The Theorem

A gate on a single qubit is a unitary operator on its state space $\mathbb{C}^2$; in the computational basis it is a unitary $2 \times 2$ complex matrix $U$. Theorem 4.1 of Nielsen and Chuang says that rotations about two fixed axes of the Bloch sphere, $\hat z$ and $\hat y$, are enough to build every such gate.

The library states the theorem for the matrix $U$, with each rotation the exponential of a combination of the three Pauli matrices $X$, $Y$ and $Z$ defined next.

Rotations About an Axis

The Pauli matrices $X$, $Y$ and $Z$ 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) The Pauli vector is $\vec\sigma = (X, Y, Z)$, and 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 $$\vec n \cdot \vec\sigma = n_x X + n_y Y + n_z Z .$$(2)^✦

The rotation $R_{\vec n}(\theta)$ about the axis $\vec n$ 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 written $R_{\hat n}(\theta)$.

The rotation $R_z(\theta)$ about the $z$ axis is the rotation about the unit vector $\hat z = (0,0,1)$, and the rotation $R_y(\theta)$ about the $y$ axis is the one about $\hat y = (0,1,0)$. Since $\hat z \cdot \vec\sigma = Z$ and $\hat y \cdot \vec\sigma = Y$, $$R_z(\theta) = e^{-i\theta Z/2}, \qquad R_y(\theta) = e^{-i\theta Y/2}.$$(4)

The Claim

What Is Checked

The claim is stated over an arbitrary unitary $2 \times 2$ complex matrix $U$ and over the rotations $R_z$ and $R_y$ of (4), which rest on the rotation (3) about an axis, the matrix $\vec n \cdot \vec\sigma$ of (2) and the Pauli matrices (1).