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Quantum Distance Approximation for Persistence Diagrams

Bernardo Ameneyro, Rebekah Herrman, George Siopsis, Vasileios Maroulas

TL;DR

This work explores the potential of quantum computers to estimate the distance between persistence diagrams as the next step in the design of a fully quantum framework for TDA, and proposes variational quantum algorithms for the Wasserstein distance as well as the dpc distance.

Abstract

Topological Data Analysis methods can be useful for classification and clustering tasks in many different fields as they can provide two dimensional persistence diagrams that summarize important information about the shape of potentially complex and high dimensional data sets. The space of persistence diagrams can be endowed with various metrics such as the Wasserstein distance which admit a statistical structure and allow to use these summaries for machine learning algorithms. However, computing the distance between two persistence diagrams involves finding an optimal way to match the points of the two diagrams and may not always be an easy task for classical computers. In this work we explore the potential of quantum computers to estimate the distance between persistence diagrams, in particular we propose variational quantum algorithms for the Wasserstein distance as well as the $d^{c}_{p}$ distance. Our implementation is a weighted version of the Quantum Approximate Optimization Algorithm that relies on control clauses to encode the constraints of the optimization problem.

Quantum Distance Approximation for Persistence Diagrams

TL;DR

This work explores the potential of quantum computers to estimate the distance between persistence diagrams as the next step in the design of a fully quantum framework for TDA, and proposes variational quantum algorithms for the Wasserstein distance as well as the dpc distance.

Abstract

Topological Data Analysis methods can be useful for classification and clustering tasks in many different fields as they can provide two dimensional persistence diagrams that summarize important information about the shape of potentially complex and high dimensional data sets. The space of persistence diagrams can be endowed with various metrics such as the Wasserstein distance which admit a statistical structure and allow to use these summaries for machine learning algorithms. However, computing the distance between two persistence diagrams involves finding an optimal way to match the points of the two diagrams and may not always be an easy task for classical computers. In this work we explore the potential of quantum computers to estimate the distance between persistence diagrams, in particular we propose variational quantum algorithms for the Wasserstein distance as well as the distance. Our implementation is a weighted version of the Quantum Approximate Optimization Algorithm that relies on control clauses to encode the constraints of the optimization problem.
Paper Structure (16 sections, 6 theorems, 23 equations, 12 figures, 2 tables)

This paper contains 16 sections, 6 theorems, 23 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminary Results
  3. Persistent Homology
  4. Persistence Diagrams
  5. Quantum Computing Basics
  6. Quantum Persistent Homology
  7. QAOA
  8. Methods
  9. Quantum Algorithm for Wasserstein Distance
  10. A More Efficient Distance
  11. Implementations
  12. An Illustrative Example
  13. Circles vs Noisy Circles
  14. Conclusion and Discussion
  15. Proofs of Theorems \ref{['thm:feasibility-dcp']} and \ref{['thm:completeness-dcp']}
  16. ...and 1 more sections

Key Result

Lemma 1

The solution to the optimization problem as defined in Eqs. eq:cost-was and eq:constraints-was using the weights is the distance $d_p^W (\mathcal{D}_1, \mathcal{D}_2)$ introduced in Def. def:was. Here, $P x = \frac{1}{2} (a + b, a + b)^{T}$ denotes the projection to the diagonal of $x = (a, b)^{T}$.

Figures (12)

  • Figure 1: Examples of simplices. A vertex for dimension zero, a line for dimension one, a triangle for dimension two, and a tetrahedron for dimension three.
  • Figure 2: Examples of persistence diagrams for point cloud data sets.
  • Figure 3: Example of the Wasserstein (a) and $d^{c}_{p}$ (b) distances between a persistence diagram with two points (blue dots) and another with one point (orange triangle). Disks around the points illustrate the penalization mechanism of each distance. The optimal matching in each case is given by the dashed lines.
  • Figure 4: Graph connecting the points of two persistence diagrams.
  • Figure 5: Graph connecting the points of two persistence diagrams.
  • ...and 7 more figures

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Definition 3
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1
  • ...and 5 more