Evidential Quantum Vertical Federated Learning

1. Problem Statement

Context:

• Vertical Federated Learning (VFL): In VFL, multiple parties hold different features of the same set of samples (e.g., a bank and an e-commerce platform sharing customer data). The goal is to train a joint model without sharing raw data.
• Quantum Federated Learning (QFL): While QFL has been explored, existing research primarily focuses on Horizontal FL (HFL), where parties share the same features but different samples.
• Challenges in Quantum VFL:
  1. Privacy: Transmitting intermediate quantum states between parties and a server risks information leakage if not handled correctly.
  2. Barren Plateaus: Training deep Variational Quantum Circuits (VQCs) for cross-party fusion often leads to the "barren plateau" problem, where gradients vanish exponentially with qubit count, making training impossible.
  3. Information Loss: Measuring quantum states at the party level before transmission destroys quantum coherence and loses information.
  4. Complexity: Classical evidence theory (Dempster-Shafer theory) for fusing uncertain information has exponential computational complexity ($2^n$) regarding the power set of hypotheses.

2. Methodology: The eviQVFL Framework

The authors propose eviQVFL, a hybrid classical-quantum framework tailored for VFL. It integrates Evidence Theory with Quantum Machine Learning to address the challenges above.

A. Overall Architecture

The framework consists of $K$ parties and one central server.

1. Party-Side Processing (Local):

   • Input: Local feature vectors $x_{i,k}$.
   • Dimension Reduction: A Tensor-Train Network (TTN) reduces high-dimensional classical features into a low-dimensional vector. This reduces parameter count while maintaining expressiveness.
   • Quantum Encoding: The reduced features are encoded into a quantum state using Angle Encoding (Rotation-Y gates).
   • Local Evolution: A Variational Quantum Circuit (VQC) processes the encoded state. The output is a quantum state $|\psi_{i,k}\rangle$ representing local "evidence" (Basic Belief Assignment) regarding the classification.
   • Transmission: Instead of measuring, the party transmits the quantum state $|\psi_{i,k}\rangle$ to the server via Quantum Teleportation. This preserves the full quantum information and ensures privacy (no raw data or classical measurements are exposed).

2. Server-Side Processing (Global Fusion):

   • Fusion: The server receives quantum states from all $K$ parties. It fuses them using a non-parametric Evidential Fusion Circuit.
   • Mechanism: This circuit implements the Conjunctive Combination Rule (CCR) of Evidence Theory using Multi-Controlled X gates.
     • Mathematically, it maps the intersection of evidence from different parties onto the amplitudes of the final quantum state.
     • Crucially, this fusion layer is fixed (non-trainable) and shallow, avoiding the need for deep parameterized circuits that cause barren plateaus.
   • Inference: The fused state is measured using projective operators. The resulting probabilities correspond to the Plausibility Function in evidence theory, which is then passed through a Softmax function to generate the final prediction.

B. Theoretical Foundation

• Evidence-Quantum Equivalence: The paper leverages the fact that an $n$-qubit system has $2^n$ basis states, which naturally maps to the $2^n$ elements of the power set in Evidence Theory. This allows the exponential complexity of evidence fusion to be handled efficiently in quantum superposition.
• Privacy: By using quantum teleportation, the server reconstructs the state without ever measuring the intermediate states, adhering to the No-Cloning Theorem and preventing data leakage.

3. Key Contributions

1. First Quantum VFL Framework: Proposed eviQVFL, the first QFL framework specifically designed for the Vertical Federated Learning setting.
2. Hybrid Architecture: Introduced a TTN-VQC architecture for efficient local feature extraction and encoding, balancing parameter efficiency with representational power.
3. Non-Parametric Evidential Fusion: Designed a fixed-depth quantum circuit based on Evidence Theory (Dempster-Shafer) to fuse multi-party information. This avoids the barren plateau problem common in trainable deep VQCs.
4. Privacy-Preserving Transmission: Utilized Quantum Teleportation to transfer local evidence states to the server, ensuring that local features and intermediate outputs remain private and unmeasured until the final fusion.
5. Theoretical Analysis: Provided bounds showing that the joint model's approximation error decays with the aggregate number of qubits across all parties, rather than being limited by individual party resources.

4. Experimental Results

The authors evaluated eviQVFL on four datasets: MNIST, FashionMNIST, Breast Cancer Wisconsin, and Credit Card Fraud Detection.

• Baselines: Compared against classical averages, classical fusion, and various quantum baselines (including "measure-then-average" and "measure-then-VQC").
• Performance:
  • Accuracy: eviQVFL consistently achieved higher classification accuracy than all classical and quantum baselines across all datasets and parameter budgets.
  • Trainability: Unlike the "teleported-then-VQC" baseline (which used a trainable VQC for fusion and suffered from barren plateaus), eviQVFL trained stably and converged quickly.
  • Loss: While quantum models showed higher loss values due to the theoretical lower bound of cross-entropy loss for quantum outputs, eviQVFL achieved the lowest loss among quantum models.
• Specific Findings:
  • In the Breast Cancer and Credit Card Fraud tasks (tabular data), eviQVFL significantly outperformed the "measure-then-VQC" baseline, which failed to learn due to barren plateaus.
  • The framework demonstrated that using the total quantum resources of all parties (via the joint fusion) leads to better representation capacity than averaging local models.

5. Significance

• Solving the Barren Plateau in VFL: By replacing the trainable fusion layer with a theoretically grounded, non-parametric evidential circuit, the paper offers a practical solution to the trainability issues of large-scale quantum federated learning.
• Enhanced Privacy: The use of quantum teleportation provides a stronger theoretical guarantee of privacy compared to classical VFL, which often relies on complex encryption schemes (like homomorphic encryption) that can be computationally expensive.
• Bridging Theory and Practice: The work successfully integrates Evidence Theory (for handling uncertainty and fusion) with Quantum Computing (for efficient state manipulation), demonstrating a new paradigm for interpretable and robust quantum machine learning.
• Scalability: The framework suggests that quantum advantage in VFL is achievable even with limited local resources, as the global model leverages the combined quantum capacity of all participating parties.

In summary, eviQVFL represents a significant step forward in quantum machine learning by addressing the specific constraints of vertical data partitioning, offering a robust, private, and trainable solution that outperforms existing classical and quantum approaches.