Quantum Portfolio Optimization QAOA Project Summary

Overview

This project focuses on solving the portfolio optimization problem using both classical and quantum-inspired methods. The goal is to find an optimal allocation of assets that maximizes returns while minimizing risk. The workflow progresses from data preparation to quantum optimization via QAOA, followed by a comprehensive comparison of results.

The main steps covered across the notebooks are:

1. Data Preparation and QUBO Formulation (Data and QUBO.ipynb)
2. Brute Force Solution (Brute Force Method.ipynb)
3. Classical Optimization using CVXPY (CVXPy Method.ipynb)
4. Quantum Approximate Optimization Algorithm (QAOA) (QAOA GPU.ipynb)
5. Result Comparison and Visualization (Comparison.ipynb)

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1. Data and QUBO Formulation

We begin by defining the problem through historical asset data. The expected returns and covariance matrix are computed to quantify performance and risk.

Mathematical Model

The classical portfolio optimization problem is defined as:

$$ \max_x ; (\mu^T x - \lambda x^T \Sigma x) $$

subject to:

$$ \sum_i x_i = 1, \quad x_i \in {0, 1} $$

where:

• ($\mu$) is the vector of expected returns.
• ($\Sigma$) is the covariance matrix.
• ($\lambda$) is the risk-aversion coefficient.

This objective function is then reformulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem:

$$ H(x) = x^T Q x $$

where $(Q)$ encodes both the return and risk terms, suitable for quantum optimization methods.

Key Outputs:

• Asset statistics and correlation heatmaps.
• Constructed QUBO matrix visualizations.

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2. Brute Force Method

The brute force approach evaluates all possible combinations of portfolio allocations, serving as the ground truth for benchmarking.

Methodology

For $(n)$ assets, all $(2^n)$ combinations are tested to compute their objective values based on the QUBO function.

Key Outputs:

• Optimal portfolio configuration.
• Stored results in .npz format.

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3. CVXPY Classical Optimization

Using the CVXPY library, we solve the same problem via convex optimization, under continuous relaxation or binary constraints.

Classical Formulation

CVXPY solves:

$$ \max_x ; (\mu^T x - \lambda x^T \Sigma x) $$

subject to linear or quadratic constraints.

After solving the optimization, the main results — such as objective function value, annualized return, annualized risk, sharpe ratio, and optimal weights — are stored in a compressed NumPy file (.npz) for later use.

Key Outputs:

• Optimal allocation vector.
• Stored results in .npz format.

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4. Quantum Optimization via QAOA

We apply the Quantum Approximate Optimization Algorithm (QAOA) using Qiskit to solve the QUBO on a simulated quantum backend (GPU-accelerated environment).

Workflow

1. Define the cost Hamiltonian from the QUBO matrix.
2. Configure QAOA parameters (layers ($p$), optimizer, etc.).
3. Run the optimization and measure probabilities.

Quantum Objective

The cost function for QAOA corresponds to the expectation value:

$$ C(\gamma, \beta) = \langle \psi(\gamma, \beta) | H | \psi(\gamma, \beta) \rangle $$

where $| \psi(\gamma, \beta) \rangle$ is the parameterized quantum state.

Key Outputs:

• QAOA energy convergence.
• Risk vs. return energy for QAOA.

5. Comparison and Analysis

Finally, results from all methods (Brute Force, CVXPY, and QAOA) are compared in terms of accuracy and solution quality.

Metrics

• Optimal portfolio composition.
• Objective function value.

Key Visualizations:

• Comparative plots of performance.

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Conclusions

• Brute-force:
  Provides the exact optimal solution by exhaustively exploring all possible portfolios. However, it is computationally expensive and scales poorly as the number of assets increases.

• Classical (CVXPY):
  Finds a near-optimal solution much faster using convex optimization. In this case, its result nearly coincides with the brute-force optimum, confirming its accuracy and efficiency as a classical benchmark.

• Quantum (QAOA):
  Achieves a feasible but suboptimal portfolio, with a lower expected return than the classical and brute-force results. This reflects current limitations in parameter tuning and circuit depth, though it illustrates the potential of quantum methods for combinatorial optimization.

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Repository Structure

├── Data and QUBO.ipynb
├── Brute Force Method.ipynb
├── CVXPy Method.ipynb
├── QAOA GPU.ipynb
├── Comparison.ipynb
├── figures/
│   ├── qubo_heatmap.png
│   ├── brute_force_plot.png
│   ├── cvxpy_frontier.png
│   ├── qaoa_convergence.png
│   ├── qaoa_probabilities.png
│   ├── comparison.png
│   └── risk_vs_return_qaoa.png
└── README.md

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Project: Quantum Portfolio Optimization (QAOA) Tools: Python, Qiskit, CVXPY, NumPy, Matplotlib