In this work, we benchmark two prominent quantum algorithms: Quantum Imaginary-Time Evolution (QITE) and the Quantum Approximate Optimization Algorithm (QAOA) for obtaining the ground state of Ising-type Hamiltonians. Specifically, we apply them to the Markowitz portfolio optimization problem in quantitative finance, on both digital quantum computers and local quantum simulators with controllable two-qubit errors (noise). In noiseless settings, we find that QAOA achieves excellent convergence to the optimal results. Under noisy conditions, the QITE method exhibits greater robustness and stability, though it incurs substantially more classical numerical cost. In contrast, we demonstrate that QAOA offers better scalability and can still yield robust results if the noise can be effectively mitigated. Our findings provide valuable insights into the trade-offs between scalability and noise tolerance and demonstrate the practical potential of quantum algorithms for solving real-world optimization problems on near-term quantum devices.
In this work, we benchmark two prominent quantum algorithms: Quantum Imaginary-Time Evolution (QITE) and the Quantum Approximate Optimization Algorithm (QAOA) for obtaining the ground state of Ising-type Hamiltonians. Specifically, we apply them to the Markowitz portfolio optimization problem in quantitative finance, on both digital quantum computers and local quantum simulators with controllable two-qubit errors (noise). In noiseless settings, we find that QAOA achieves excellent convergence to the optimal results. Under noisy conditions, the QITE method exhibits greater robustness and stability, though it incurs substantially more classical numerical cost. In contrast, we demonstrate that QAOA offers better scalability and can still yield robust results if the noise can be effectively mitigated. Our findings provide valuable insights into the trade-offs between scalability and noise tolerance and demonstrate the practical potential of quantum algorithms for solving real-world optimization problems on near-term quantum devices.
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.
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.
Quantum computing is no longer in its early stages. There already exists quantum computers with more qubits than a classical computer is capable of efficiently simulating. This current stage is considered intermediate and is therefore called the NISQ era (noisy intermediate-scale quantum). The main feature of this current stage is that there are still not enough qubits to perform quantum error correction, hence the noisy name. In this context of quantum computing without quantum error correction and with an intermediate number of qubits, variational algorithms gained prominence and, among them, there is one called QAOA (quantum approximate optimization algorithm). As the name suggests, this is a quantum algorithm that approximates the solution of optimization problems. The objective of this work was to apply this algorithm to solve an optimization problem in the finance area known as portfolio optimization. This application took place both in an ideal way (without noise) and in a way consistent with the current capacity of quantum computers (with noise). Both were simulated using IBM’s Python tool for simulation and access of quantum computers via cloud called Qiskit. The results suggest that the QAOA performance with noise was, as expected, worse than the ideal case, but still satisfactory within the limitations of the method.
Quantum annealing has emerged as a promising computational paradigm for solving large-scale combinatorial optimization problems that are intractable for classical algorithms. Financial portfolio optimization, which involves balancing expected returns against risk under multiple practical constraints, represents one of the most challenging optimization tasks in quantitative finance. Traditional approaches based on mean-variance theory, heuristic meta-optimization, and classical quadratic programming often struggle to scale efficiently as the dimensionality of assets and constraints increases. This study explores the application of quantum annealing techniques to financial portfolio optimization by formulating the problem as a quadratic unconstrained binary optimization model suitable for quantum hardware. The paper provides a systematic mapping of portfolio objectives and constraints into the Ising Hamiltonian framework and analyzes how quantum tunneling and probabilistic state transitions can enhance convergence toward near-global optima. Comparative insights between classical solvers and quantum annealing-based approaches are discussed in terms of solution quality, computational efficiency, and scalability. Furthermore, the study highlights current hardware limitations, embedding challenges, and noise effects that influence optimization performance. The findings indicate that quantum annealing offers substantial potential for accelerating portfolio optimization in high-dimensional financial systems, particularly in scenarios involving complex constraints and dynamic market conditions, thereby establishing a foundation for future quantum-driven financial decision-support systems.
The Quantum Approximate Optimization Algorithm (QAOA) is a leading candidate for solving combinatorial optimization problems on near-term quantum hardware. However, its practical deployment remains severely limited by hardware-induced noise and decoherence. In the absence of full fault tolerance, Quantum Error Mitigation (QEM) techniques offer a viable approach to enhance QAOA performance on noisy intermediate-scale quantum (NISQ) devices. In this work, we present a unified classification of QEM strategies specifically tailored for QAOA, systematically categorized into preprocessing, during execution, and post-processing phases. We further classify QAOA deployment settings—such as hardware-constrained devices, stochastic noise-dominated environments, and dense graphs problem—and recommend appropriate QEM techniques for each. A comparative analysis highlights the trade-offs among mitigation accuracy, resource overhead, and robustness to different noise types. By synthesizing recent advances and identifying open challenges, our paper provides both a comprehensive survey and a practical framework for designing robust, noise-resilient QAOA pipelines on near-term quantum platforms.