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.
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, a core combinatorial optimization problem in quantitative finance, requires balancing return maximization and risk minimization within discrete asset sets. Classical algorithms often struggle with scalability and local optima traps when handling large-scale asset pools, limiting their practicality in dynamic markets. This study proposes a complete Quantum Approximate Optimization Algorithm (QAOA) workflow for portfolio optimization, from problem formalization to experimental validation. We first transform the portfolio optimization problem into a Quadratic Unconstrained Binary Optimization (QUBO) framework, then design a parameterized QAOA circuit with adjustable layers to balance expressiveness and noise resilience. We propose a methodology using Qiskit for quantum simulation and classical optimizers for parameter tuning, we propose implementing the algorithm on real-world S&P 500 stock data. We could also analyze the impact of quantum hardware noise and propose mitigation strategies, providing actionable insights for Noisy intermediate-scale quantum computing (NISQ)-era implementations. This work bridges quantum algorithm design with real-world financial applications, offering a theoretical framework for quantum-driven portfolio optimization.
Quantum finance represents a pivotal and cutting-edge application domain within the burgeoning field of quantum computing. In this work, we propose a two-step quantum approximate optimization algorithm (two-step QAOA) for portfolio optimization and risk assessment. The algorithm initiates by formulating the stock selection problem as a quadratic unconstrained binary optimization (QUBO) problem and employs a classical-quantum hybrid method to find the ground state of the Hamiltonian. We then introduce an energy-based characteristic indicator U∈[0,1), which quantitatively evaluates portfolio performance under customizable investment preferences, effectively capturing the trade-off between expected return and risk. The number of qubits required scales with the number of stocks N in the pool, and the number of Hamiltonian terms is O(N2). Numerical simulations show that the algorithm provides consistent and reasonable assessment results on both training and test datasets under different investment preferences (aggressive or conservative), validating the capability of the characteristic indicator to extract intrinsic information from the portfolios. Additionally, by incorporating warm-starting and digitized counterdiabatic techniques, the algorithm achieves improved scalability and faster convergence. Our work presents a flexible and practical algorithmic framework for applying quantum computing in the financial domain.
Abstract The Quantum Approximate Optimization Algorithm (QAOA) is a quantum algorithm proposed for Noisy Intermediate-Scale Quantum (NISQ) devices and is regarded as a promising approach to combinatorial optimization problems, with potential applications in the financial sector. In this study, we apply QAOA to the portfolio optimization problem, which is one of the central challenges in financial engineering. A portfolio consists of a combination of multiple assets, and the portfolio optimization problem aims to determine the optimal asset allocation by balancing expected return and risk. In the context of quantum optimization, portfolio optimization is often formulated using discrete variables. Unlike conventional binary formulations, we consider a ternary portfolio optimization problem that accounts for three states—holding, not holding, and short selling—and compare its performance using different mixer operators. Specifically, we implement QAOA with the standard mixer and several XY Mixers (XY Ring, XY Parity Ring, XY Full, and QAMPA), and conducted simulations using real data based on the German stock index (DAX 30) for portfolios consisting of 5 and 8 assets. Furthermore, we introduce noise based on a depolarizing channel to investigate the behavior of the algorithm in realistic environments. The results show that while XY Mixers exhibit superiority in noiseless settings, their advantage degrades in noisy environments, and the optimal choice of mixer depends on both the number of QAOA depths and the noise strength.
Portfolio optimization is a primary component of the decision-making process in finance, aiming to tactfully allocate assets to achieve optimal returns while considering various constraints. Herein, we proposed a method that uses the knapsack-based portfolio optimization problem and incorporates the quantum computing capabilities of the quantum walk mixer with the quantum approximate optimization algorithm (QAOA) to address the challenges presented by the NP-hard problem. Additionally, we present the sequential procedure of our suggested approach and demonstrate empirical proof to illustrate the effectiveness of the proposed method in finding the optimal asset allocations across various constraints and asset choices. Moreover, we discuss the effectiveness of the QAOA components in relation to our proposed method. Consequently, our study successfully achieves the approximate ratio of the portfolio optimization technique using a circuit layer ofp⩾3, compared to the classical best-known solution of the knapsack problem. Our proposed methods potentially contribute to the growing field of quantum finance by offering insights into the potential benefits of employing quantum algorithms for complex optimization tasks in financial portfolio management.
The Quantum Approximate Optimization Algorithm (QAOA) is a quantum algorithm proposed for Noisy Intermediate-Scale Quantum (NISQ) devices and is regarded as a promising approach to combinatorial optimization problems, with potential applications in the financial sector. In this study, we apply QAOA to the portfolio optimization problem, which is one of the central challenges in financial engineering. A portfolio consists of a combination of multiple assets, and the portfolio optimization problem aims to determine the optimal asset allocation by balancing expected return and risk. In the context of quantum optimization, portfolio optimization is often formulated using discrete variables. Unlike conventional binary formulations, we consider a ternary portfolio optimization problem that accounts for three states-holding, not holding, and short selling-and compare its performance using different mixer operators. Specifically, we implement QAOA with the standard mixer and several XY Mixers (XY Ring, XY Parity Ring, XY Full, and QAMPA), and conducted simulations using real data based on the German stock index (DAX 30) for portfolios consisting of 5 and 8 assets. Furthermore, we introduce noise based on a depolarizing channel to investigate the behavior of the algorithm in realistic environments. The results show that while XY Mixers exhibit superiority in noiseless settings, their advantage degrades in noisy environments, and the optimal choice of mixer depends on both the number of QAOA depths and the noise strength.
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.