The Exercise

A gate on one qubit is a unitary operator on its two dimensional state space. Every such gate is a rotation of the qubit about some axis in three dimensions by some angle, multiplied by a phase factor. The exercise is to show this, and to find the phase, the angle and the axis for two particular gates, the Hadamard gate and the phase gate.

The Qubit and Its Gates

A quantum system $Q$ 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 $|0\rangle$, $|1\rangle$.^✦ A unitary $U$ of a quantum system $Q$, a closed system evolution of $Q$, is an operator on the state space of $Q$ with $U^\dagger U = U U^\dagger = 1$.^✦

Each gate on this page is the unitary of the qubit that acts on $\mathbb{C}^2$ as a $2 \times 2$ complex matrix in the computational basis. The Pauli matrices are $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$(1) and for a real vector $\vec{n} = (n_x, n_y, n_z)$ in three dimensions the matrix $\vec{n} \cdot \vec{\sigma}$ is $n_x X + n_y Y + n_z Z$, with $\vec{\sigma} = (X, Y, Z)$ the Pauli vector.

The Hadamard gate $H$ and the phase gate $S$ are the unitaries of the qubit with matrices $$H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, \qquad S = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}.$$(2)

Rotations

The rotation $R_{\vec{n}}(\theta)$ about the axis $\vec{n}$, a real vector in three dimensions, by a real angle $\theta$ is the unitary of the qubit whose matrix is the matrix exponential $$R_{\vec{n}}(\theta) = \exp\!\left( -\frac{i\theta}{2}\, \vec{n} \cdot \vec{\sigma} \right).$$(3)^✦ For a unit vector $\hat{n}$ it is written $R_{\hat{n}}(\theta)$. The rotation $R_z(\theta)$ about the $\hat{z}$ axis is the rotation about $\hat{n} = (0, 0, 1)$, with matrix $$R_z(\theta) = e^{-i\theta Z/2}.$$(4)^✦

The Claim

What Is Checked

The claims are stated over unitaries of the qubit: an arbitrary one, $U$, in (5), the gates $H$ and $S$ of (2) in (6) and (7), and on the right of each equation the rotations $R_{\hat{n}}(\theta)$ and $R_z(\theta)$ of (3) and (4).

The library proves that the matrix of each of these gates is unitary, the rotation $R_{\vec{n}}(\theta)$ for every real vector $\vec{n}$ and every angle $\theta$ and $R_z(\theta)$ for every angle, so that each of them is a unitary of the qubit.