Quantum Neural Network

A parameterized quantum circuit used as a machine learning model, similar to a classical neural network.

A Quantum Neural Network (QNN) is a parameterized quantum circuit trained to perform machine learning tasks like classification or regression.

Structure

Input x → [Encoding S(x)] → [Variational U(θ)] → Measure → Output

1. Data Encoding

Convert classical data to quantum state: $$|x⟩=S(x)|0⟩$$

Methods:

• Angle encoding: $R_y(x_i)$ gates
• Amplitude encoding: Data in amplitudes
• Feature maps: More complex encodings

2. Variational Layers

Parameterized circuit with trainable parameters: $$U(\theta )=\prod _l{U}_l({\theta }_l)$$

3. Measurement

Extract prediction from quantum state: $$\hat{y}=⟨O⟩=⟨x|U^{†}(\theta )OU(\theta )|x⟩$$

Training

Like classical neural networks:

1. Forward pass (circuit execution)
2. Compute loss
3. Compute gradients (parameter shift rule)
4. Update parameters

$$\theta \leftarrow \theta -\eta {\nabla }_{\theta }\mathcal{L}$$

Architectures

Variational Quantum Classifier (VQC)

Binary classification: $$P(y=1|x)=⟨Z⟩$$

Quantum Convolutional NN

Inspired by CNNs:

• Local operations (like convolutions)
• Pooling via measurement

Hybrid Networks

Classical layers + quantum layers:

Classical NN → Quantum circuit → Classical NN

Comparison to Classical NNs

Advantages (Potential)

• Exponentially large state space
• Natural quantum data handling
• Possible expressibility advantages
• Different inductive biases

Challenges

Current Applications

• Quantum chemistry (with quantum data)
• Small classification benchmarks
• Proof-of-concept demonstrations
• Hybrid quantum-classical models

Open Questions

• When do QNNs outperform classical?
• How to avoid barren plateaus?
• What problems are best suited?
• How do they scale?

See also: Parameterized Quantum Circuit, Quantum Machine Learning