Quantum Nearest-Neighbor Models: Leveraging Quantum Metrics for Pattern Recognition

Table of Contents

1. Introduction
2. Classical k-Nearest Neighbors (k-NN) Overview
3. Motivation for Quantum k-NN (QkNN)
4. Quantum State Similarity Measures
5. Encoding Classical Data into Quantum States
6. Distance Metrics in Quantum Space
7. Quantum Fidelity and Inner Products
8. Amplitude Encoding for Input Vectors
9. Variational k-NN Quantum Models
10. Quantum Kernel k-NN Approaches
11. Oracle-Based Quantum Nearest Neighbor Search
12. Quantum k-NN via Grover’s Search
13. QkNN for Classification Tasks
14. QkNN for Anomaly and Outlier Detection
15. Experimental Results and Toy Datasets
16. QkNN with Hybrid Classical-Quantum Pipelines
17. Implementation in Qiskit and PennyLane
18. Strengths and Limitations
19. Future Directions
20. Conclusion

1. Introduction

Quantum nearest-neighbor (QkNN) models are adaptations of the classical k-nearest neighbor algorithm, reformulated for quantum computing. They leverage quantum metrics such as fidelity and kernel-based distance to classify or cluster data based on similarity in quantum Hilbert space.

2. Classical k-Nearest Neighbors (k-NN) Overview

• Lazy, non-parametric algorithm
• Classifies based on majority vote of nearest neighbors in feature space
• Sensitive to distance metric and data dimensionality

3. Motivation for Quantum k-NN (QkNN)

• High-dimensional vector spaces naturally map to quantum states
• Inner products (e.g., fidelity) computed efficiently on quantum hardware
• Speedups in similarity search via amplitude estimation and Grover-like methods

4. Quantum State Similarity Measures

• Fidelity: \( F(\psi, \phi) = |\langle \psi | \phi
  angle|^2 \)
• Bures Distance, Trace Distance
• Inner product as proxy for Euclidean similarity

5. Encoding Classical Data into Quantum States

• Normalize data vectors to unit length
• Use amplitude or angle encoding
• Each data point corresponds to a quantum state

6. Distance Metrics in Quantum Space

• Define proximity using:
• State overlap (inner product)
• Fidelity-based measures
• Distance between feature embeddings

7. Quantum Fidelity and Inner Products

• Estimable via Swap Test:
  \[
  F(\psi, \phi) = ext{Tr}[
  ho_\psi
  ho_\phi]
  \]
• Implemented using quantum circuits to compare states

8. Amplitude Encoding for Input Vectors

• Map \( x = (x_1, …, x_n)
  ightarrow \sum_i x_i |i
  angle \)
• Efficient encoding critical for QkNN practicality

9. Variational k-NN Quantum Models

• Use VQC to generate quantum embeddings
• Compute fidelity between embedded query and training examples

10. Quantum Kernel k-NN Approaches

• Compute kernel matrix using quantum circuits
• Apply classical k-NN using quantum-enhanced distances

11. Oracle-Based Quantum Nearest Neighbor Search

• Quantum RAM (qRAM) enables loading training data
• Use Grover-based search to identify closest matches

12. Quantum k-NN via Grover’s Search

• Use amplitude amplification to find nearest neighbor with high probability
• Provides quadratic speedup over classical search

13. QkNN for Classification Tasks

• Apply on encoded datasets (e.g., MNIST patches, synthetic parity)
• Label predicted by majority class of top-k nearest quantum states

14. QkNN for Anomaly and Outlier Detection

• Use quantum metrics to detect low-density regions
• Estimate anomaly score based on inverse fidelity

15. Experimental Results and Toy Datasets

• Iris dataset, XOR, circle vs square classification
• Small quantum circuits for proof-of-concept studies

16. QkNN with Hybrid Classical-Quantum Pipelines

• Classical preprocessing → quantum distance evaluation
• Useful in resource-constrained NISQ setups

17. Implementation in Qiskit and PennyLane

• Qiskit: Statevector, SwapTestCircuit
• PennyLane: qml.state_fidelity, variational embeddings

18. Strengths and Limitations

Strengths:

• Conceptually simple
• Quantum advantage in similarity evaluation

Limitations:

• Encoding overhead
• Fidelity computation requires repeated measurements
• Limited by qRAM availability

19. Future Directions

• Hardware-efficient encoding schemes
• Fault-tolerant QkNN at scale
• Application in graph-based QML

20. Conclusion

Quantum nearest-neighbor models offer an intuitive and potentially powerful extension of classical k-NN by utilizing quantum state similarity. While still in early development, they show promise for tasks involving classification, clustering, and anomaly detection in quantum-enhanced learning systems.