The Exercise

A single photon shared between two optical modes carries a qubit: the photon is in one mode or in the other, or in a superposition of the two. Gates on this qubit are built from phase shifters and beamsplitters. The exercise is to build the Hadamard gate from them: a phase shifter on one mode followed by a beamsplitter on both acts on the qubit as the Hadamard matrix $H$, up to a global phase.

The library states the claim for a phase shifter of phase $\pi$ and a beamsplitter of angle $\pi/2$, which are defined below.

Qubit Gates

The Pauli matrices 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) and the Hadamard matrix is $$H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}.$$(2)

For a real vector $\vec{n}$ in three dimensions, with components $n_x$, $n_y$, $n_z$, and the Pauli vector $\vec{\sigma} = (X, Y, Z)$, the matrix $\vec{n} \cdot \vec{\sigma}$ is the complex $2 \times 2$ matrix that combines the Pauli matrices with these components as coefficients, and the rotation $R_{\vec{n}}(\theta)$ about $\vec{n}$ by a real angle $\theta$ is its exponential: $$\vec{n} \cdot \vec{\sigma} = n_x X + n_y Y + n_z Z, \qquad R_{\vec{n}}(\theta) = \exp\Bigl(-\frac{i\theta}{2}\, \vec{n} \cdot \vec{\sigma}\Bigr).$$(3) For a unit vector $\hat{n}$ the rotation is written $R_{\hat{n}}(\theta)$. The rotation $R_y(\theta)$ of a qubit about the $\hat{y}$ axis is $R_{\hat{n}}(\theta)$ with $\hat{n} = (0, 1, 0)$, the matrix $e^{-i\theta Y/2}$.^✦

Optical Elements

On $d$ modes, the state of one photon is a column of $d$ complex amplitudes, one for each mode, and an optical element acts on it as a complex $d \times d$ matrix.

For a complex $2 \times 2$ matrix $G$ and two modes $p$ and $q$ among $d$, the two-level embedding $[G]_{pq}$ is the complex $d \times d$ matrix that acts as $G$ on the amplitudes of $p$ and $q$ and as the identity on every other mode.^✦ The beamsplitter $B_{pq}(\theta)$ of two modes $p$ and $q$ and a real angle $\theta$ is the two-level embedding of the rotation $R_y(\theta)$ on those modes, $$B_{pq}(\theta) = \bigl[R_y(\theta)\bigr]_{pq}.$$(4)^✦

The phase shifter $P_p(\varphi)$ of a mode $p$ among $d$ and a real phase $\varphi$ is the diagonal complex $d \times d$ matrix with diagonal entries $$P_p(\varphi)_{jj} = \begin{cases} e^{i\varphi}, & j = p, \\ 1, & j \neq p. \end{cases}$$(5)^✦

On the two modes $0$ and $1$, the column with amplitude one in mode $0$ is the state $|01\rangle$, the logical $|0\rangle$ of the qubit, and the column with amplitude one in mode $1$ is $|10\rangle$, the logical $|1\rangle$. On these two modes $d = 2$, and the two-level embedding $[G]_{01}$ is $G$ itself. The optical Hadamard circuit $C$ on the modes $0$ and $1$ is the phase shifter of phase $\pi$ on mode $0$ followed by the beamsplitter of angle $\pi/2$ on the two modes: $$C = B_{01}(\pi/2)\, P_0(\pi) = R_y(\pi/2) \begin{pmatrix} e^{i\pi} & 0 \\ 0 & 1 \end{pmatrix}.$$(6)^✦

The Claim

What Is Checked

The claim is stated over complex $2 \times 2$ matrices and columns of amplitudes: the circuit (6), whose beamsplitter is the exponential (3) about the $\hat{y}$ axis, the Hadamard matrix (2), and the two columns $|01\rangle$ and $|10\rangle$.