Parameter-Shift Rule: Training a Quantum Neural Network

Quantum Neural Network

Target ⟨X⟩ 0.60

Learning rate η 0.35

Show ±π/2 shift-rule evaluation points

Training State

Cost C(θ)

0.000

⟨X⟩ output

0.000

θ₁ (Rᵧ)

0.0°

θ₂ (R_z)

0.0°

|∇C| (shift rule)

0.000

Iteration

How it works

The circuit is a single qubit: R_z(θ₂) R_y(θ₁) |0⟩, measured in the observable X. For this circuit the expectation value has a closed form:

⟨X⟩(θ₁,θ₂) = sin(θ₁)·cos(θ₂)
Cost:  C(θ) = (⟨X⟩(θ) − target)²

A real quantum computer cannot backpropagate through a circuit — there is no classical computation graph inside the hardware. Instead, for any gate of the form R(θ)=exp(−iθP/2) with P a Pauli operator, the exact gradient of an expectation value is obtained by running the same circuit twice, with that one parameter shifted by ±π/2:

∂⟨O⟩/∂θᵢ = [ ⟨O⟩(θᵢ+π/2) − ⟨O⟩(θᵢ−π/2) ] / 2

This parameter-shift rule is not a finite-difference approximation — it is mathematically exact for Pauli rotation gates, which is why it is the standard way variational quantum circuits (QNNs, VQE, QAOA) are trained on real hardware. Each training step below evaluates the circuit at four shifted points (θ₁±π/2 and θ₂±π/2) to build the exact gradient ∇C, then takes a step θ ← θ − η∇C, exactly as gradient descent would on a classical network.

Surface — the full cost landscape C(θ₁,θ₂) over one period of each angle.
Ball — the QNN's current parameters, rolling downhill as training proceeds.
Small markers — the four parameter-shifted circuit evaluations used to compute this step's gradient.
Trail — the last training steps, fading with age.

Current parameters θ Shift-rule eval points High cost Low cost

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