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Sample efficient graph classification using binary Gaussian boson sampling

Amanuel Anteneh, Olivier Pfister

TL;DR

This work proposes a binary-detector Gaussian boson sampling (GBS) approach for graph classification, enabling room-temperature, hardware-friendly implementations. It analyzes how sample complexity and coarse-graining affect learning, showing that raw outcome space grows as $| abla| 2^M$, but mu-coarse-graining reduces this to $O(M)$ and nu-coarse-graining to $O(1)$ samples, with corresponding feature-vector dimensions. Empirically, the binary GBS kernel achieves competitive or superior accuracies compared with classical graph kernels, notably on ENZYMES, and demonstrates robustness to graph-size imbalance, while avoiding the need for displacement. The results highlight the practicality of GBS-based graph kernels and point to future work on incorporating vertex/edge labels and refining coarse-graining strategies to further improve performance and hardware feasibility.

Abstract

We present a variation of a quantum algorithm for the machine learning task of classification with graph-structured data. The algorithm implements a feature extraction strategy that is based on Gaussian boson sampling (GBS) a near term model of quantum computing. However, unlike the currently proposed algorithms for this problem, our GBS setup only requires binary (light/no light) detectors, as opposed to photon number resolving detectors. These detectors are technologically simpler and can operate at room temperature, making our algorithm less complex and less costly to implement on the physical hardware. We also investigate the connection between graph theory and the matrix function called the Torontonian which characterizes the probabilities of binary GBS detection events.

Sample efficient graph classification using binary Gaussian boson sampling

TL;DR

This work proposes a binary-detector Gaussian boson sampling (GBS) approach for graph classification, enabling room-temperature, hardware-friendly implementations. It analyzes how sample complexity and coarse-graining affect learning, showing that raw outcome space grows as , but mu-coarse-graining reduces this to and nu-coarse-graining to samples, with corresponding feature-vector dimensions. Empirically, the binary GBS kernel achieves competitive or superior accuracies compared with classical graph kernels, notably on ENZYMES, and demonstrates robustness to graph-size imbalance, while avoiding the need for displacement. The results highlight the practicality of GBS-based graph kernels and point to future work on incorporating vertex/edge labels and refining coarse-graining strategies to further improve performance and hardware feasibility.

Abstract

We present a variation of a quantum algorithm for the machine learning task of classification with graph-structured data. The algorithm implements a feature extraction strategy that is based on Gaussian boson sampling (GBS) a near term model of quantum computing. However, unlike the currently proposed algorithms for this problem, our GBS setup only requires binary (light/no light) detectors, as opposed to photon number resolving detectors. These detectors are technologically simpler and can operate at room temperature, making our algorithm less complex and less costly to implement on the physical hardware. We also investigate the connection between graph theory and the matrix function called the Torontonian which characterizes the probabilities of binary GBS detection events.
Paper Structure (26 sections, 2 theorems, 32 equations, 7 figures, 2 tables)

This paper contains 26 sections, 2 theorems, 32 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Reminders about Gaussian Boson Sampling (GBS) and graph theory
  3. Gaussian Boson Sampling
  4. Graph theory
  5. Sample complexity of GBS
  6. The problem with using GBS beyond sampling
  7. Coarse graining of sample distributions
  8. Sample complexity of previously proposed GBS graph kernels
  9. The Algorithm
  10. GBS with binary detectors and its relation to graph theory
  11. Constructing the feature vectors
  12. Complexity analysis
  13. Sample Complexity
  14. Space Complexity
  15. Time Complexity
  16. ...and 11 more sections

Key Result

Lemma 1

$\frac{(n+M-1)!}{n!(M-1)!}\in \omega(\lfloor\sqrt{M}\rfloor^{\lfloor{\sqrt{M}}\rfloor})$for$n = \lfloor\sqrt{M}\rfloor$

Figures (7)

  • Figure 1: In the original input space $\mathbb{R}^2$ the data points, which belong either to the class 'red' or 'blue', are not separable by a linear function (the decision boundary) but after mapping the points to feature vectors in a higher dimensional space $\mathbb{R}^3$ a linear function is able to separate the two classes. This linear decision boundary can be calculated by supervised machine learning models such as a support vector machine. In our case the input space is the set of all undirected graphs which we denote as $\mathbb{G}$.
  • Figure 2: Example of a 3-mode Gaussian boson sampler. Mode $i\in\{1,2,3\}$ starts in the vacuum state $\ket{0}$, is then squeezed by $\hat{S}(r_i)$ and passes through the network of two beamsplitters (the interferometer) before the number of photons in each mode is measured by the detectors $D_{i\in\{1,2,3\}}$.
  • Figure 3: 4-vertex graph and its corresponding adjacency matrix
  • Figure 4: Results of the principal component analysis (PCA) on the $\nu$ feature vector entries for the ENZYMES, MUTAG, IMDB_BINARY and FINGERPRINT datasets. The heatmaps show the weight/coefficient associated with each feature with regard to the first four principal components.
  • Figure 5: Distribution of graph sizes according to class for each dataset.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Lemma 2
  • proof