We employ a variational neural annealing approach for solving large-scale constrained portfolio optimization problems, mapped onto classical Ising-like Hamiltonians, and study its performance under finite annealing times. The method enables efficient exploration of high-dimensional feasible spaces and achieves competitive solutions on standard financial indices. To characterize the underlying optimization dynamics, we analyze the decay of the residual energy as a function of annealing steps and observe a power-law behavior for fixed problem size. Extending this analysis across different portfolio sizes, we examine whether the optimization dynamics are consistent with a dynamical finite-size scaling form and find that the residual energy data collapse onto a single scaling curve within the studied range. This scaling analysis yields an effective annealing-time exponent that quantifies how computational effort grows with problem size. Our results demonstrate that finite-size scaling concepts provide a useful framework for understanding and forecasting the performance of neural annealing-based optimization methods in realistic, constrained settings.
We employ a variational neural annealing approach for solving large-scale constrained portfolio optimization problems, mapped onto classical Ising-like Hamiltonians, and study its performance under finite annealing times. The method enables efficient exploration of high-dimensional feasible spaces and achieves competitive solutions on standard financial indices. To characterize the underlying optimization dynamics, we analyze the decay of the residual energy as a function of annealing steps and observe a power-law behavior for fixed problem size. Extending this analysis across different portfolio sizes, we examine whether the optimization dynamics are consistent with a dynamical finite-size scaling form and find that the residual energy data collapse onto a single scaling curve within the studied range. This scaling analysis yields an effective annealing-time exponent that quantifies how computational effort grows with problem size. Our results demonstrate that finite-size scaling concepts provide a useful framework for understanding and forecasting the performance of neural annealing-based optimization methods in realistic, constrained settings.
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.
Portfolio optimization is a cornerstone of financial decision-making, traditionally relying on classical algorithms to balance risk and return. Recent advances in quantum computing offer a promising alternative, leveraging quantum algorithms to efficiently explore complex solution spaces and potentially outperform classical methods in high-dimensional settings. However, conventional quantum approaches typically assume a one-to-one correspondence between qubits and variables (e.g. financial assets), which severely limits the applicability of gate-based quantum systems due to current hardware constraints. As a result, only quantum annealing-like methods have been used in realistic scenarios. In this work, we show how a gate-based variational quantum algorithm can be applied to a real-world portfolio optimization problem by assigning multiple variables per qubit. Specifically, we address a problem involving over 250 variables, where the market graph representing a real stock market is iteratively partitioned into sub-portfolios of highly correlated assets. This approach enables improved scalability compared to traditional variational methods and opens new possibilities for quantum-enhanced financial applications.
Portfolio optimization is an important process in finance that consists in finding the optimal asset allocation that maximizes expected returns while minimizing risk. When assets are allocated in discrete units, this is a combinatorial optimization problem that can be addressed by quantum and quantum-inspired algorithms. In this work we present an integer simulated annealing method to find optimal portfolios in the presence of discretized convex and non-convex cost functions. Our algorithm can deal with large size portfolios with hundreds of assets. We introduce a performance metric, the time to target, based on a lower bound to the cost function obtained with the continuous relaxation of the combinatorial optimization problem. This metric allows us to quantify the time required to achieve a solution with a given quality. We carry out numerical experiments and we benchmark the algorithm in two situations: (i) Monte Carlo instances are started at random, and (ii) the algorithm is warm-started with an initial instance close to the continuous relaxation of the problem. We find that in the case of warm-starting with convex cost functions, the time to target does not grow with the size of the optimization problem, so discretized versions of convex portfolio optimization problems are not hard to solve using classical resources. We have applied our method to the problem of re-balancing in the presence of non-convex transaction costs, and we have found that our algorithm can efficiently minimize those terms.
Portfolio optimization (PO) is extensively employed in financial services to assist in achieving investment objectives. By providing an optimal asset allocation, PO effectively balances the risk and returns associated with investments. However, it is important to note that as the number of involved assets and constraints increases, the portfolio optimization problem can become increasingly difficult to solve, falling into the category of NP-hard problems. In such scenarios, classical algorithms, such as the Monte Carlo method, exhibit limitations in addressing this challenge when the number of stocks in the portfolio grows. Quantum annealing algorithm holds promise for solving complex portfolio optimization problems in the NISQ era. Many studies have demonstrated the advantages of various quantum annealing algorithm variations over the standard quantum annealing approach. In this work, we conduct a numerical investigation of randomly generated unconstrained single-period discrete mean-variance portfolio optimization instances. We explore the application of a variety of unconventional quantum annealing algorithms, employing both forward annealing and reverse annealing schedules. By comparing the time-to-solution(TTS) and success probabilities of diverse approaches, we show that certain methods exhibit advantages in enhancing the success probability when utilizing conventional forward annealing schedules. Furthermore, we find that the implementation of reverse annealing schedules can significantly improve the performance of select unconventional quantum annealing algorithms.
This paper investigates the application of quantum annealing to address the challenges of combinatorial optimization within large-scale network routing. Traditional classical algorithms frequently encounter significant computational bottlenecks when attempting to determine optimal routes across complex networks, often failing to scale effectively with increasing network size and complexity. Quantum annealing offers a potentially transformative approach by leveraging quantum mechanical effects to explore solution spaces more efficiently. This work outlines a methodology for mapping network routing problems – characterized by a vast number of possible routes – onto the quantum annealing hardware. We present a model for representing the problem, formulating it as an Ising model, and then detail the annealing process. The core claim is that quantum annealing can provide a viable solution for optimizing network routes, particularly in scenarios where classical methods are computationally intractable. We explore the underlying principles of quantum annealing, focusing on its ability to escape local minima and achieve a lower energy state, representing the optimal solution. The research aims to demonstrate the feasibility and potential advantages of employing quantum annealing in practical network routing applications, paving the way for more efficient and robust network designs.
Portfolio optimization is a well-established problem in the field of finance and has recently gained attention in the context of quantum computing. The hardware limitations of quantum computers prevent the direct application of quantum algorithms to large-scale problems. We propose a novel two-stage approach that combines the strengths of both quantum annealing and gate-based quantum computing to efficiently solve large-scale portfolio optimization problems. Building upon the existing Large System Sampling Approximation (LSSA) approach, our work introduces a novel modification to the decomposition and aggregation steps of LSSA. We divide the problem into sub-problems of smaller sizes by solving the Maximum Independent Set (MIS) of the market graph and use a Parameterized Quantum Circuit (PQC) to aggregate the sub-problem solutions. We conduct extensive experiments on real-world data from the US stock market on up to 128 assets on simulators. Our results demonstrate that the proposed approach performs better with the same hardware resources. The outcomes of our research suggest that hybrid annealer-gate quantum computing can provide a practical and scalable solution to large-scale portfolio optimization problems, bridging the gap between theoretical advancements in quantum computing and real-world applications in finance.