Loan portfolio optimization (LPO) seeks portfolios that minimize credit risk while satisfying practical selection constraints. Accounting for portfolio-level risk requires modeling not only expected losses but also loss variability and borrower-default correlations, leading to a large-scale combinatorial optimization problem whose complexity grows rapidly with portfolio size. In this work, we formulate LPO as a Quadratic Constrained Binary Optimization (QCBO) problem that jointly captures expected and unexpected losses and transform the resulting formulation into a Quadratic Unconstrained Binary Optimization (QUBO) model suitable for variational quantum optimization. To address the limited qubit resources of current quantum hardware, we employ Pauli Correlation Encoding (PCE), which enables a large number of portfolio-selection variables to be represented using comparatively few qubits. The proposed framework is evaluated on a real-world microfinance dataset comprising 2012 borrowers. For problem instances with up to 1000 portfolio-selection variables, we benchmark classical simulations of a variational quantum algorithm (VQA) against Google OR-Tools under matched time budgets, obtaining feasible portfolios with an objective-value gap of about 40% across all sizes. We further implement the framework on superconducting quantum hardware for instances containing up to 1500 variables. Across these instances, the experimentally obtained objective values are within 10% of the corresponding simulation results. These findings demonstrate the feasibility of applying PCE-based variational quantum optimization to large-scale loan portfolio optimization under current quantum hardware limitations.
Loan portfolio optimization (LPO) seeks portfolios that minimize credit risk while satisfying practical selection constraints. Accounting for portfolio-level risk requires modeling not only expected losses but also loss variability and borrower-default correlations, leading to a large-scale combinatorial optimization problem whose complexity grows rapidly with portfolio size. In this work, we formulate LPO as a Quadratic Constrained Binary Optimization (QCBO) problem that jointly captures expected and unexpected losses and transform the resulting formulation into a Quadratic Unconstrained Binary Optimization (QUBO) model suitable for variational quantum optimization. To address the limited qubit resources of current quantum hardware, we employ Pauli Correlation Encoding (PCE), which enables a large number of portfolio-selection variables to be represented using comparatively few qubits. The proposed framework is evaluated on a real-world microfinance dataset comprising 2012 borrowers. For problem instances with up to 1000 portfolio-selection variables, we benchmark classical simulations of a variational quantum algorithm (VQA) against Google OR-Tools under matched time budgets, obtaining feasible portfolios with an objective-value gap of about 40% across all sizes. We further implement the framework on superconducting quantum hardware for instances containing up to 1500 variables. Across these instances, the experimentally obtained objective values are within 10% of the corresponding simulation results. These findings demonstrate the feasibility of applying PCE-based variational quantum optimization to large-scale loan portfolio optimization under current quantum hardware limitations.
Variational quantum algorithms offer a promising route to combinatorial optimization, but their applicability is limited by the challenge of encoding large-scale problems within restricted qubit resources. In this work, we introduce a qubit-efficient variational framework based on Pauli correlation encoding (PCE) and apply it to electric power demand portfolio optimization. Binary variables are represented through expectation values of Pauli correlation operators, which encode multi-body correlations of the quantum state and provide a continuous relaxation enabling compact representations with few qubits. We further propose a two-stage hybrid formulation, in which a time-averaged problem provides initialization for a time-resolved optimization. Numerical simulations demonstrate near-optimal performance across problem sizes ranging from m=18 to 10,296, with normalized cost gaps on the order of 10−4 relative to solutions with certified optimality. We show that the performance is governed by the interplay between continuous relaxation and discretization: the effective resolution of the correlator representation determines how reliably improvements in the continuous loss translate into better discrete solutions, with larger systems exhibiting more consistent behavior. Finally, we demonstrate robustness on a trapped-ion quantum processor, where high-quality solutions are obtained despite noise and finite sampling. These results establish PCE as a physically motivated and qubit-efficient framework for large-scale combinatorial optimization.
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, 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.
We present a scalable, hardware-aware methodology for extending the Variational Quantum Eigensolver (VQE) to large, realistic Dynamic Portfolio Optimization (DPO) problems. Building on the scaling strategy from our previous work, where we tailored a VQE workflow to both the DPO formulation and the target QPU, we now put forward two significant advances. The first is the implementation of the Ising Sample-based Quantum Configuration Recovery (ISQR) routine, which improves solution quality in Quadratic Unconstrained Binary Optimization problems. The second is the use of the VQE Constrained method to decompose the optimization task, enabling us to handle DPO instances with more variables than the available qubits on current hardware. These advances, which are broadly applicable to other optimization problems, allow us to address a portfolio with a size relevant to the financial industry, consisting of up to 38 assets and covering the full Spanish stock index (IBEX 35). Our results, obtained on a real Quantum Processing Unit (IBM Fez), show that this tailored workflow achieves financial performance on par with classical methods while delivering a broader set of high-quality investment strategies, demonstrating a viable path towards obtaining practical advantage from quantum optimization in real financial applications.
In this study, the portfolio optimization problem is explored, using a combination of classical and quantum computing techniques. The portfolio optimization problem with specific objectives or constraints is often a quadratic optimization problem, due to the quadratic nature of, for example, risk measures. Quantum computing is a promising solution for quadratic optimization problems, as it can leverage quantum annealing and quantum approximate optimization algorithms, which are expected to tackle these problems more efficiently. Quantum computing takes advantage of quantum phenomena like superposition and entanglement. In this paper, a specific problem is introduced, where a portfolio of loans need to be optimized for 2030, considering ‘Return on Capital’ and ‘Concentration Risk’ objectives, as well as a carbon footprint constraint. This paper introduces the formulation of the problem and how it can be optimized using quantum computing, using a reformulation of the problem as a quadratic unconstrained binary optimization (QUBO) problem. Two QUBO formulations are presented, each addressing different aspects of the problem. The QUBO formulation succeeded in finding solutions that met the emission constraint, although classical simulated annealing still outperformed quantum annealing in solving this QUBO, in terms of solutions close to the Pareto frontier. Overall, this paper provides insights into how quantum computing can address complex optimization problems in the financial sector. It also highlights the potential of quantum computing for providing more efficient and robust solutions for portfolio management.
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.