The Exercise

A qubit passes through two Hadamard gates, one after the other. The exercise is to show that it comes out in the state it went in with, whatever that state was.

A Quantum System

A quantum system $S$ is a physical object whose state space is a finite dimensional complex Hilbert space.^✦ For a finite dimensional complex Hilbert space $\mathcal{H}$, the system with model space $\mathcal{H}$ is the quantum system whose state space is $\mathcal{H}$ itself.^✦ The qubit is the two level quantum system with model space $\mathbb{C}^2$, the standard two dimensional complex Euclidean space.^✦

A unitary $U$ of a quantum system $S$, the evolution of $S$ as a closed system, is an operator on the state space of $S$ with $$U^\dagger U = U U^\dagger = I,$$(1)^✦ where $I$ is the identity operator.

The Hadamard Gate

The Hadamard gate $H$ is the unitary of the qubit whose operator has, in the standard basis of $\mathbb{C}^2$, the matrix $$H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}.$$(2)^✦

The Claim

The square $H^2$ is the Hadamard gate applied twice in succession.

What Is Checked

The claim (3) is stated over the Hadamard gate as a unitary of the qubit, the operator on $\mathbb{C}^2$ with matrix (2). About this construction the library states that the matrix (2) is unitary, that is, it satisfies $M^\dagger M = M M^\dagger = I$, and that the operator of a unitary matrix is unitary, so that $H$ satisfies (1).