The Exercise

The Hadamard gate $H$ acts on a single qubit. The exercise asks for single qubit unitaries $A$, $B$ and $C$ whose product $ABC$ is the identity, together with a phase $\alpha$, such that $H = e^{i\alpha} AXBXC$, where $X$ is the Pauli $X$ gate. The library answers it with rotations about the $\hat y$ and $\hat z$ axes.

A Quantum System

A quantum system $S$ is a physical object whose state space is a finite dimensional complex Hilbert space.^✦ The qubit is the two level system whose state space is $\mathbb{C}^2$, with the computational basis $|0\rangle$, $|1\rangle$.^✦

A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$, pure or mixed.^✦ A unitary $U$ of $S$, an evolution of $S$ as a closed system, is an operator on the state space of $S$ with $U^\dagger U = U U^\dagger = I$.^✦ The evolution of a state $\rho$ of $S$ under $U$ is $$\rho \mapsto U \rho U^\dagger .$$(1)^✦ For two unitaries $U$ and $V$ of $S$, $U$ after $V$ is the unitary of $S$ with operator $UV$: $V$ acts first, then $U$.^✦

Gates of the Qubit

In the computational basis the Pauli matrices are $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$(2) and the Hadamard matrix is $$H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}.$$(3)

For a real vector $\vec n = (n_x, n_y, n_z)$ in three dimensions, the matrix $\vec n \cdot \vec\sigma$ is $n_x X + n_y Y + n_z Z$.^✦ The rotation by a real angle $\theta$ about $\vec n$ is the $2 \times 2$ complex matrix $$R_{\vec n}(\theta) = \exp\Bigl( -\frac{i\theta}{2}\, \vec n \cdot \vec\sigma \Bigr),$$(4)^✦ written $R_{\hat n}(\theta)$ when the axis is a unit vector $\hat n$. About the $\hat y$ axis, $\hat y = (0, 1, 0)$, and about the $\hat z$ axis, $\hat z = (0, 0, 1)$, the rotations are $$R_y(\theta) = e^{-i\theta Y/2}, \qquad R_z(\theta) = e^{-i\theta Z/2}.$$(5)

The Hadamard gate $H$ is the unitary of the qubit whose operator acts on $\mathbb{C}^2$ as the matrix (3) in the computational basis.^✦ The Pauli $X$ gate, the quantum NOT, and the rotation gates $R_y(\theta)$ and $R_z(\theta)$, for every real angle $\theta$, are the unitaries of the qubit given in the same way by $X$ and by the matrices (5).

The Three Matrices

The library’s matrices are $$A = R_y(\pi/4),$$(6) $$B = R_y(-\pi/4)\, R_z(-\pi/2),$$(7) $$C = R_z(\pi/2).$$(8) As a gate of the qubit, $AXBXC$ is the composite of six gates, $$AXBXC = R_y(\pi/4)\, X\, R_y(-\pi/4)\, R_z(-\pi/2)\, X\, R_z(\pi/2).$$(9)

The Claim

What Is Checked

The claim is stated over the qubit and its states, the evolution (1) of a state under a unitary, the Hadamard, Pauli $X$ and rotation gates and their composites, and the matrices $A$, $B$ and $C$ built from the rotations $R_y$ and $R_z$ of (5), which rest on the Pauli matrices and the exponential (4).

The library proves that the Hadamard matrix, $X$, $R_y(\theta)$ and $R_z(\theta)$ are unitary matrices, so that each gate is a unitary of the qubit, and that a product of unitaries is unitary. Conjugating a density operator by a unitary gives a density operator, so the evolution (1) takes states to states.