The Exercise

The three qubit phase flip code stores one qubit in three, written in the eigenstates $|\pm\rangle$ of the Pauli $X$: it encodes $|0\rangle$ as $|0_L\rangle = |{+}{+}{+}\rangle$ and $|1\rangle$ as $|1_L\rangle = |{-}{-}{-}\rangle$. Write $X_i$ for $X$ acting on the $i$th qubit. The exercise is to show that the stabilizer of the code, the Pauli operators on the three qubits that leave every code state unchanged, is generated by $X_1X_2$ and $X_2X_3$.

A Qubit and Its Observables

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ A pure state $\psi$ of $S$ is a unit vector $|\psi\rangle$ in the state space of $S$.^✦ The qubit is the two level system whose state space is $\mathbb{C}^2$.^✦

An observable $A$ on a system $S$ is a self adjoint operator on the state space of $S$, $A^\dagger = A$.^✦ The Pauli matrices are $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(1) For a real vector $\vec{n} = (n_x, n_y, n_z)$ in three dimensions, the Pauli observable $\vec{n} \cdot \vec{\sigma}$ is the observable on the qubit whose matrix is $n_x X + n_y Y + n_z Z$^✦, and the observable $X$ is $\hat{n} \cdot \vec{\sigma}$ for the unit vector $\hat{n} = (1, 0, 0)$.^✦ The identity observable $I$ on $S$ is the observable whose operator is the identity on the state space of $S$.^✦

A pure state $\psi$ of $S$ is an eigenstate of an observable $A$ on $S$ with eigenvalue $\mu$, a real number, when $A|\psi\rangle = \mu|\psi\rangle$.^✦

The states $|+\rangle$ and $|-\rangle$ are the pure states of the qubit with vectors $$|\pm\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ \pm 1 \end{pmatrix} = \frac{|0\rangle \pm |1\rangle}{\sqrt{2}},$$(2) where the normalization $1/\sqrt{2}$ makes each a unit vector.

The Phase Flip Code

The composite system $S \otimes T$ of two systems $S$ and $T$ is the quantum system whose state space is the tensor product of their state spaces.^✦ For pure states $\psi$ of $S$ and $\varphi$ of $T$, the product $\psi \otimes \varphi$ is the pure state of $S \otimes T$ with vector $|\psi\rangle \otimes |\varphi\rangle$.^✦ Observables combine the same way: the tensor product $A \otimes B$ of an observable $A$ on $S$ and an observable $B$ on $T$ is the observable on $S \otimes T$ whose operator is the tensor product of theirs.^✦ The three qubit register is the composite system of three qubits, the quantum system whose state space is $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$.^✦

Label the basis $\{|+\rangle, |-\rangle\}$ of the qubit by $\{0, 1\}$, $|+\rangle$ by $0$ and $|-\rangle$ by $1$, and write $f_s$ for its member with label $s$.^✦ For labels $s_1, s_2, s_3 \in \{0, 1\}$, the basis state $|f_{s_1} f_{s_2} f_{s_3}\rangle = |f_{s_1}\rangle \otimes |f_{s_2}\rangle \otimes |f_{s_3}\rangle$ is a vector of the register’s state space. The codeword $|a_L\rangle$ with label $a \in \{0, 1\}$ is the basis state whose three labels all equal $a$^✦; as a pure state of the register it is the product $f_a \otimes f_a \otimes f_a$ of three copies of the state $f_a$.^✦

The code projector $P$ of the phase flip code is the operator on the register’s state space $$P = \sum_{a \in \{0,1\}} |a_L\rangle\langle a_L| = |{+}{+}{+}\rangle\langle{+}{+}{+}| + |{-}{-}{-}\rangle\langle{-}{-}{-}|.$$(3)^✦

On the register, the observables $X_1X_2$, $X_2X_3$ and $X_1X_3$ are tensor products of $X$ and the identity: $$X_1X_2 = X \otimes X \otimes I, \qquad X_2X_3 = I \otimes X \otimes X, \qquad X_1X_3 = X \otimes I \otimes X.$$(4)

The projector $P_+(A)$ of an observable $A$ on a system $S$ is the operator $$P_+(A) = \tfrac{1}{2}\,(I + A),$$(5)^✦ which projects onto the $+1$ eigenspace of $A$ when $A^2 = I$.

The Claim

What Is Checked

The claim is stated over the codeword states $|a_L\rangle$ of the three qubit register, the observables $X_1X_2$, $X_2X_3$ and $X_1X_3$ of (4), built from the Pauli observable $X$ and the identity, the code projector $P$ of (3) and the projectors $P_+$ of (5).

The library proves that a product of pure states is a pure state, the norm of a product vector being the product of the norms. It proves that the Pauli observable is self adjoint, its matrix being Hermitian, and that the identity and every tensor product of observables are self adjoint as well; $X_1X_2$, $X_2X_3$ and $X_1X_3$ are therefore observables of the register.