The Exercise

A gate on a single qubit is a unitary operator $U$ on its state space $\mathbb{C}^2$ : in the computational basis $|0\rangle$ , $|1\rangle$ , a complex $2 \times 2$ matrix with $U^\dagger U = I$ . Corollary 4.2 builds every such gate from five gates on the same qubit. Three are single-qubit gates $A$ , $B$ and $C$ whose product is the identity; the other two are bit flips $X$ , the gate that exchanges $|0\rangle$ and $|1\rangle$ , one between $C$ and $B$ and one between $B$ and $A$ . Without the flips the circuit does nothing. With them it performs $U$ , up to a global phase $e^{i\alpha}$ .

The Bit Flip

The Pauli matrix $X$ is the complex $2 \times 2$ matrix $X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.$ (1)^✦

The Claim

Theorem 1. Let $U$ be a unitary complex $2 \times 2$ matrix. Then there are a real number $\alpha$ and unitary complex $2 \times 2$ matrices $A$ , $B$ and $C$ with $ABC = I$ such that $U = e^{i\alpha} \, A X B X C .$ (2)^✦

What Is Checked

The claim is stated over unitary complex $2 \times 2$ matrices, the gates of a single qubit, and over the Pauli matrix $X$ of (1).