Quantum computing, from zero to my thesis

A classical bit is either 0 or 1. A qubit's state is written α|0⟩ + β|1⟩ — a mixture of both possibilities at once, where α and β are numbers called amplitudes. Squaring an amplitude gives the probability of finding the qubit in that state when you look: P(0) = |α|², P(1) = |β|².

Every possible qubit state maps to a point on a sphere — the Bloch sphere. North pole is |0⟩, south pole is |1⟩, and everywhere in between is a genuine quantum state. Two dials move the arrow: θ tilts it between the poles (changing the probabilities), φ spins it around the axis (changing something subtler — you'll meet it in the next lesson).

Press the H (Hadamard) gate on a fresh |0⟩ qubit and the arrow swings from the north pole to the equator. Now P(0) = P(1) = ½ — a perfect coin toss. That's superposition: not magic, just a definite arrow pointing sideways.

But here's the part most explanations skip. Once on the equator, the arrow can still spin around the axis — that angle is called phase. Spin it anywhere you like: the probabilities refuse to move. Phase is invisible to a single measurement — yet it's the fuel of every quantum algorithm, because gates can convert hidden phase differences back into visible probability differences.

Take two qubits. Put the first in superposition (H), then apply CNOT — a gate that flips the second qubit only when the first is 1. The result is the Bell state: (|00⟩ + |11⟩)/√2. Measured together, they only ever agree — both 0 or both 1, never mixed.

The eerie part shows up on the spheres. Each qubit's own arrow shrinks to nothing: ask either qubit alone "what state are you in?" and the honest answer is "no state at all — 50/50, no direction." All the information has moved into the correlation between them. The ⟨Z⊗Z⟩ meter reads +1: perfect agreement.

Until you measure, the qubit is its arrow: definite, continuous, spinnable. The moment you measure, two things happen at once. First, you get a plain classical answer — 0 or 1 — drawn randomly with probabilities |α|² and |β|². Second, the arrow snaps to the pole matching your answer. The superposition is gone; measuring again just repeats the same answer.

That's why quantum algorithms are shaped the way they are: compute in superposition, interfere phases, and only measure at the very end — one shot at extracting an answer from an arrow you spent the whole circuit preparing.

A variational circuit is a circuit whose rotation angles are trainable parameters — the quantum cousin of a neural network's weights. To classify a data point: encode it as a rotation, pass it through the parameterized gates, measure ⟨Z⟩, and read the sign as the predicted class.

Training needs gradients, and quantum circuits have a beautiful trick: the parameter-shift rule. Evaluate the circuit twice — once with a dial turned +π/2, once −π/2 — and half the difference is the exact gradient: ∂⟨Z⟩/∂θ = [⟨Z⟩(θ+π/2) − ⟨Z⟩(θ−π/2)] / 2. No approximation. Then it's ordinary gradient descent, just like any ML model.

In a classical CNN, a small filter sweeps the image, computing one number per patch — stacked filters produce feature maps that reveal edges, textures, shapes. A quanvolutional layer keeps the sweep but replaces the filter's arithmetic with a 2-qubit circuit: the four pixels of each 2×2 patch become four RY rotation angles, a CNOT entangles the qubits, and the measured ⟨Z⟩ is the output pixel. Four different rotation offsets give four channels — four quantum feature maps from one drawing.

Why bother? The quantum filter computes a nonlinear, entanglement-mixed function of the patch that classical filters don't naturally express — and exploring where that helps is an open research question.