Quantum Machine Learning Tutorials

Step 1: Feature Encoding via RY Rotations

Purpose: Convert classical input features into quantum states

Optimization: Each feature is encoded as a rotation angle (θ = feature × π/4), mapping classical data into the quantum Hilbert space (2^29 dimensions)

Why: Enables quantum superposition to explore multiple feature combinations simultaneously

Step 2: Variational Ansatz with 3 Layers

Purpose: Create quantum entanglement and learn compressed representations

Optimization: Each layer applies:

• UY Rotations: Single-qubit gates (RY) that rotate quantum states
• RZZ Entanglement: Two-qubit gates creating quantum correlations between adjacent qubits
• UZ Phases: Phase rotations (RZ) adding quantum interference effects

Why: Quantum entanglement captures complex feature interactions that classical layers miss, enabling better compression

Step 3: Knowledge Distillation Loss Optimization

Purpose: Transfer knowledge from teacher (1.1B params) to quantum student (~110M params)

Optimization: Uses temperature-scaled KL divergence with proper normalization:

distillation_loss = (temperature ** 2 * kl_div_loss) / seq_len
total_loss = alpha * distillation_loss + (1 - alpha) * student_loss

Why: Normalization by sequence length ensures stable training and prevents loss explosion

Step 4: Gradient Accumulation and Clipping

Purpose: Stabilize training with limited memory

Optimization:

• Batch size: 1 (memory efficient)
• Gradient accumulation: 4 steps (effective batch size: 4)
• Gradient clipping: max_norm = 1.0 (prevents exploding gradients)

Why: Quantum circuits require careful gradient management to maintain quantum state fidelity

Step 5: Exponential Moving Average Loss Tracking

Purpose: Smooth loss tracking for better monitoring

Optimization: Uses EMA with α = 0.99 to track loss trends:

class ExponentialMovingAverage:
    def update(self, new_value):
        self.value = self.alpha * self.value + (1 - self.alpha) * new_value

Why: Quantum training loss can be noisy; EMA provides stable trend tracking

Step 1: Patch Embedding with Quantum Encoding

Purpose: Convert image patches into quantum states

Optimization: Image patches are encoded into quantum feature vectors using amplitude encoding, where pixel values are mapped to quantum state amplitudes

Why: Quantum encoding enables exploration of exponentially large feature spaces (2^n dimensions for n qubits)

Step 2: Quantum Self-Attention Mechanism

Purpose: Replace classical attention with quantum attention

Optimization: Uses variational quantum circuits to compute attention weights:

• Query, Key, Value vectors encoded into quantum states
• Quantum gates compute attention scores via quantum interference
• Measurement extracts classical attention weights

Why: Quantum attention captures non-local correlations between image patches more efficiently

Step 3: Quantum Feed-Forward Network

Purpose: Process quantum features through variational layers

Optimization: Multi-layer quantum circuits with:

• Parameterized rotation gates (RY, RZ)
• Entangling gates (CNOT, RZZ) for feature mixing
• Measurement and classical post-processing

Why: Quantum circuits can represent complex feature transformations with fewer parameters

Step 4: Hybrid Classical-Quantum Architecture

Purpose: Combine classical ViT layers with quantum layers

Optimization: Strategic placement of quantum layers:

• Early layers: Classical (efficient feature extraction)
• Middle layers: Quantum (complex pattern recognition)
• Final layers: Classical (task-specific output)

Why: Hybrid approach balances quantum advantage with computational efficiency

Step 5: Quantum Gradient Optimization

Purpose: Optimize quantum circuit parameters

Optimization: Uses parameter-shift rule for quantum gradients:

# Parameter-shift rule for quantum gradients
def quantum_gradient(circuit, param_idx):
    shifted_plus = circuit(param + π/2)
    shifted_minus = circuit(param - π/2)
    gradient = (shifted_plus - shifted_minus) / 2
    return gradient

Why: Enables gradient-based optimization of quantum circuits while maintaining quantum state fidelity

Step 6: Quantum Data Augmentation Stack (Aaronson's Quantum Supremacy Approach)

Purpose: Generate non-classical training data using Matrix Product State (MPS) tensor networks

Optimization: Based on Scott Aaronson's quantum supremacy framework, QVIT uses an MPS-based augmentation stack:

MPS Tensor Network Structure

The augmentation stack uses Matrix Product States to represent quantum-entangled feature correlations:

# MPS Tensor Structure (O(n) complexity, not O(2^n))
|ψ⟩ = Σ A₁[i₁] A₂[i₂] ... Aₙ[iₙ] |i₁i₂...iₙ⟩

# Entanglement Entropy
S = -Tr(ρ log ρ) ≤ log(bond_dim)

# For bond_dim=4: S ≤ log(4) ≈ 1.4 bits of entanglement

Augmentation Pipeline Steps

1. Feature Encoding: Map image patches to virtual qubit states using amplitude encoding
2. MPS Tensor Initialization: Create tensor network with bond dimension controlling entanglement capacity
3. Quantum-Correlated Noise Generation: Sample from MPS probability distributions P(i) = |ψ_i|²
4. Phase Mixing: Apply quantum phase relationships: x' = x·cos(φ) + x_shifted·sin(φ)
5. Synthetic Sample Generation: Create augmented samples preserving quantum correlations

Non-Classical Data Generation

The MPS augmentation stack generates non-classical data that classical augmentation methods cannot produce:

• Entangled Feature Correlations: Features are correlated through quantum entanglement, not just linear combinations
• Non-Local Dependencies: MPS captures correlations between distant image patches via bond dimension
• Quantum Superposition: Augmented samples explore multiple feature combinations simultaneously
• Phase-Based Relationships: Quantum phases create interference patterns that enhance or suppress features
• Exponential State Space: n qubits explore 2^n dimensional Hilbert space, enabling richer data distributions

Impact on Quantum vs Classical Performance

The non-classical data generated by the MPS augmentation stack provides quantum methods with advantages that classical methods cannot exploit:

Why Quantum Methods Outperform Classical

The non-classical data generated by the MPS stack enables quantum methods to outperform classical methods through:

• Quantum Entanglement Advantage: The augmented data contains entangled correlations that quantum circuits can process natively, while classical models require exponentially many parameters to approximate
• Superposition Benefits: Quantum models can process multiple augmented samples simultaneously through superposition, enabling better generalization
• Phase Interference: Quantum phases in the augmented data create interference patterns that quantum circuits can exploit for feature selection
• Information-Theoretic Superiority: The MPS augmentation increases mutual information I(X:Y) between features, which quantum models leverage more effectively than classical models
• Regularization Effect: Quantum entanglement acts as a form of quantum-inspired regularization, preventing overfitting while maintaining expressiveness

Aaronson's Quantum Supremacy Insight

Key Principle: Quantum systems explore probability distributions that are computationally hard to sample classically.

Application to QVIT: The MPS tensor network approximates these hard-to-sample quantum distributions, generating augmented data with correlation structures that:

• Classical augmentation methods (SMOTE, ADASYN) cannot easily produce
• Quantum circuits can process efficiently through native entanglement
• Enable quantum models to learn patterns inaccessible to classical models

Configuration Parameters