Hybrid Quantum-Classical Ridgelet Neural Networks for Portfolio Optimization

1. Problem Statement

The paper addresses two fundamental challenges in quantitative finance:

1. Financial Time-Series Forecasting: Predicting stock returns is difficult due to the non-linear, chaotic, and volatile nature of financial markets. Traditional statistical models (ARIMA, GARCH) often fail to capture complex patterns, while standard Deep Learning (DL) models (LSTM, Transformers) struggle with interpretability, overfitting, and capturing directional singularities.
2. Portfolio Optimization: Selecting an optimal subset of assets from a large universe to maximize returns while minimizing risk is a combinatorial optimization problem. Solving this under cardinality constraints (selecting a fixed number of assets) is NP-hard for classical computers as the search space grows exponentially.

The authors propose a Hybrid Quantum-Classical framework that leverages the expressive power of quantum computing for feature extraction and optimization, specifically tailored to handle the non-linearity of financial data.

2. Methodology

The proposed framework consists of two integrated modules: a Quantum Ridgelet Neural Network (QRNN) for forecasting and a Quantum Approximate Optimization Algorithm (QAOA) for portfolio selection.

A. Quantum Ridgelet Neural Network (QRNN)

The QRNN is designed to forecast asset returns by integrating Ridgelet transforms with Parametrized Quantum Circuits (PQCs).

• Ridgelet Transform: Unlike standard wavelets, Ridgelet transforms are sensitive to singularities and directional information in data. They decompose time-series data into multi-resolution components, capturing both local and global trends.
• Quantum Encoding: Classical input data is encoded into quantum states using angle encoding via $R_y(\theta)$ rotation gates.
• Single-Qubit Formulation: The authors mathematically prove that a sequence of quantum rotations driven by Ridgelet functions collapses into a single effective rotation. The expectation value of the Pauli-Z operator ($\langle Z \rangle$) maps to a non-linear function of the classical Ridgelet output: $\cos(2g_J(x))$. This creates a quantum analogue of a non-linear neuron.
• Multi-Qubit Extension: The model is generalized to $n$-qubit systems to handle high-dimensional feature spaces, utilizing tensor products of single-qubit states.
• Architecture: The pipeline involves Ridgelet feature extraction $\rightarrow$ Linear mapping to qubit space $\rightarrow$ Variational Quantum Circuit (VQC) with entangling gates $\rightarrow$ Measurement $\rightarrow$ Classical post-processing.

B. QAOA-Based Portfolio Optimization

Once the QRNN predicts expected returns ($\hat{\mu}$), the portfolio selection problem is formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem.

• Objective: Minimize portfolio variance while maximizing expected return, subject to a cardinality constraint ($K$ assets).
• QUBO Formulation: The constrained problem is converted into an unconstrained form by adding a quadratic penalty term for the cardinality constraint:
  $$Q(x) = -\hat{\mu}^T x + x^T \Sigma x + P \left( \sum x_i - K \right)^2$$
• QAOA Solver: The QUBO problem is mapped to an Ising Hamiltonian ($H_C$). The QAOA algorithm prepares a variational quantum state $|\psi(\gamma, \beta)\rangle$ using alternating cost and mixer unitaries. Classical optimizers tune parameters $\gamma$ and $\beta$ to minimize the expected energy, yielding the optimal binary portfolio vector $x^*$.

C. Noise and Error Correction Analysis

To evaluate robustness, the authors simulated the model under three conditions:

1. Noisy Model: Incorporating bit-flip, phase-flip, and depolarizing channels.
2. Full Quantum Error Correction (QEC): Using a 3-qubit repetition code to encode logical qubits.
3. Partial QEC: Applying error correction only to specific parts of the circuit to balance complexity and stability.

3. Key Contributions

1. Novel QRNN Architecture: Introduction of a hybrid model combining Ridgelet transforms with PQCs, enabling efficient representation learning of directional patterns in financial data.
2. Theoretical Formulation: A rigorous mathematical derivation extending the QRNN from a single-qubit system to multi-qubit systems, proving the equivalence between the quantum circuit output and the classical Ridgelet expansion.
3. Integrated Optimization Framework: A seamless pipeline where QRNN forecasts feed directly into a QAOA-based portfolio optimizer, solving the asset selection problem in QUBO format.
4. Robustness Analysis: Comprehensive evaluation of the model under realistic noise conditions, demonstrating that Partial QEC offers the best trade-off between accuracy and computational complexity compared to Full QEC or no correction.
5. Empirical Validation: Extensive testing on real-world financial data (12 stocks, 2010–2019) using NVIDIA H100 NVL GPU clusters for high-performance simulation.

4. Experimental Results

The model was tested against classical baselines (LSTM, GRU, Transformer, CRNN) and quantum baselines (QLSTM, QGRU, QTransformer).

• Forecasting Accuracy:
  • The QRNN achieved the lowest RMSE and MAE in both training and testing phases.
  • Multi-qubit results: QRNN Testing RMSE was 0.0293, significantly outperforming CRNN (0.0862), LSTM (0.2149), and quantum baselines like QLSTM (0.1482).
  • The model demonstrated superior generalization across diverse sectors (Tech, Auto, Energy).
• Noise Resilience:
  • Noise drastically increased error rates (e.g., RMSE jumped from 0.071 to 0.298 in single-qubit scenarios).
  • Partial QEC reduced errors effectively (RMSE ~0.217) with lower circuit complexity than Full QEC, which showed marginal gains but higher complexity.
• Portfolio Performance:
  • The QRNN-QAOA strategy achieved a Sharpe Ratio of 0.9281, outperforming Top-K (0.6311), Equal Weight (0.4632), and Random selection (0.1293).
  • It maintained a closer alignment between predicted and actual returns and exhibited significantly lower variance.

5. Significance and Limitations

Significance:

• Superior Predictive Power: The integration of Ridgelet transforms with quantum entanglement allows the model to capture complex non-linearities and directional trends that classical DL models miss.
• Scalability of Optimization: By converting portfolio selection to QUBO and solving it with QAOA, the approach offers a potential pathway to solving combinatorial financial problems that are intractable for classical solvers at scale.
• Practical Feasibility: The study provides a realistic assessment of quantum advantages in the NISQ (Noisy Intermediate-Scale Quantum) era, highlighting the necessity of error mitigation strategies like Partial QEC.

Limitations:

• Computational Complexity: The simulation of quantum states (especially with error correction) scales exponentially ($4^n$ for density matrices). The authors noted that scaling beyond 3 logical qubits (9 physical qubits) led to GPU memory exhaustion (OOM) due to the density matrix size.
• Simulation vs. Hardware: The results are based on high-fidelity simulations on classical GPUs, not actual quantum hardware, though the noise models simulate realistic hardware imperfections.
• Training Dynamics: Increasing qubit count introduces challenges like vanishing gradients and noise accumulation, making training difficult for large-scale systems.

Conclusion

The paper presents a robust, theoretically grounded hybrid framework that successfully leverages quantum computing for financial forecasting and portfolio optimization. By combining the feature extraction capabilities of Ridgelet transforms with the optimization power of QAOA, the QRNN model demonstrates superior accuracy and risk-adjusted returns compared to state-of-the-art classical and quantum deep learning models. The study emphasizes that while noise is a significant hurdle, strategic error mitigation (Partial QEC) can make such hybrid models viable for real-world financial applications.