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Quantum Topological Graph Neural Networks for Detecting Complex Fraud Patterns

Mohammad Doost, Mohammad Manthouri

TL;DR

This work presents QTGNN, a novel framework that fuses quantum embeddings, non-linear variational graph convolutions, and persistence-based topological invariants to detect complex fraud patterns in large financial networks. By integrating quantum-state representations with higher-order topology and hybrid quantum-classical optimization, the approach achieves state-of-the-art performance on PaySim while providing interpretable topological attributions. The authors supply convergence guarantees, NISQ-oriented implementation strategies, and a thorough ablation and cost-benefit analysis, underscoring practical viability despite quantum overhead. The framework promises robust, scalable, and auditable fraud detection suitable for regulatory environments and real-world deployment.

Abstract

We propose a novel QTGNN framework for detecting fraudulent transactions in large-scale financial networks. By integrating quantum embedding, variational graph convolutions, and topological data analysis, QTGNN captures complex transaction dynamics and structural anomalies indicative of fraud. The methodology includes quantum data embedding with entanglement enhancement, variational quantum graph convolutions with non-linear dynamics, extraction of higher-order topological invariants, hybrid quantum-classical anomaly learning with adaptive optimization, and interpretable decision-making via topological attribution. Rigorous convergence guarantees ensure stable training on noisy intermediate-scale quantum (NISQ) devices, while stability of topological signatures provides robust fraud detection. Optimized for NISQ hardware with circuit simplifications and graph sampling, the framework scales to large transaction networks. Simulations on financial datasets, such as PaySim and Elliptic, benchmark QTGNN against classical and quantum baselines, using metrics like ROC-AUC, precision, and false positive rate. An ablation study evaluates the contributions of quantum embeddings, topological features, non-linear channels, and hybrid learning. QTGNN offers a theoretically sound, interpretable, and practical solution for financial fraud detection, bridging quantum machine learning, graph theory, and topological analysis.

Quantum Topological Graph Neural Networks for Detecting Complex Fraud Patterns

TL;DR

This work presents QTGNN, a novel framework that fuses quantum embeddings, non-linear variational graph convolutions, and persistence-based topological invariants to detect complex fraud patterns in large financial networks. By integrating quantum-state representations with higher-order topology and hybrid quantum-classical optimization, the approach achieves state-of-the-art performance on PaySim while providing interpretable topological attributions. The authors supply convergence guarantees, NISQ-oriented implementation strategies, and a thorough ablation and cost-benefit analysis, underscoring practical viability despite quantum overhead. The framework promises robust, scalable, and auditable fraud detection suitable for regulatory environments and real-world deployment.

Abstract

We propose a novel QTGNN framework for detecting fraudulent transactions in large-scale financial networks. By integrating quantum embedding, variational graph convolutions, and topological data analysis, QTGNN captures complex transaction dynamics and structural anomalies indicative of fraud. The methodology includes quantum data embedding with entanglement enhancement, variational quantum graph convolutions with non-linear dynamics, extraction of higher-order topological invariants, hybrid quantum-classical anomaly learning with adaptive optimization, and interpretable decision-making via topological attribution. Rigorous convergence guarantees ensure stable training on noisy intermediate-scale quantum (NISQ) devices, while stability of topological signatures provides robust fraud detection. Optimized for NISQ hardware with circuit simplifications and graph sampling, the framework scales to large transaction networks. Simulations on financial datasets, such as PaySim and Elliptic, benchmark QTGNN against classical and quantum baselines, using metrics like ROC-AUC, precision, and false positive rate. An ablation study evaluates the contributions of quantum embeddings, topological features, non-linear channels, and hybrid learning. QTGNN offers a theoretically sound, interpretable, and practical solution for financial fraud detection, bridging quantum machine learning, graph theory, and topological analysis.

Paper Structure

This paper contains 31 sections, 8 theorems, 56 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Foundational Work in Graph-Based Anomaly Detection
  4. Advances in Graph Neural Networks for Anomaly Detection
  5. Quantum and Topological Approaches in Learning Systems
  6. Research Gaps and Motivation
  7. Methodology
  8. Quantum Topological Graph Neural Network
  9. Quantum Data Embedding with Entanglement Enhancement
  10. Variational Quantum Graph Convolution with Non-Linear Dynamics
  11. Topological Quantum Signature Extraction with Higher-Order Invariants
  12. Quantum-Classical Hybrid Anomaly Learning with Adaptive Optimization
  13. Decision and Interpretability with Topological Attribution
  14. Implementation Considerations
  15. NISQ Optimizations
  16. ...and 16 more sections

Key Result

Lemma 3.1

Let $U_\theta = \prod_{l=1}^L e^{-i \theta_l H_l}$ with bounded Hermitian generators: and analytic readout: for a fixed observable $M$. Then the map is real-analytic and in particular $C^\infty$; on any compact subset of $\Theta$, it is $L$-smooth for some $L < \infty$.

Figures (7)

  • Figure 1: Flowchart of the QTGNN Framework Illustrating the Pipeline from Graph Preprocessing to Fraud Detection Decisions.
  • Figure 2: Radar Plot for Performance Metrics of QTGNN and Ablation Studies.
  • Figure 3: Cost-Benefit Analysis of Quantum and Classical Approaches in Fraud Detection.
  • Figure 4: Complexity-Accuracy Tradeoff in Quantum GNNs.
  • Figure 5: Barren Plateau Analysis of Quantum Circuit Training Dynamics.
  • ...and 2 more figures

Theorems & Definitions (14)

  • Lemma 3.1: Smoothness of variational quantum losses
  • proof : Proof sketch
  • Theorem 3.1: SGD convergence to stationarity in nonconvex setting
  • proof : Proof sketch
  • Corollary 3.1: Linear (geometric) convergence under PL
  • Remark 3.1: Why PL can be reasonable here
  • Remark 3.2: Convergence under weaker assumptions
  • Theorem 3.2: Convergence of Riemannian SGD
  • Theorem 3.3: Convergence of block coordinate descent
  • Remark 3.3
  • ...and 4 more