Encoding Classical Data into Quantum States: Foundations and Techniques

Table of Contents

1. Introduction
2. Why Encoding Matters in Quantum Machine Learning
3. Characteristics of Quantum Data Representations
4. The Challenge of Data Input in QML
5. Types of Quantum Data Encoding
6. Basis Encoding
7. Amplitude Encoding
8. Angle Encoding
9. Phase Encoding
10. Hybrid Encoding Strategies
11. Comparing Encoding Techniques
12. Hardware Implications of Encoding
13. Encoding Schemes and Quantum Circuits
14. Practical Examples with Qiskit and PennyLane
15. Fidelity Preservation and Feature Scaling
16. Embedding Classical Features into Hilbert Space
17. Expressivity vs Efficiency Trade-offs
18. Encoding Strategies for Time-Series and Images
19. Noise and Error Mitigation in Encoded Circuits
20. Conclusion

1. Introduction

Encoding classical data into quantum states is a foundational step in quantum computing and especially in Quantum Machine Learning (QML). It determines how effectively a quantum model can access and process classical information.

2. Why Encoding Matters in Quantum Machine Learning

Quantum circuits process quantum states. Classical datasets must be embedded into quantum Hilbert space before quantum gates can operate on them. Encoding directly impacts model accuracy and scalability.

3. Characteristics of Quantum Data Representations

• Quantum states are unit vectors in Hilbert space
• Measurement probabilities reflect amplitude magnitudes
• Quantum systems require normalization: \( \sum |a_i|^2 = 1 \)

4. The Challenge of Data Input in QML

• High classical-to-quantum overhead
• Gate depth increases with data dimensionality
• Complex embeddings may require ancilla qubits or long circuits

5. Types of Quantum Data Encoding

• Basis Encoding
• Amplitude Encoding
• Angle (Rotation) Encoding
• Phase Encoding
• Hybrid or Tensor Product Encodings

6. Basis Encoding

• Each classical bit maps to a qubit basis state (0 or 1)
• Simple and efficient
• Not expressive for continuous or high-dimensional data

Example:

from qiskit import QuantumCircuit
qc = QuantumCircuit(3)
qc.x(0)  # Encoding '1' in first qubit

7. Amplitude Encoding

• Data values mapped to amplitudes of a quantum state
• Compact: stores \( 2^n \) values in \( n \) qubits
• Hard to prepare circuits efficiently

State: \( |x
angle = \sum_i x_i |i
angle \)

8. Angle Encoding

• Classical values used as rotation angles on quantum gates
• Used in VQC, QNNs

Example:

qc.ry(x, qubit)

9. Phase Encoding

• Encodes data in the relative phase between quantum basis states
• Useful for phase estimation tasks
• Not directly measurable; used in interference-based models

10. Hybrid Encoding Strategies

• Combine amplitude + angle or basis + phase
• Encodes different feature types across different qubits

11. Comparing Encoding Techniques

12. Hardware Implications of Encoding

• Amplitude encoding has deep circuits
• Angle/basis encoding easier to transpile for NISQ devices
• Encoding depth affects noise sensitivity

13. Encoding Schemes and Quantum Circuits

Each encoding maps data into quantum states using gate operations:

• Basis: X gates
• Angle: RX/RY/RZ
• Amplitude: complex initialization
• Phase: U1, RZ with controlled rotations

14. Practical Examples with Qiskit and PennyLane

Qiskit:

qc.rx(data[0], 0)
qc.ry(data[1], 1)

PennyLane:

qml.AngleEmbedding(data, wires=[0, 1, 2], rotation='Y')

15. Fidelity Preservation and Feature Scaling

• Normalize inputs to match unit vector constraint
• Scale features to angle bounds (e.g., [0, π])

16. Embedding Classical Features into Hilbert Space

Encoding = feature map:

• Maps classical x → \( |\phi(x)
  angle \)
• Quantum kernel = \( |\langle \phi(x) | \phi(x’)
  angle|^2 \)

17. Expressivity vs Efficiency Trade-offs

• Richer encodings provide better model performance
• Deeper circuits introduce more noise
• Choose encoding based on hardware and dataset size

18. Encoding Strategies for Time-Series and Images

• Time-series: use sequences of angle encodings
• Images: patch-wise encoding or PCA + amplitude map

19. Noise and Error Mitigation in Encoded Circuits

• Use short-depth encoders
• Apply error mitigation post-measurement
• Cross-validate on simulators and hardware

20. Conclusion

Data encoding is the bridge between classical information and quantum computing. Choosing the right encoding technique is vital for model performance, hardware efficiency, and real-world viability of quantum machine learning systems.