🎛️ Parameterized Circuits

Master variational quantum algorithms with trainable circuit parameters

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Measurement Operations

🎯 What are Parameterized Circuits?

Parameterized quantum circuits (PQCs) are quantum circuits containing gates with tunable parameters that can be optimized classically. They form the foundation of variational quantum algorithms, enabling hybrid quantum-classical computation on near-term quantum devices.

⚡ The Variational Principle

PQCs use a quantum processor to evaluate circuit outputs for given parameters, while a classical optimizer adjusts parameters to minimize a cost function. This hybrid approach maximizes the utility of noisy intermediate-scale quantum (NISQ) devices.

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Tunable Parameters

Rotation angles θ in gates like RX(θ), RY(θ), RZ(θ) can be adjusted during optimization

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Iterative Optimization

Parameters are updated iteratively to minimize cost function through gradient descent

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Hybrid Workflow

Quantum circuit evaluates cost, classical optimizer proposes new parameters

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Expressibility

Circuit architecture determines range of quantum states that can be represented

📝 Circuit Structure

Layer 1:RY(θ₁) ⊗ RY(θ₂) ⊗ RY(θ₃)
Layer 2:CNOT(q₀,q₁) • CNOT(q₁,q₂)
Layer 3:RY(θ₄) ⊗ RY(θ₅) ⊗ RY(θ₆)
Output:|ψ(θ)⟩ → Measure → Cost function

💡 Key Insight

Parameterized circuits enable quantum machine learning and optimization on current hardware. By encoding problems into cost functions and optimizing circuit parameters, we can solve practical problems before fault-tolerant quantum computers become available.