The Exercise

A gate on a single qubit can be built from rotations about two fixed axes. The exercise asks for the Hadamard gate $H$ in this form: a product of rotations about the $\hat x$ and $\hat z$ axes, times a global phase $e^{i\varphi}$.

The library’s answer uses three rotations by the angle $\pi/2$, about the $\hat x$, $\hat z$ and $\hat x$ axes in turn, and holds both for the gates themselves and for the way they act on the states of the qubit.

A Quantum System

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ The qubit is the two level quantum system whose state space is $\mathbb{C}^2$, with the computational basis $|0\rangle$, $|1\rangle$ in which every matrix below is written.^✦ A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$, pure or mixed.^✦

A unitary $U$ of $S$ is an operator on the state space of $S$ with $U^\dagger U = U U^\dagger = 1$; it describes the evolution of $S$ as a closed system.^✦ For unitaries $U$ and $V$ of $S$, the unitary $U$ after $V$ is the product $UV$, which applies $V$ first and then $U$.^✦ The evolution of a state $\rho$ of $S$ under a unitary $U$ of $S$ is the state $$\rho \mapsto U \rho U^\dagger .$$(1)^✦

Gates of the Qubit

The Pauli matrices are the complex $2 \times 2$ matrices $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(2) For a real vector $\vec n = (n_x, n_y, n_z)$ in three dimensions, the matrix $\vec n \cdot \vec\sigma$ is the combination $n_x X + n_y Y + n_z Z$ of the Pauli matrices, with the components of $\vec n$ as coefficients.^✦ The rotation operator $R_{\vec n}(\theta)$ about $\vec n$ by a real angle $\theta$ is the complex $2 \times 2$ matrix $$R_{\vec n}(\theta) = \exp\!\left( -\frac{i\theta}{2}\, \vec n \cdot \vec\sigma \right),$$(3)^✦ a matrix exponential; for a unit vector $\hat n$ it is the rotation $R_{\hat n}(\theta)$ by $\theta$ about the axis $\hat n$. The rotations about the $\hat x$ and $\hat z$ axes are the cases $\hat n = (1, 0, 0)$ and $\hat n = (0, 0, 1)$: $$R_x(\theta) = e^{-i\theta X/2}, \qquad R_z(\theta) = e^{-i\theta Z/2}.$$(4)

For a real angle $\theta$, the rotation gates $R_x(\theta)$ and $R_z(\theta)$ of the qubit are the unitaries of the qubit whose matrices are $e^{-i\theta X/2}$ and $e^{-i\theta Z/2}$.

The Hadamard gate $H$ is the unitary of the qubit whose matrix is the Hadamard matrix $$H = \frac{1}{\sqrt 2} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}.$$(5)^✦

The Claim

Let $U$ be the unitary of the qubit that applies $R_x(\pi/2)$, then $R_z(\pi/2)$, then $R_x(\pi/2)$ again, so that $U = R_x(\pi/2)\, R_z(\pi/2)\, R_x(\pi/2)$.

What Is Checked

The claim is stated over the Hadamard gate $H$ and the rotation gates $R_x(\pi/2)$ and $R_z(\pi/2)$ of the qubit, over the unitary $U$ that composes them, and over the evolution (1) of a state of the qubit under $H$ and under $U$.

About these constructions the library proves that the Hadamard matrix (5) and the matrices $R_x(\theta)$ and $R_z(\theta)$ of (4) are unitary for every angle $\theta$, so that each gate is a unitary of the qubit. It proves as well that the product of two unitaries is unitary, and that conjugating a density operator by a unitary gives a density operator, so that $U \rho U^\dagger$ is again a state.