Abstract We present a detailed study of portfolio optimization using different versions of the quantum approximate optimization algorithm (QAOA). For a given list of assets, the portfolio optimization problem is formulated as quadratic binary optimization constrained on the number of assets contained in the portfolio. QAOA has been suggested as a possible candidate for solving this problem (and similar combinatorial optimization problems) more efficiently than classical computers in the case of a sufficiently large number of assets. However, the practical implementation of this algorithm requires a careful consideration of several technical issues, not all of which are discussed in the present literature. The present article intends to fill this gap and thereby provides the reader with a useful guide for applying QAOA to the portfolio optimization problem (and similar problems). In particular, we will discuss several possible choices of the variational form and of different classical algorithms for finding the corresponding optimized parameters. Viewing at the application of QAOA on error-prone NISQ hardware, we also analyse the influence of statistical sampling errors (due to a finite number of shots) and gate and readout errors (due to imperfect quantum hardware). Finally, we define a criterion for distinguishing between ‘easy’ and ‘hard’ instances of the portfolio optimization problem.
Abstract We present a detailed study of portfolio optimization using different versions of the quantum approximate optimization algorithm (QAOA). For a given list of assets, the portfolio optimization problem is formulated as quadratic binary optimization constrained on the number of assets contained in the portfolio. QAOA has been suggested as a possible candidate for solving this problem (and similar combinatorial optimization problems) more efficiently than classical computers in the case of a sufficiently large number of assets. However, the practical implementation of this algorithm requires a careful consideration of several technical issues, not all of which are discussed in the present literature. The present article intends to fill this gap and thereby provides the reader with a useful guide for applying QAOA to the portfolio optimization problem (and similar problems). In particular, we will discuss several possible choices of the variational form and of different classical algorithms for finding the corresponding optimized parameters. Viewing at the application of QAOA on error-prone NISQ hardware, we also analyse the influence of statistical sampling errors (due to a finite number of shots) and gate and readout errors (due to imperfect quantum hardware). Finally, we define a criterion for distinguishing between ‘easy’ and ‘hard’ instances of the portfolio optimization problem.
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.
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.
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.
Recently, several researchers proposed portfolio optimization as a potential use case for quantum optimization. However, the literature is lacking an extensive benchmark quantifying the potential of quantum computers for portfolio optimization. In this work, we contribute to closing this gap. We provide a computational study, comparing quantum approaches against state-of-the-art classical methods on a meaningful, real-world instance set. In particular, we compare quantum annealing and the quantum approximate optimization algorithm against classical mixed-integer programming, simulated annealing, steepest descent local search, tabu search and a problem-tailored heuristic. We consider a variance-minimizing variant of portfolio optimization which we show to be more difficult to solve for classical optimizers than return-maximizing or multi-objective formulations. Our benchmark data set comprises 270 instances with up to 1,000 assets from actual stock data. Due to hardware limitation, quantum methods could only be tested for instances with at most 30 assets. The results show that all instances can be solved to proven optimality by mixed-integer programming in the order of seconds. Moreover, the problem-tailored heuristic consistently outperforms quantum approaches in terms of solution quality for fixed runtime. Thus, we conclude that there is only very limited room for a potential quantum advantage for the considered variant of portfolio optimization.
A quantum approach for portfolio optimization involves using quantum computing to solve complex optimization problems related to portfolio selection. The approach involves constructing a mathematical model of the portfolio optimization problem using various algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) or the Variational Quantum Eigensolver (VQE). This approach has the potential to provide more accurate and efficient solutions than classical approaches, particularly for large and complex portfolio optimization problems. Moreover, this paper also dives into discussing and comparing the results obtained from both these approaches. Our work is under the assumptions required for the proposed formulation deriving meaningful considerations about the solution quality found, measured as the value of Sharpe Ratio.
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.