Parameterized Quantum Circuits

Parameterized Quantum Circuits

The key innovation that makes Quantum Machine Learning possible is the parameterized quantum circuit (PQC): a quantum circuit where some gate angles are tunable variables rather than fixed constants. Just as a neural network learns by adjusting weights, a PQC learns by adjusting rotation angles.

Why Parameterization Matters

A fixed quantum circuit always produces the same output for the same input state. A parameterized circuit produces different outputs depending on $\theta$: this is what enables learning.

The simplest example: the Ry gate.

When $\theta =0$: $R_y|0⟩=|0⟩$ (no rotation) When $\theta =\pi$: $R_y|0⟩=|1⟩$ (full flip) When $\theta =\pi /2$: $R_y|0⟩=(|0⟩+|1⟩)/\sqrt{2}$ (superposition)

By tuning $\theta$, a single-qubit PQC can prepare any state on the Bloch sphere.

Multi-Parameter Circuits

Real QML models use many parameters. A typical layer structure:

The total parameter count is: (parameters per layer) x (number of layers). For n qubits with one Ry per qubit per layer and L layers: n*L parameters.

Gradient Computation: Parameter Shift Rule

Unlike neural networks where you can use backpropagation through differentiable operations, quantum circuits require a special technique: the parameter shift rule.

For a gate $G(\theta )=\exp (-i\theta P/2)$ generated by a Pauli operator $P$, the gradient of an expectation-value cost is:

This means you can compute exact gradients by running the circuit twice with shifted parameters. The fine print: the rule is exact when $L$ is the expectation value of a measured observable and the gate's generator has two eigenvalues, which every Pauli rotation satisfies. If your loss applies nonlinear post-processing to that expectation value (a squared error, a cross-entropy), you apply the shift rule to the expectation value and finish the derivative with the classical chain rule.