A portfolio optimization problem is highly important for the financial sector and critical in portfolio management. To solve the mean-variance portfolio optimization classically, the time resources needed grow exponentially with the number of stocks used. In practice, the mean-variance portfolio optimization can be solved for a few hundred assets with classical computers. Due to the exponential nature of the portfolio optimization problem, quantum computing can be utilized to improve the computation process. For solving the portfolio optimization with quantum computers we use hybrid algorithms, build of quantum and classical computing parts. More specifically, we consider the most promising hybrid algorithm, called Variational Quantum Eigensolver (VQE), which is particularly suitable for near-term quantum hardware. To maximize the potential of the near-term quantum computers, we aim to discover the suitable form of the most crucial components of the VQE algorithm. In addition to this, to further improve the performance of the VQE algorithm, we investigate the utilization of various warm-start techniques for maximizing the efficient use of the limited quantum resources. The crucial components of the VQE algorithm are the quantum circuit, i.e. ansatz, and classical optimizer. We approach the problem of finding the suitable ansatz for portfolio optimization from two different directions, with theoretical and hardware-efficient approaches. We start with the theoretically best possible, but complex structured chemistry-inspired ansatz and continue to study the capabilities of hardware efficient, but ad hoc, Dicke state ansatzes. For optimizing the classical component of the VQE algorithm, we will briefly consider the effect of various classical optimizers on the performance of the most suitable ansatzes. When the most promising components of the VQE algorithm are known, the application of various warm-start techniques of parameter initialization can begin. In initialization strategies, we investigate how the special features of Dicke state ansatzes can be used to develop tailored initialization strategies. We also investigate how optimal parameters of related problems can be utilized in parameter initialization. Our results draw clear differences in warm-startability between various ansatzes and optimizers. The results on the warm-startability of different ansatzes are important for the quantum algorithm design, because often when solving especially real-world optimization problems, the original problem can change just a little, and therefore warm-start techniques can be highly beneficial. Based on the results the Dicke state ansatzes can be surprisingly useful and their special features allow us to develop a successful initialization strategy.
A portfolio optimization problem is highly important for the financial sector and critical in portfolio management. To solve the mean-variance portfolio optimization classically, the time resources needed grow exponentially with the number of stocks used. In practice, the mean-variance portfolio optimization can be solved for a few hundred assets with classical computers. Due to the exponential nature of the portfolio optimization problem, quantum computing can be utilized to improve the computation process. For solving the portfolio optimization with quantum computers we use hybrid algorithms, build of quantum and classical computing parts. More specifically, we consider the most promising hybrid algorithm, called Variational Quantum Eigensolver (VQE), which is particularly suitable for near-term quantum hardware. To maximize the potential of the near-term quantum computers, we aim to discover the suitable form of the most crucial components of the VQE algorithm. In addition to this, to further improve the performance of the VQE algorithm, we investigate the utilization of various warm-start techniques for maximizing the efficient use of the limited quantum resources. The crucial components of the VQE algorithm are the quantum circuit, i.e. ansatz, and classical optimizer. We approach the problem of finding the suitable ansatz for portfolio optimization from two different directions, with theoretical and hardware-efficient approaches. We start with the theoretically best possible, but complex structured chemistry-inspired ansatz and continue to study the capabilities of hardware efficient, but ad hoc, Dicke state ansatzes. For optimizing the classical component of the VQE algorithm, we will briefly consider the effect of various classical optimizers on the performance of the most suitable ansatzes. When the most promising components of the VQE algorithm are known, the application of various warm-start techniques of parameter initialization can begin. In initialization strategies, we investigate how the special features of Dicke state ansatzes can be used to develop tailored initialization strategies. We also investigate how optimal parameters of related problems can be utilized in parameter initialization. Our results draw clear differences in warm-startability between various ansatzes and optimizers. The results on the warm-startability of different ansatzes are important for the quantum algorithm design, because often when solving especially real-world optimization problems, the original problem can change just a little, and therefore warm-start techniques can be highly beneficial. Based on the results the Dicke state ansatzes can be surprisingly useful and their special features allow us to develop a successful initialization strategy.
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.
Combinatorial Optimization problems play an important role in various domains such as logis- tics, life science or the financial sector. As their complexity in general grows exponentially with their problem size, finding the optimal solution classically becomes infeasible very fast. This raises the central question of whether quantum computers are in principle suitable for solving combinatorial optimization problems better and faster than classical computers. In order to find answers to that question, one needs to study quantum algorithms designed for those problems under ideal conditions (i.e., simulations, which are however limited due to exponential scaling) to investigate their potential, but one also needs real quantum hardware to test its viability and evaluate whether practical advantages are reasonable. In this regard, we consider variational quantum algorithms, especially the Quantum Approximate Optimization Algorithm (QAOA) consisting of a problem and a mixer Hamiltonian with freely selectable parameters, originally proposed as variational hybrid ansatz with classical optimization. This algorithm structure of- fers flexible and relatively shallow circuits, which makes it suitable for current NISQ hardware. However, optimizing the variational circuit parameters can be an NP-hard problem at worst. In this thesis, we study different versions of QAOA, especially with a focus on limitations, perfor- mance and potential advantages opposite classical methods. The first part of the thesis concerns the investigation of various aspects regarding the imple- mentation of the variational QAOA protocol such as: the incorporation of a problem constraint either by adding a suitable penalty term of by choosing a suitable mixer (that restricts the search space), the scaling of the problem Hamiltonian in comparison to the mixer Hamiltonian, the choice of the initial values for the classical optimization of the circuit parameters, and finally the choice of the function to be minimized (i.e., CVaR vs. expectation value). We apply this proto- col to both the portfolio optimization and the feature selection and evaluate the performance. In order to improve QAOA, we consider the warm-start QAOA for different problem instances and analyze its improved performance with regard to involved quantum effects. For this purpose, we propose a classical pre-processing method and demonstrate that the warm-start results can be reproduced or even exceeded by this method. The second part addresses the issue that the standard variational form requires a high number of circuit parameters that have to be optimized at a sufficiently large depth, which is a major limitationforachievingpotentialscalingadvantagescomparedtoclassicalmethods. Toovercome this limitation, we investigate and further develop fixed schedules for QAOA, which only rely on two free parameters. We start with introducing the continuous adiabatic theorem as it motivates the form of QAOA and show its limitations, which we address by applying a discrete version of the theorem. With our findings, a new version of QAOA based on linear-ramps (LR-QAOA) is introduced and further developed. In order to derive suitable parameter values for this schedule, we propose a method based on extrapolation from smaller towards larger problem sizes. With our results obtained on a noiseless emulator, we evaluate the quantum runtime scaling and discuss whether a scaling advantage can be achieved compared to classical methods.
The possibility of adopting quantum computing in the field of financial services can bring a new breakthrough in portfolio management since the optimization of this method requires large numbers for the solution that exceed the capabilities of classical computing algorithms. This study focuses on using a hybrid quantum-classical computational method known as the Variational Quantum Eigensolver (VQE) in optimizing portfolio selection. The VQE aims at finding the best allocation of the portfolio; the best weights that correspond to minimum risk and maximum return as predicted by the modern portfolio theory. The study assesses the suitability and speed of VQE in solving matters involving portfolio management tasks such as handling vast data and nonlinear constraints which are hard to solve through other optimization methods. Thus, under the theoretical framework of quantum computing, the study seeks to address some of the following challenges: computational time and scalability in portfolio optimization. The VQE algorithm work on different portfolio configurations; their performance in comparison with classical techniques; special attention is paid to the rate of convergence and the achievement of the optimal solution. Based on the results of this work we can conclude that there is a considerable literature on quantum finance and that this field can benefit from utilization of quantum algorithms as a tool to improve decision-making in the financial market. That is why the study proves that quantum optimization techniques, to wit VQE, can provide better solutions when it comes to portfolio management within the sphere of finance.
In finance, portfolio optimization aims at finding optimal investments maximizing a trade-off between return and risks, given some constraints. Classical formulations of this quadratic optimization problem have exact or heuristic solutions, but the complexity scales up as the market dimension increases. Recently, researchers are evaluating the possibility of facing the complexity scaling issue by employing quantum computing. In this paper, the problem is solved using the Variational Quantum Eigensolver (VQE), which in principle is very efficient. The main outcome of this work consists of the definition of the best hyperparameters to set, in order to perform Portfolio Optimization by VQE on real quantum computers. In particular, a quite general formulation of the constrained quadratic problem is considered, which is translated into Quadratic Unconstrained Binary Optimization by the binary encoding of variables and by including constraints in the objective function. This is converted into a set of quantum operators (Ising Hamiltonian), whose minimum eigenvalue is found by VQE and corresponds to the optimal solution. In this work, different hyperparameters of the procedure are analyzed, including different ansatzes and optimization methods by means of experiments on both simulators and real quantum computers. Experiments show that there is a strong dependence of solutions quality on the sufficiently sized quantum computer and correct hyperparameters, and with the best choices, the quantum algorithm run on real quantum devices reaches solutions very close to the exact one, with a strong convergence rate towards the classical solution, even without error-mitigation techniques. Moreover, results obtained on different real quantum devices, for a small-sized example, show the relation between the quality of the solution and the dimension of the quantum processor. Evidences allow concluding which are the best ways to solve real Portfolio Optimization problems by VQE on quantum devices, and confirm the possibility to solve them with higher efficiency, with respect to existing methods, as soon as the size of quantum hardware will be sufficiently high.
This research investigates the efficacy of quantum and classical algorithms in the context of portfolio optimization, focusing on a dataset comprising 20 equities from India's National Stock Exchange over a duration of three years. The study employs quantum methodologies such as the Quantum Approxi-mate Optimization Algorithm (QAOA), the Variational Quantum Eigensolver (VQE), and the Exact Minimum Eigen Solver, alongside conventional strategies including Mean-Variance Op-timization, Genetic Algorithms, and Simulated Annealing. The methodology for this study includes data preprocessing, issue designing, and performance evaluation based on metrics such as MSE, RMSE, and Mean Absolute Error. The results obtained show quantum algorithms exhibit better performance over the classical methods regarding prediction accuracy and risk-return profiles, especially the Exact Minimum Eigen hSolver and QAOA. The Exact Minimum Eigen Solver and QAOA had the lowest MSE values (0.006 and 0.005, respectively), whereas the best-performing classical technique, the Genetic Algorithm, had an MSE of 0.0105. These findings imply that quantum algorithms provide significant advantages in portfolio optimization problems, with the potential to revolutionize financial optimization tactics. The research can be further unfolded into avenues for future inquiry, including expansion of the choice of assets, alternative data sources integration, as well as hybrid quantum-classical models to efficiently enhance portfolio management strategies.
Variational Quantum Algorithms are promising candidates for near-term quantum advantage, but face significant scalability challenges due to barren plateaus. In this work, we address the trainability of the Variational Quantum Eigensolver for combinatorial portfolio optimization by evaluating three warm-start initialization strategies: Clifford, near-Clifford and a mixed approach. We compare these against a cold-start baseline initialized with the initial parameters close to zero across system sizes from [Formula: see text] to [Formula: see text]. Our results show that while the uninformed cold start fails to converge for larger systems, all warm-start strategies achieve a [Formula: see text] success rate and high initial-state fidelities. Crucially, we observe no clear difference in convergence behavior among the warm-start methods. Consequently, for this class of QUBO-based portfolio optimization problems, we identify the purely Clifford-based approach as the most efficient strategy; it consistently places the optimizer in a region from which the variational optimization converges successfully while remaining polynomially simulable on classical hardware during the quantum phase of the algorithm, indicating that the additional computational cost of non-Clifford resources offers negligible benefits for initialization in this specific context.