Quantum Machine Learning for Colorectal Cancer Data: Anastomotic Leak Classification and Risk Factors

1. Problem Statement

The study addresses the critical clinical challenge of predicting Anastomotic Leaks (AL) in colorectal cancer surgery. AL is a severe postoperative complication where surgically reconnected bowel tissue fails to heal, leading to life-threatening peritonitis and sepsis.

• Data Characteristics: The dataset consists of 200 patients (28 with AL, 172 without), resulting in a highly imbalanced class distribution (14% prevalence of the positive class).
• Clinical Challenge: In medical risk prediction, false negatives (missing a leak) are far more dangerous than false positives. However, classical machine learning models often struggle with imbalanced data, typically biasing toward the majority class (high specificity) at the expense of sensitivity (recall).
• Objective: To evaluate whether Quantum Machine Learning (QML), specifically Variational Quantum Classifiers (VQC), can outperform classical models in identifying minority class risks (AL) by leveraging high-dimensional quantum feature spaces.

2. Methodology

A. Data Preprocessing and Feature Selection

• Source: Clinical data from Hospital Nový Jičín (2015–2016).
• Initial Variables: Included patient comorbidities (Diabetes, Smoking) and intraoperative techniques (ICG fluorescence, transanal drain/NoCoil, left colic artery preservation/ACSP).
• Selection Process: Statistical analysis (Chi-squared tests) identified Diabetes (DM), Smoking, NoCoil, and ACSP as significant predictors. Multivariate logistic regression using Akaike's Information Criterion (AIC) excluded ICG due to collinearity with NoCoil.
• Final Feature Space: Four binary/continuous features: DM, Smoking, NoCoil, ACSP.

B. Quantum Machine Learning Architecture

The authors implemented a Variational Quantum Classifier (VQC) using a 4-qubit circuit (one qubit per feature).

1. Feature Map (Data Encoding):
   • Used ZZFeatureMap to map classical data $x \in \mathbb{R}^4$ into a high-dimensional Hilbert space.
   • This encoding utilizes second-order expansions and entangling gates (CNOT) to capture non-linear interactions between features, theoretically making the minority class linearly separable in the quantum space.
2. Variational Ansatz (Circuit Structure):
   Two architectures were compared:
   • RealAmplitudes (RA): Layers of $R_y$ rotations and CNOT gates. Shallow depth (11), restricted to real amplitudes.
   • EfficientSU2 (ESU2): Layers of $R_y$ and $R_z$ rotations followed by CNOTs. Deeper depth (15), higher expressivity, capable of capturing more complex quantum states.
3. Measurement & Output:
   • The expectation value of the Pauli-Z operator on the first qubit was measured.
   • A classical sigmoid function mapped the output $[-1, 1]$ to a probability $[0, 1]$.
   • Loss Function: Binary Cross-Entropy.

C. Simulation and Optimization

• Environment: Simulated using Qiskit Aer with a depolarizing noise model (gate error probability $p=0.05$) and sampling noise (1024 shots) to mimic Noisy Intermediate-Scale Quantum (NISQ) hardware.
• Optimizers: Tested both gradient-based (BFGS, SLSQP) and gradient-free/metaheuristic (CMA-ES, COBYLA, SPSA) algorithms.
• Classical Baselines: Logistic Regression, AdaBoost, Linear Discriminant Analysis (LDA), Gaussian Naive Bayes (GNB), and Multi-Layer Perceptron (MLP).
• Evaluation Metric: Models were optimized using an $F_\beta$-score strategy to prioritize sensitivity (recall) over specificity, reflecting clinical priorities.

3. Key Results

A. Performance Comparison

• Sensitivity (Recall): This is the most significant finding.
  • Classical Models: Stagnated at a maximum sensitivity of 66.7%, regardless of the algorithm. They maintained high specificity (82–94%) but failed to detect a significant portion of leaks.
  • Quantum Models (QNN): Uniformly achieved a sensitivity of 83.3% across various optimizers and ansatze, while maintaining a robust specificity of ~79.4%.
• Discrimination (AUC):
  • The best classical model (GNB) achieved an AUC of 0.809.
  • The ESU2-BFGS quantum configuration matched this upper bound with an AUC of 0.809.
• Calibration:
  • The BFGS-optimized ESU2 model achieved the lowest Brier Score (0.111) and Log Loss (0.372) among all models, indicating superior probability calibration compared to classical baselines (e.g., MLP had a Log Loss of 0.403).

B. Optimizer Behavior

• Gradient-based vs. Gradient-free: While gradient-based methods (BFGS, SLSQP) achieved lower continuous loss values, they were prone to overfitting noise-distorted local gradients.
• CMA-ES: The gradient-free CMA-ES optimizer demonstrated superior stability and resilience to quantum noise, effectively navigating the global loss landscape to find better decision boundaries for the minority class.

C. Statistical Significance

• McNemar tests confirmed that the QNNs produced fundamentally different prediction matrices compared to classical baselines (specifically Logistic Regression and LDA), validating that the quantum models are not merely acting as dequantizable surrogates but are utilizing a distinct inductive bias.

4. Key Contributions

1. Demonstration of Quantum Advantage in Imbalanced Data: The study provides empirical evidence that QNNs can resolve the "sensitivity-specificity trade-off" better than classical models in low-prevalence clinical settings, specifically by prioritizing the detection of rare, high-risk events.
2. Noise-Robust Optimization: It highlights that gradient-free optimizers (like CMA-ES) may be more suitable for NISQ-era hardware than gradient-based methods when dealing with noisy, discrete decision boundaries.
3. Feature Space Analysis: The work suggests that the ZZFeatureMap effectively maps clinical risk factors into a Hilbert space where the minority class (leak) becomes more linearly separable, a capability classical hyperplanes failed to replicate.
4. Benchmarking Framework: Establishes a rigorous protocol for comparing QML against hyperparameter-tuned classical models under realistic noise conditions, including metrics for calibration (Brier score) and decision threshold optimization ($F_\beta$).

5. Significance and Future Outlook

• Clinical Impact: A 16.6% increase in sensitivity (from 66.7% to 83.3%) could significantly reduce mortality by identifying more patients at risk of anastomotic leaks, allowing for preventative interventions.
• Hardware Readiness: While the study used simulation, the results suggest that even with noise, QML offers a distinct advantage. Future work must address barren plateaus (vanishing gradients) and integrate error mitigation techniques for physical hardware deployment.
• Theoretical Implication: The findings challenge the assumption that near-term quantum models suffer from poor calibration. Instead, they suggest that the expressive capacity of quantum feature maps can capture complex, non-linear biological relationships better than classical linear boundaries.

In conclusion, this paper serves as a proof-of-concept that Quantum Machine Learning offers a viable pathway to solving critical bottlenecks in medical risk stratification, specifically where classical models fail to detect rare but life-threatening complications due to data imbalance.