Recently, Quantum Approximate Optimization Algorithm (QAOA) and Quantum Alternating Operator ansatz (QAOAz) are utilized to solve the Mean Variance (MV) model for the portfolio optimization problem (QAOAz-MV), which shows performance advantages over classical algorithms on this huge search space. For the more complex and comprehensive risk parity (RP) model, a novel QAOAz solution (QAOAz-RP) is proposed. We begin by defining the RP model for the portfolio optimization problem. Next, we detail the QAOAz algorithm process, where the problem Hamiltonian with the ZZZZ term is derived, the corresponding quantum circuit is ingeniously constructed using the parity check method, and the whole quantum circuit containing the ring XY-mixer is given. Finally, to improve the optimization performance of QAOAz, a Particle Swarm Optimization (PSO) optimizer is introduced to tune the parameters of the quantum circuits, which is applicable to both QAOAz-MV and QAOAz-RP. The experiment conducted on multiple financial markets (e.g. Chinese, U.S., and European) demonstrate that PSO-QAOAz-RP is significantly better for portfolio optimization than ABC-LP, GWO, GA and QAOAz-RP on eight portfolios in all metrics. PSO-QAOAz-MV also has advantage for MV model over the quantum algorithms, including QAOA (improves approximate ratio by 54.79% on average) and QAOAz (improves approximate ratio by 15.38% on average). This study not only provides a breakthrough quantum solution for portfolio optimization, but also provides a reusable technology paradigm for the deep integration of quantum computing and financial engineering. • It is the first QAOAz solution for the RP model of portfolio optimization problem. • The problem Hamiltonian and quantum circuit are derived and ingeniously constructed. • A PSO optimizer is proposed to optimize the parameters of QAOAz’s quantum circuit. • Our QAOAz-RP and PSO-QAOAz-RP outperform state-of-the-art classical algorithms. • Our PSO optimizer is effective on both MV and RP models.
Recently, Quantum Approximate Optimization Algorithm (QAOA) and Quantum Alternating Operator ansatz (QAOAz) are utilized to solve the Mean Variance (MV) model for the portfolio optimization problem (QAOAz-MV), which shows performance advantages over classical algorithms on this huge search space. For the more complex and comprehensive risk parity (RP) model, a novel QAOAz solution (QAOAz-RP) is proposed. We begin by defining the RP model for the portfolio optimization problem. Next, we detail the QAOAz algorithm process, where the problem Hamiltonian with the ZZZZ term is derived, the corresponding quantum circuit is ingeniously constructed using the parity check method, and the whole quantum circuit containing the ring XY-mixer is given. Finally, to improve the optimization performance of QAOAz, a Particle Swarm Optimization (PSO) optimizer is introduced to tune the parameters of the quantum circuits, which is applicable to both QAOAz-MV and QAOAz-RP. The experiment conducted on multiple financial markets (e.g. Chinese, U.S., and European) demonstrate that PSO-QAOAz-RP is significantly better for portfolio optimization than ABC-LP, GWO, GA and QAOAz-RP on eight portfolios in all metrics. PSO-QAOAz-MV also has advantage for MV model over the quantum algorithms, including QAOA (improves approximate ratio by 54.79% on average) and QAOAz (improves approximate ratio by 15.38% on average). This study not only provides a breakthrough quantum solution for portfolio optimization, but also provides a reusable technology paradigm for the deep integration of quantum computing and financial engineering. • It is the first QAOAz solution for the RP model of portfolio optimization problem. • The problem Hamiltonian and quantum circuit are derived and ingeniously constructed. • A PSO optimizer is proposed to optimize the parameters of QAOAz’s quantum circuit. • Our QAOAz-RP and PSO-QAOAz-RP outperform state-of-the-art classical algorithms. • Our PSO optimizer is effective on both MV and RP models.
With the development of intelligent algorithms, new challenges have emerged in the financial field in optimizing the investment portfolio. In view of the current problems existing in portfolio optimization, the particle swarm algorithm (Particle Swarm Optimization, PSO) and quantum approximation optimization algorithm (Quantum Approximate Optimization Algorithm, QAOA) of the initial population Qanta coding is improved, and a hybrid algorithm suitable for solving enterprise portfolio optimization problems is proposed. The results found that the average performance of this method is below 0.06, which is better compared with the comparison algorithm. At the same time, with the increase of the risk level, the maximum, minimum and average increase of the yield are 10.99%,19.80% and 7.98% respectively, stabilizing. The results show that the research algorithm is effective in solving the enterprise portfolio optimization problem, has better optimization ability in practical application, and can provide reference for enterprise investors to make investment decisions.
Portfolio Optimization (PO) is a financial problem aiming to maximize the net gains while minimizing the risks in a given investment portfolio. The novelty of Quantum algorithms lies in their acclaimed potential and capability to solve complex problems given the underlying Quantum Computing (QC) infrastructure. Utilizing QC's applicable strengths to the finance industry's problems, such as PO, allows us to solve these problems using quantum-based algorithms such as Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization Algorithm (QAOA). While the Quantum potential for finance is highly impactful, the architecture and composition of the quantum circuits have not yet been properly defined as robust financial frameworks/algorithms as state of the art in present literature for research and design development purposes. In this work, we propose a novel scalable framework, denoted PO-QA, to systematically investigate the variation of quantum parameters (such as rotation blocks, repetitions, and entanglement types) to observe their subtle effect on the overall performance. In our paper, the performance is measured and dictated by convergence to similar ground-state energy values for resultant optimal solutions by each algorithm variation set for QAOA and VQE to the exact eigensolver (classical solution). Our results provide effective insights into comprehending PO from the lens of Quantum Machine Learning in terms of convergence to the classical solution, which is used as a benchmark. This study paves the way for identifying efficient configurations of quantum circuits for solving PO and unveiling their inherent inter-relationships.
Abstract This paper presents an improved Quantum Approximate Optimization Algorithm variant based on Conditional Value-at-Risk for addressing portfolio optimization problems. Portfolio optimization is a NP-hard combinatorial problem that aims to select an optimal set of assets and their quantities to balance risk against expected return. The proposed approach uses the QAOA to find the optimal asset combination that maximizes returns while minimizing risk, with a focus on the tail end of the loss distribution. An enhanced QAOA ansatz introduced that offers a balance between optimization quality and circuit depth, leading to faster convergence and higher probabilities of obtaining optimal solutions. Experiments are conducted using historical stock data from Nasdaq, optimizing portfolios with varying numbers of stocks. In the case of 16 stocks, our method achieves optimal cost values with merely 35 iterations, whereas the standard QAOA requires 700 iterations. Our method outperforms other approaches, particularly as the size of the problem increases.
Portfolio optimization is a fundamental problem in quantitative finance that seeks to maximize risk-adjusted returns while managing investment risk. With the advent of quantum computing, there is growing interest in applying quantum algorithms to solve complex optimization problems. This paper presents a comprehensive comparative study of quantum and classical approaches to portfolio optimization, specifically comparing the Quantum Approximate Optimization Algorithm (QAOA), Variational Quantum Eigensolver (VQE), and classical mean-variance optimization. We evaluate these methods on both real market data and synthetic datasets with varying correlation structures. Our experimental results show that while classical methods currently outperform quantum approaches on near-term quantum devices in terms of solution quality and computational efficiency, quantum algorithms demonstrate conceptual viability and promise for future applications as quantum hardware improves. The classical method achieves optimal Sharpe ratios of 1.2-1.8, while QAOA and VQE achieve 80-95% and 75-90% of classical performance respectively. This work provides insights into the current state and future potential of quantum computing for financial optimization.
The increasing complexity of financial markets has heightened the demand for advanced computational methods capable of optimizing portfolios under multi-dimensional constraints involving risk, return, diversification, and liquidity. Traditional optimization frameworks, such as mean-variance analysis and heuristic meta-optimization, are limited by their scalability and sensitivity to non-convex, high-dimensional data structures. This study explores the application of quantum algorithms to portfolio optimization, demonstrating how Quantum Approximate Optimization Algorithm (QAOA), Quantum Annealing, and Variational Quantum Eigensolver (VQE) can address computational bottlenecks inherent in classical approaches. Quantum computation offers the potential to encode complex financial systems as Hamiltonian functions, allowing parallel exploration of vast solution spaces through superposition and entanglement. This enables faster convergence toward optimal portfolio allocations while simultaneously accounting for multi-factor correlations and dynamic market conditions. The research further investigates the role of hybrid quantum–classical optimization pipelines, in which quantum solvers perform core optimization tasks while classical systems manage data preprocessing and constraint validation. Empirical modeling and simulation indicate that quantum-based approaches can outperform conventional methods in risk-adjusted return optimization, efficient frontier computation, and real-time portfolio rebalancing. By bridging the gap between quantum computing and financial engineering, this study contributes to the development of next-generation financial analytics systems, aligning with the broader goal of achieving more adaptive, transparent, and data-resilient financial decision-making.
Large-scale portfolio optimization is hard because investors must balance risk, return, diversification, and liquidity across many assets with complex constraints. Classical methods, such as quadratic programming or metaheuristics, often become slow or less effective as the number of assets and factors grows. In this project, I explore the use of the Quantum Approximate Optimization Algorithm (QAOA) as a hybrid quantum–classical approach to portfolio selection. The framework encodes portfolio decisions as qubits and represents financial objectives—such as mean–variance trade-offs or risk-adjusted return—inside a parameterized quantum circuit. Classical optimization is then used to tune the circuit parameters so that the quantum system converges toward low-risk, high-return portfolios. Multi-factor constraints, including sector limits, liquidity rules, ESG scores, and macroeconomic risk factors, are integrated into the Hamiltonian as penalty terms. A qualitative comparison with classical solvers (e.g., mixed-integer programming and simulated annealing) suggests that QAOA can scale more gracefully in high-dimensional settings and handle complex constraint structures. Overall, the project shows how quantum hybrid optimization could support future portfolio management, linking theoretical quantum algorithms with practical, sustainability-aware investment strategies.