Single-Qubit Gates with Linear Optics

Here is a sentence almost no other quantum-computing platform gets to say: you have already built the gates. On a superconducting machine a single-qubit gate is a microwave pulse, shaped and calibrated to the nanosecond; on an ion it is a laser flash with its phase servo-locked. On a photonic chip a single-qubit gate is a piece of the chip — the very Mach–Zehnder interferometer whose 2×2 unitary you spent Module 4 programming. This lesson performs the reread: take the MZI-as-SU(2) result, replace "field amplitude" with "probability amplitude", and watch the classical transfer function turn into the Hadamard, the phase gates and every rotation of the Bloch sphere — with literally zero new hardware.

The dual-rail dictionary

A dual-rail qubit is one photon in two waveguides, state vector \alpha|0\rangle_L + \beta|1\rangle_L with |0\rangle_L = |1,0\rangle, |1\rangle_L = |0,1\rangle. Because the photon count is fixed at one, the two-mode field state lives in a two-dimensional Hilbert space, and every lossless linear-optical element acts on (\alpha, \beta)^\mathsf{T} as a 2×2 unitary — the same matrix that Module 4 applied to classical amplitudes. The dictionary has two entries:

Chain them — coupler, internal phase \theta, coupler — and multiply the matrices, exactly as in Module 4:

U_{\text{MZI}}(\theta) \;=\; \tfrac12\begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix} \begin{pmatrix} e^{i\theta} & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix} \;=\; i\,e^{i\theta/2} \begin{pmatrix} \sin\tfrac{\theta}{2} & \cos\tfrac{\theta}{2} \\[2pt] \cos\tfrac{\theta}{2} & -\sin\tfrac{\theta}{2} \end{pmatrix}.

Global phases are unobservable, so strip the prefactor and read the dial:

That last bullet deserves a pause. Module 4's transfer function P_{\text{bar}} = \sin^2(\theta/2) described how a watt of laser power divides between two ports. The identical formula now states the probability that one photon takes one exit. Nothing about the chip changed; what changed is that the quantity flowing through it is now a probability amplitude — and that, when we later put two photons in, those amplitudes will do things power never could.

Watching the Born rule on the bench

Prepare a photon in the real superposition \cos\alpha\,|0\rangle_L + \sin\alpha\,|1\rangle_L and push it through an MZI of internal phase \theta. The output amplitude in rail 0 is \sin(\tfrac\theta2 + \alpha) (multiply it out — one line), so the two detectors fire with probabilities \sin^2(\tfrac\theta2+\alpha) and \cos^2(\tfrac\theta2+\alpha). Scan \theta and you trace a fringe; move the slider to change the input state and the whole fringe slides — the interferometer cannot tell "phase applied by the gate" from "phase already in the qubit", which is precisely why it can be used to measure a qubit in any basis you like.

Worked example: dialling up H, X and Z

Set \theta = \pi/2. The matrix becomes \tfrac{1}{\sqrt2}\bigl(\begin{smallmatrix} 1 & 1 \\ 1 & -1 \end{smallmatrix}\bigr) up to global phase — the Hadamard. Check its action: H|0\rangle_L puts amplitude 1/\sqrt2 in each rail, and a detector pair confirms 50:50 clicks. Now cascade two identical MZIs: the matrices multiply, H \cdot H = I, and the photon exits the rail it entered with certainty — a striking bench experiment, since each MZI alone randomises the exit. Only amplitudes, not probabilities, explain it: the second MZI undoes the first because the two 50:50 splits interfere, just as in the classical mesh.

Set \theta = 0 (a bare pair of couplers, no phase): the matrix is anti-diagonal — every photon swaps rails, the Pauli X. Set \theta = \pi: the rails are preserved but the |1\rangle_L amplitude picks up a minus sign, the Pauli Z. One device, three textbook gates, selected by one heater current — and since Module 4's mesh is tiled from exactly these MZIs, a photonic quantum processor's entire single-qubit layer is the classical matrix mesh, unmodified.

Polarisation: waveplates and the Poincaré sphere

For polarisation qubits the same SU(2) story is told with birefringent crystal instead of interference. A waveplate is a slab whose two crystal axes have different refractive indices: light polarised along the fast axis outruns light along the slow axis, so the slab applies a relative phase between two polarisation modes — the same \mathrm{diag}(1, e^{i\varphi}) as a phase shifter, in a rotated basis set by the plate's orientation. Two standard cuts do almost everything:

Long before qubits, opticians mapped polarisation states onto a sphere: linear polarisations around the equator, circular at the poles — the Poincaré sphere, drawn by Henri Poincaré in 1892. It is the Bloch sphere, a century early, axis for axis: same geometry, same rotations, different century's notation. Every intuition you own about Bloch rotations transfers to waveplates without translation, and vice versa.

A tempting reading of the 50:50 fringes: "half the photons went one way and half the other, and the two crowds interfered." Wrong on both counts. Run the MZI at one photon per second and the fringe pattern still builds up, click by click, in the statistics of single events — each photon's amplitude takes both arms and interferes with itself at the recombiner. Dirac's famous summary ("each photon interferes only with itself") oversells slightly — the next lesson is precisely about two photons interfering — but for linear gates on one photon it is exactly right, and it kills a second misconception too: the beamsplitter does not split the photon into two half-energy fragments. Put a detector in each arm and you always get one click in one arm, never two half-clicks. What divides is the amplitude; what arrives is always a whole photon; and if you do detect which arm, the superposition — and the fringe — is gone.

Henri Poincaré introduced his sphere in 1892 to tame the zoo of elliptical polarisation states; Felix Bloch drew the same picture for spin-½ in 1946; quantum information adopted it decades later. That one sphere serves all three is no coincidence — polarisation, spin-½ and the qubit are each a two-dimensional complex state space, and SU(2) has only one two-dimensional story to tell. The practical payoff for a photonics lab is very concrete: a century of polarimetry hardware — waveplates, compensators, polarimeters that read out a point on the Poincaré sphere — is, without modification, a full toolkit for preparing, rotating and tomographing polarisation qubits. Quantum optics didn't need to invent single-qubit control; it inherited it.

Where this goes next

Single-qubit control, then, comes free with the photonic platform: the mesh layouts of Module 4 already route, rotate and measure photonic qubits in any basis. Everything now hangs on making two photons respond to each other. The next lesson presents the one two-photon effect linear optics does provide — the Hong–Ou–Mandel interference of indistinguishable photons on a beamsplitter — which is not yet a gate, but is the raw material every photonic entangling scheme is machined from.