rangle = |w + v\rangle, \qquad Z^u|w\rangle = (-1)^{u \cdot w}|w\rangle,$$(2) with $u \cdot w$ the dot product of $u$ and $w$ modulo $2$.

The Steane Code

Let $H$ be the matrix over $\mathbb{Z}_2$ with three rows and seven columns whose $j$th column is the binary numeral of $j$, most significant bit on top: $$H = \begin{pmatrix} 0&0&0&1&1&1&1 \\ 0&1&1&0&0&1&1 \\ 1&0&1&0&1&0&1 \end{pmatrix}.$$(3)^✦ The Hamming code $C$ is the linear code of length $7$ over $\mathbb{Z}_2$ whose codewords are the words annihilated by $H$. The dual Hamming code $C_2 = C^\perp$ of $C$ is the linear code of length $7$ over $\mathbb{Z}_2$ whose codewords are the words orthogonal to every codeword of $C$; $C_2$ has dimension $3$ and so eight codewords.

For a binary linear code $C_2$ of length $n$ and a bit vector $x \in \mathbb{Z}_2^n$, the coset state $|x + C_2\rangle$ 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 $C_2$: $$|x + C_2\rangle = \frac{1}{\sqrt{|C_2|}} \sum_{y \in C_2} |x + y\rangle .$$(4)^✦ The Steane logical zero $|0_L\rangle$ is the coset state of the dual Hamming code at the zero word, a pure state of seven qubits: $$|0_L\rangle = |0 + C_2\rangle = \frac{1}{\sqrt 8} \sum_{y \in C_2} |y\rangle .$$(5)^✦

The rows $h_1, h_2, h_3$ of $H$ give the six stabilizer generators $g_1, \dots, g_6$ of the Steane code, unitaries of the seven qubits. The three of $X$ type are the bit flip strings $g_i = X^{h_i}$; the three of $Z$ type are the phase flip strings $g_{i+3} = Z^{h_i}$, for $i = 1, 2, 3$.^✦ The logical $\bar Z = Z_1 Z_2 \cdots Z_7$, where $Z_i$ is $Z$ acting on the $i$th qubit, is the phase flip string $Z^{1111111}$ on the all ones word.^✦ Appended to the six generators, it makes up the list of seven operators $$(g_1, \dots, g_6, \bar Z) = (X^{h_1}, X^{h_2}, X^{h_3}, Z^{h_1}, Z^{h_2}, Z^{h_3}, Z^{1111111}).$$(6)^✦

The Claim

What Is Checked

The claim is stated over the register of seven qubits, the seven operators (6) built from the rows of $H$, the logical zero (5), the eigenvalue condition (1) and, in its second part, the states $|\psi\rangle\langle\psi|$ and $|0_L\rangle\langle 0_L|$.

On these constructions the library proves that every coset state (4) has unit norm, so that $|0_L\rangle$ is a pure state, and that the projector onto a unit vector is a density operator, so that $|\psi\rangle\langle\psi|$ is a state. It proves each bit flip string unitary as a linear isometry of the state space onto itself, and each phase flip string unitary as a diagonal operator whose entries $\pm 1$ have modulus one.