The Exercise

A gate $U$ on a single qubit is a unitary operator on the state space $\mathbb{C}^2$ of the qubit, that is, a complex $2 \times 2$ matrix with $U^\dagger U = I$. The exercise asks for a decomposition of an arbitrary such gate into rotations about two fixed axes of the Bloch sphere, $\hat{x}$ and $\hat{y}$: first a rotation about $\hat{x}$, then one about $\hat{y}$, then one about $\hat{x}$ again, with the three angles chosen to fit $U$ and an overall phase factor in front.

Pauli Matrices and Rotations

The Pauli matrices $X$, $Y$ and $Z$, the components of the Pauli vector $\vec{\sigma} = (X, Y, Z)$, 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) with $i$ the imaginary unit. For a real vector $\vec{n} = (n_x, n_y, n_z)$ in three dimensions, the matrix $\vec{n} \cdot \vec{\sigma}$ is the complex $2 \times 2$ matrix formed from (1) with the components of $\vec{n}$ as coefficients, $$\vec{n} \cdot \vec{\sigma} = n_x X + n_y Y + n_z Z.$$(2)^✦

For a real vector $\vec{n}$ in three dimensions and a real angle $\theta$, the rotation $R_{\vec{n}}(\theta)$ 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 $\hat{n}$ this is the rotation operator by $\theta$ about $\hat{n}$.

The rotation $R_x(\theta)$ about the $\hat{x}$ axis by an angle $\theta$ is (3) with axis $\hat{n} = (1, 0, 0)$, and the rotation $R_y(\theta)$ about the $\hat{y}$ axis is the one with axis $\hat{n} = (0, 1, 0)$: $$R_x(\theta) = e^{-i\theta X/2}, \qquad R_y(\theta) = e^{-i\theta Y/2}.$$(4)

The Claim

What Is Checked

The claim is stated over an arbitrary unitary complex $2 \times 2$ matrix $U$ and over the rotations $R_x$ and $R_y$ of (4), which rest on the general rotation (3), on the matrix $\vec{n} \cdot \vec{\sigma}$ of (2) and on the Pauli matrices (1). The library states the decomposition (5) about these constructions.