The Exercise

The $\pi/8$ gate $T$ and the rotation $R_z(\pi/4)$, which turns the Bloch vector of a qubit by $\pi/4$ about the $z$ axis, are two gates on a single qubit. The exercise is to show that they have the same effect: whatever state $\rho$ the qubit is in, applying $T$ leaves it in the same state as applying $R_z(\pi/4)$.^✦

A Quantum System

A quantum system $S$ is given by its state space, a finite dimensional complex Hilbert space.^✦ A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$.^✦ A unitary $U$ of $S$ is an operator on the state space of $S$ with $U^\dagger U = U U^\dagger = 1$, a closed evolution of the system, under which a state $\rho$ of $S$ evolves as $$\rho \longmapsto U \rho U^\dagger .$$(1)

The qubit is the quantum system whose state space is $\mathbb{C}^2$ itself, the two dimensional complex Euclidean space, with the computational basis $|0\rangle$, $|1\rangle$.^✦ A gate on the qubit is a unitary of the qubit, given by its $2 \times 2$ matrix in that basis.

Rotations and the $\pi/8$ Gate

The Pauli matrices $X$, $Y$ and $Z$ are the complex $2 \times 2$ matrices $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(2) Together they form the Pauli vector $\vec{\sigma} = (X, Y, Z)$. For a real vector $\vec{n} = (n_x, n_y, n_z)$ in three dimensions, $\vec{n} \cdot \vec{\sigma}$ is the complex $2 \times 2$ matrix $n_x X + n_y Y + n_z Z$.^✦

The rotation $R_{\vec{n}}(\theta)$ about the axis $\vec{n}$ by a real angle $\theta$ is 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 $\hat{n}$. About the $z$ axis, $\hat{z} = (0, 0, 1)$ and $\hat{z} \cdot \vec{\sigma} = Z$, so that $$R_z(\theta) = R_{\hat{z}}(\theta) = e^{-i\theta Z/2}.$$(4)^✦

The rotation gate $R_z(\theta)$ is the unitary of the qubit whose matrix in the computational basis is (4).^✦ The $\pi/8$ gate $T$ is the unitary of the qubit whose matrix is $$T = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{pmatrix}.$$(5)^✦

The Claim

Since $Z$ is diagonal, so is $R_z(\pi/4)$, and comparing its entries with (5), $$R_z(\pi/4) = \begin{pmatrix} e^{-i\pi/8} & 0 \\ 0 & e^{i\pi/8} \end{pmatrix} = e^{-i\pi/8}\, T.$$(6) In the evolution (1) this global phase meets its complex conjugate and cancels.

What Is Checked

The claim is stated over the states of the qubit and their evolution (1) under two unitaries of the qubit: the $\pi/8$ gate, with the matrix (5), and the rotation gate $R_z(\pi/4)$, whose matrix (4) is built from the Pauli matrices (2) through the exponential (3).

About these constructions the library proves that the matrices of $T$ and of $R_z(\theta)$, for every angle $\theta$, are unitary, and that conjugating a state of any quantum system by a unitary of the system gives a positive operator of unit trace, so that each side of (7) is again a state of the qubit.