Parameterized Quantum Circuit

A quantum circuit with adjustable parameters, similar to a neural network layer.

A Parameterized Quantum Circuit (PQC), also called a quantum circuit ansatz or variational circuit, is a quantum circuit whose gates depend on tunable parameters.

Structure

$$U(\theta )=U_L({\theta }_L)\cdots U_2({\theta }_2)U_1({\theta }_1)$$

Each layer contains:

• Fixed entangling gates (e.g., CNOT)
• Parameterized rotation gates (e.g., $R_y(\theta )$, $R_z(\theta )$)

Example Circuit

         θ₁      θ₂      θ₃
|0⟩ ──Ry(θ₁)──●──Ry(θ₃)──●──
              │          │
|0⟩ ──Ry(θ₂)──⊕──Ry(θ₄)──⊕──

Parameters $\theta =({\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4)$ are optimized.

Common Building Blocks

Rotation Gates

$$R_x(\theta )=e^{-i\theta X/2},R_y(\theta )=e^{-i\theta Y/2},R_z(\theta )=e^{-i\theta Z/2}$$

Entangling Layers

• CNOT ladders
• CZ layers
• All-to-all connectivity

Ansatz Types

Training PQCs

Like neural networks, PQCs are trained by:

1. Forward pass: Run circuit, measure expectation
2. Compute cost: Compare to target
3. Backward pass: Estimate gradients
4. Update: Adjust parameters

Gradient Methods

Parameter shift rule: $$\frac{\partial ⟨O⟩}{\partial \theta }=\frac{1}{2}[⟨O{⟩}_{\theta +\pi /2}-⟨O{⟩}_{\theta -\pi /2}]$$

Requires two circuit evaluations per parameter.

Expressibility

How much of Hilbert space can the PQC reach?

Trade-off between expressibility and trainability.

Applications

• VQE: Ansatz for molecular wave functions
• QAOA: Alternating problem/mixer layers
• QML: Feature maps and variational classifiers

Comparison to Neural Networks

See also: Variational Quantum Algorithm, Quantum Machine Learning