The Exercise

The three qubit phase flip code encodes one qubit into three, with the logical states $|0_L\rangle = |{+}{+}{+}\rangle$ and $|1_L\rangle = |{-}{-}{-}\rangle$. The errors in view are no error at all and a phase flip $Z$ on one of the three qubits, the set $\{I, Z_1, Z_2, Z_3\}$. The exercise is to verify that the code satisfies the quantum error correction conditions for this set of errors.

In the library the code is given by its projector $P$ and the errors by four operators $E_0, \dots, E_3$, all on the state space of three qubits.

Three Qubits

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 in the state space of $S$, $\lVert \psi \rVert = 1$, determined physically up to a global phase.^✦ The composite system $S \otimes T$ of two systems $S$ and $T$ is the system whose state space is the tensor product of their state spaces.^✦

The qubit $Q$ is the two level system whose state space is $\mathbb{C}^2$.^✦ Write $|0\rangle$ and $|1\rangle$ for the standard basis vectors of $\mathbb{C}^2$. The states $|+\rangle$ and $|-\rangle$ of the qubit $Q$ are the pure states of $Q$ with the vectors $$|+\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \frac{|0\rangle + |1\rangle}{\sqrt{2}}, \qquad |-\rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -1 \end{pmatrix} = \frac{|0\rangle - |1\rangle}{\sqrt{2}},$$(1) the eigenvectors of the Pauli operator $X$ with eigenvalues $+1$ and $-1$. Three qubits form the composite system $Q \otimes (Q \otimes Q)$, on which the code below is defined.

The Code

Label the two eigenvectors of $X$ by $s \in \{0, 1\}$: $$|x_0\rangle = |+\rangle, \qquad |x_1\rangle = |-\rangle.$$(2)^✦ For labels $s_1, s_2, s_3 \in \{0, 1\}$, the vector $|x_{s_1} x_{s_2} x_{s_3}\rangle$ of the three qubits is the product $$|x_{s_1} x_{s_2} x_{s_3}\rangle = |x_{s_1}\rangle \otimes \bigl( |x_{s_2}\rangle \otimes |x_{s_3}\rangle \bigr).$$(3)^✦ The codewords of the phase flip code are the two such vectors with all three labels equal, $$|a_L\rangle = |x_a x_a x_a\rangle, \qquad a \in \{0, 1\}.$$(4)^✦ The code projector $P$ of the phase flip code is the operator on the state space of the three qubits that sums the projectors onto the two codewords: $$P = \sum_{a \in \{0, 1\}} |a_L\rangle \langle a_L|.$$(5)^✦

Phase Flips

A unitary $U$ of a system $S$ is a closed system evolution of $S$: an operator on the state space of $S$ with $U^\dagger U = U U^\dagger = I$.^✦ The identity $I$ of $S$ is the identity operator on the state space of $S$, and it is a unitary of $S$. For unitaries $U$ of $S$ and $V$ of $T$, the product unitary $U \otimes V$ is the unitary of $S \otimes T$ whose operator is the tensor product of the two.^✦ A unitary of a system of qubits is also called a gate. With $V = I$ the product unitary is the gate $U \otimes I$, which acts on the first factor alone; with $U = I$ it is the gate $I \otimes V$, which acts on the last.

The phase flip $Z$ is the unitary of the qubit whose matrix is the Pauli matrix $$Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(6)^✦ Built from it, the four error gates are unitaries of the three qubit system, $$I, \qquad Z_1 = Z \otimes I, \qquad Z_2 = I \otimes (Z \otimes I), \qquad Z_3 = I \otimes (I \otimes Z),$$(7)^✦ where each $I$ is the identity of the qubits it stands for. The error operators are these four, labelled in this order: $$E_0 = I, \qquad E_1 = Z_1, \qquad E_2 = Z_2, \qquad E_3 = Z_3.$$(8)^✦

The Claim

An operator $P$ on the state space of a system, together with a family of operators $E_i$ on the same space, satisfies the quantum error correction conditions when there is a Hermitian complex matrix $\alpha$, with rows and columns labelled like the $E_i$, such that $$P E_i^\dagger E_j P = \alpha_{ij} P \quad \text{for all } i, j.$$(9)^✦

What Is Checked

The claim is stated over the code projector $P$ of (5), built from the codewords $|{+}{+}{+}\rangle$ and $|{-}{-}{-}\rangle$, and over the error operators of (8), the operators of the gates $I$, $Z_1$, $Z_2$, $Z_3$ on three qubits.

Of these constructions the library proves that $|+\rangle$ and $|-\rangle$ have unit norm and that the matrix of $Z$ is unitary. It also proves that the tensor product of two unitaries is unitary, so each error gate is a unitary of the three qubit system.