Quantum Persistent Homology Encoding

---
name: quantum-persistent-homology-encoding
description: "Quantum data encoding methodology that preserves persistent homology topological features. Maps point cloud data to quantum states while maintaining topological invariants (Betti numbers, persistence diagrams). Use when: topological data analysis with quantum computing, quantum machine learning with topology preservation, persistent homology quantum encoding, algebraic topology quantum features, TDA quantum pipelines."
license: Complete terms in LICENSE.txt
metadata:
  arxiv_id: "2605.28927"
  published: "2026-05-29"
  tags: [quantum, topological-data-analysis, persistent-homology, encoding, algebraic-topology]
---

# Quantum Encodings Preserving Persistent Homology

## Core Methodology

Given a dataset with a notion of distance (e.g., point cloud in Euclidean space), this methodology constructs quantum state encodings that preserve topological features captured by persistent homology. The encoding ensures that the persistence diagrams (birth-death pairs of topological features) of the original data are recoverable from the quantum state.

## Key Insights

1. **Topology Preservation in Quantum Space**: Standard amplitude/angle encodings destroy topological structure; this method constructs encodings that maintain persistent homology invariants
2. **Betti Number Recovery**: The encoded quantum states allow reconstruction of Betti numbers (connected components, holes, voids) from quantum measurements
3. **Bridges TDA and QML**: Enables topological feature extraction as a preprocessing step for quantum machine learning models

## Algorithm Overview

1. **Input**: Point cloud data with distance metric
2. **Filtration**: Build simplicial complex filtration (Vietoris-Rips or Čech)
3. **Persistence Computation**: Extract persistence diagrams/barcodes
4. **Topology-Preserving Encoding**: Map points to quantum amplitudes such that:
   - Close points in data → similar quantum states
   - Topological features → measurable quantum correlations
5. **Verification**: Check that persistence diagrams are recoverable from quantum measurements

## Encoding Construction

The encoding uses a combination of:
- Distance-preserving amplitude encoding
- Topological feature injection via ancilla qubits
- Measurement protocols for Betti number estimation

## When to Use

- Quantum machine learning on datasets with important topological structure
- Protein structure analysis, molecular topology
- Sensor network coverage hole detection
- Any QML task where topological features are predictive

## Practical Considerations

- Requires O(n) qubits for n data points in naive encoding
- Logarithmic encodings possible for structured data
- Measurement overhead for persistence recovery depends on filtration complexity