The Exercise

A circuit that can rotate a qubit about a fixed axis $\hat n$ of the Bloch sphere by a single angle $\theta$, as many times as it likes, applies the rotation $R_{\hat n}(\theta)$ any number $k$ of times and so has the powers $R_{\hat n}(\theta)^k$ at its disposal. The exercise asks how well they replace the rotation $R_{\hat n}(\alpha)$ about the same axis by an arbitrary angle $\alpha$, with the error of a replacement taken in the worst case over input states.

It has two parts. The error between two rotations about one axis, by angles $\alpha$ and $\alpha + \beta$, has a closed form that depends on $\beta$ alone. The second part uses it to show that whenever the multiples of $\theta$ come arbitrarily close to $\alpha$ modulo $4\pi$, some power of $R_{\hat n}(\theta)$ lies within any prescribed error of $R_{\hat n}(\alpha)$.^✦

Rotations of a Qubit

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) Write $\vec\sigma = (X, Y, Z)$ for the Pauli vector. 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 given by the matrix exponential $$R_{\vec n}(\theta) = \exp\Bigl( -\frac{i\theta}{2}\, \vec n \cdot \vec\sigma \Bigr),$$(3)^✦ written $R_{\hat n}(\theta)$ when the axis is a unit vector $\hat n$, with $n_x^2 + n_y^2 + n_z^2 = 1$.

On the state space $\mathbb{C}^2$ of a qubit, $R_{\vec n}(\theta)$ is regarded as the linear operator given by this matrix: $$R_{\vec n}(\theta) \colon\ |\psi\rangle \longmapsto \exp\Bigl( -\frac{i\theta}{2}\, \vec n \cdot \vec\sigma \Bigr) |\psi\rangle , \qquad |\psi\rangle \in \mathbb{C}^2 .$$(4)^✦

The error $E(U,V)$ made when an operator $V$ on $\mathbb{C}^2$ is applied in place of $U$ is the operator norm of their difference: $$E(U,V) = \| U - V \| = \max_{\| |\psi\rangle \| = 1} \bigl\| (U - V)|\psi\rangle \bigr\| .$$(5)

The Claim

What Is Checked

Both theorems are stated over the rotation operators (4) on the state space of a qubit, whose matrices (3) are built from the Pauli matrices (1) through $\vec n \cdot \vec\sigma$ in (2), and over the error (5) between two such operators.