The Exercise

Two qubits pass through a controlled-NOT (CNOT) gate $C$, the first qubit as control and the second as target. Write $X_1$ for the Pauli $X$ applied to the first qubit with the second left alone, $X_2$ for $X$ applied to the second, and $Y_1$, $Y_2$, $Z_1$, $Z_2$ in the same way. The exercise asks what each of these six operators becomes when it is sandwiched between two CNOTs, as in $C X_1 C$; in every case the answer is a Pauli on one qubit or a product of Paulis on the two. It also asks for two rotations that commute with the CNOT: the rotation $R_z(\theta)$ about the $\hat z$ axis applied to the control, and the rotation $R_x(\theta)$ about the $\hat x$ axis applied to the target.

The library states all eight identities as equalities between unitaries of the two qubits.

Two Qubits

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ For such a space $H$ there is the quantum system whose state space is $H$, and the qubit is the quantum system whose state space is $\mathbb{C}^2$. 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; the exercise takes place in $\mathrm{qubit} \otimes \mathrm{qubit}$.^✦

A pure state $\lvert \psi \rangle$ of $S$ is a unit vector in the state space of $S$, determined physically up to a global phase.^✦ The computational basis states $\lvert 0 \rangle$ and $\lvert 1 \rangle$ of the qubit are the pure states given by the standard basis vectors $e_0$ and $e_1$ of $\mathbb{C}^2$.^✦

A unitary $U$ of $S$ is an operator on the state space of $S$ with $U^\dagger U = U U^\dagger = 1$, the evolution of $S$ as a closed system.^✦ The identity evolution of $S$ is the unitary $1$.^✦ For two unitaries $U$ and $V$ of $S$, the unitary $U$ after $V$ is $UV$, first $V$ and then $U$; a product such as $C X_1 C$ is a composite in this sense.^✦ For a unitary $U$ of $S$ and a unitary $V$ of $T$, the product unitary $U \otimes V$ is the unitary of $S \otimes T$ whose operator is the tensor product of the operators of $U$ and $V$.^✦

A gate on one factor of a composite system $S \otimes T$ is a product unitary of $S \otimes T$ that acts as the identity evolution on the other factor. For a unitary $U$ of $S$ and a second system $T$, the gate $U$ on the first factor is the unitary $U \otimes 1$ of $S \otimes T$; for a unitary $V$ of $T$, the gate $V$ on the second factor is $1 \otimes V$.

The Gates

The Pauli matrices are the complex $2 \times 2$ matrices $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(1) The Pauli gates $X$, $Y$ and $Z$ of the qubit are the unitaries whose operators have the matrices (1) in the computational basis; $X$ is the quantum NOT, or bit flip, and $Z$ the phase flip.

For a real vector $\vec n$ in three dimensions, the matrix $\vec n \cdot \vec \sigma$ is $n_x X + n_y Y + n_z Z$, the combination of the Pauli matrices with the components of $\vec n$ as coefficients.^✦ The rotation $R_{\vec n}(\theta)$ about $\vec n$ by a real angle $\theta$ is the complex $2 \times 2$ matrix $$R_{\vec n}(\theta) = \exp\!\left(-\frac{i\theta}{2}\, \vec n \cdot \vec \sigma\right),$$(2)^✦ written $R_{\hat n}(\theta)$ when the axis is a unit vector $\hat n$. The rotations about the axes $\hat x = (1, 0, 0)$ and $\hat z = (0, 0, 1)$ are $R_x(\theta) = e^{-i\theta X/2}$ and $R_z(\theta) = e^{-i\theta Z/2}$. For a real angle $\theta$, the rotation gates $R_x(\theta)$ and $R_z(\theta)$ are the unitaries of the qubit whose operators have these matrices in the computational basis.

For a unitary $U$ of a system $S$, the controlled gate $C(U)$ is the unitary of $\mathrm{qubit} \otimes S$, a control qubit together with the target $S$, whose operator is $$C(U) = \lvert 0 \rangle\langle 0 \rvert \otimes 1 + \lvert 1 \rangle\langle 1 \rvert \otimes U.$$(3) The CNOT gate $C$ is the controlled gate $C(X)$ on two qubits, which flips the second qubit exactly when the first is $\lvert 1 \rangle$.^✦

The Claim

A subscript names the wire a gate of the qubit acts on: $X_1 = X \otimes 1$ is $X$ on the control, $Y_2 = 1 \otimes Y$ is $Y$ on the target, and in the same way $R_{z,1}(\theta) = R_z(\theta) \otimes 1$ and $R_{x,2}(\theta) = 1 \otimes R_x(\theta)$.

What Is Checked

The identities are stated over the CNOT gate, the controlled gate $C(X)$, and over the Pauli gates and the rotation gates $R_x(\theta)$ and $R_z(\theta)$ on either qubit.

About these constructions the library proves that the controlled gate is unitary: for every unitary $U$ of a system $S$, the operator (3) satisfies $C(U)^\dagger C(U) = C(U) C(U)^\dagger = 1$.^✦ It proves the same of the composite and of the tensor product of two unitaries.