Abstract In this note, we describe an experiment on portfolio optimization using the Quadratic Unconstrained Binary Optimization (QUBO) formulation. The dataset we use is taken from a real-world problem for which a classical solution is currently deployed and used in production. In this work, carried out in a collaboration between the Raiffeisen Bank International (RBI) and Reply, we derive a QUBO formulation, which we solve using various methods: two D-Wave hybrid solvers, that combine the employment of a quantum annealer together with classical methods, and a purely classical algorithm. Particular focus is given to the implementation of the constraint that requires the resulting portfolio’s variance to be below a specified threshold, whose representation in an Ising model is not straightforward. We find satisfactory results, consistent with the global optimum obtained by the exact classical strategy. However, since the tuning of QUBO parameters is crucial for the optimization, we investigate a hybrid method that allows for automatic tuning.
Abstract In this note, we describe an experiment on portfolio optimization using the Quadratic Unconstrained Binary Optimization (QUBO) formulation. The dataset we use is taken from a real-world problem for which a classical solution is currently deployed and used in production. In this work, carried out in a collaboration between the Raiffeisen Bank International (RBI) and Reply, we derive a QUBO formulation, which we solve using various methods: two D-Wave hybrid solvers, that combine the employment of a quantum annealer together with classical methods, and a purely classical algorithm. Particular focus is given to the implementation of the constraint that requires the resulting portfolio’s variance to be below a specified threshold, whose representation in an Ising model is not straightforward. We find satisfactory results, consistent with the global optimum obtained by the exact classical strategy. However, since the tuning of QUBO parameters is crucial for the optimization, we investigate a hybrid method that allows for automatic tuning.
In this note, we describe an experiment on portfolio optimization using the Quadratic Unconstrained Binary Optimization (QUBO) formulation. The dataset we use is taken from a real-world problem for which a classical solution is currently deployed and used in production. In this work, carried out in a collaboration between the Raiffeisen Bank International (RBI) and Reply, we derive a QUBO formulation, which we solve using various methods: two D-Wave hybrid solvers, that combine the employment of a quantum annealer together with classical methods, and a purely classical algorithm. Particular focus is given to the implementation of the constraint that requires the resulting portfolio’s variance to be below a specified threshold, whose representation in an Ising model is not straightforward. We find satisfactory results, consistent with the global optimum obtained by the exact classical strategy. However, since the tuning of QUBO parameters is crucial for the optimization, we investigate a hybrid method that allows for automatic tuning.
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 this work, we introduce a new workflow to solve portfolio optimization problems on annealing platforms. We combine a classical preprocessing step with a modified unconstrained binary optimization (QUBO) model and evaluate it using simulated annealing (classical computer), digital annealing (Fujitsu’s Digital Annealing Unit), and quantum annealing (D-Wave Advantage). Starting from Markowitz’s theory on portfolio optimization, our classical preprocessing step finds the most promising assets within a set of possible assets to choose from. We then modify existing QUBO models for portfolio optimization, such that there are no limitations on the number of assets that can be invested in. Furthermore, our QUBO model enables an investor to also place an arbitrary amount of money into each asset. We apply this modified QUBO to the set of promising asset candidates we generated previously via classical preprocessing. A solution to our QUBO model contains information about what percentage of the whole available capital should be invested into which asset. For the evaluation, we have used publicly available real-world data sets of stocks of the New York Stock Exchange as well as common ETFs. Finally, we have compared the respective annealing results with randomly generated portfolios by using the return, variance, and diversification of the created portfolios as measures. The results show that our QUBO formulation is capable of creating well-diversified portfolios that respect certain criteria given by an investor, such as maximizing return, minimizing risk, or sticking to a certain budget.
Recent advances in quantum hardware offer new approaches to solve various optimization problems that can be computationally expensive when classical algorithms are employed. We propose a hybrid quantum-classical algorithm to solve a dynamic asset allocation problem where a target return and a target risk metric (expected shortfall) are specified. We propose an iterative algorithm that treats the target return as a constraint in a Markowitz portfolio optimization model, and dynamically adjusts the target return to satisfy the targeted expected shortfall. The Markowitz optimization is formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem. The use of the expected shortfall risk metric enables the modeling of extreme market events. We compare the results from D-Wave's 2000Q and Advantage quantum annealers using real-world financial data. Both quantum annealers are able to generate portfolios with more than 80% of the return of the classical optimal solutions, while satisfying the expected shortfall. We observe that experiments on assets with higher correlations tend to perform better, which may help to design practical quantum applications in the near term.
We test whether quantum-inspired optimization, specifically Quadratic Unconstrained Binary Optimization (QUBO) solved via simulated annealing, can improve implementable portfolio performance compared to classical baselines when realistic frictions are included. Our experimental design uses daily returns from the Center for Research in Security Prices (CRSP) for S&P500 constituents spanning 2014–2025, with monthly rebalancing, explicit transaction costs, and turnover accounting. We compare QUBO portfolios against equal-weight and convex mean–variance portfolios across multiple universe sizes (20, 50, 67 assets), lookback windows, and transaction-cost levels. Statistical significance is assessed via paired moving-block bootstrap on out-of-sample net returns. A hyperparameter sweep over encoding depth, penalty coefficients, and turnover reveals 19 of 81 QUBO configurations beat the equal-weight baseline, with the best achieving a Sharpe of 1.59 versus 1.42. However, under a fair walk forward model-selection protocol that eliminates look-ahead bias, the QUBO strategy does not achieve statistically significant superiority over either classical benchmark (∆Sharpe vs. equal weight: −0.13, 95% CI [−0.46,0.16]; vs. mean–variance: −0.29, 95% CI [−0.63,0.06]). Excessive turnover (averaging ∼1.0 per rebalance) is identified as the primary bottleneck. We conclude that current QUBO formulations with simulated annealing do not yet deliver a robust, implementable edge over well-tuned convex optimizers, though the framework remains a promising research direction as quantum hardware and hybrid solvers mature.
This paper investigates and compares quantum and classical investment strategies for portfolio construction under realistic trading and allocation constraints. The study considers a multi-period portfolio re-balancing setting in which asset weights are subject to lower and upper bounds, class-level exposure restrictions, and turnover limitations. On the quantum side, the portfolio selection problem is reformulated as a quadratic unconstrained binary optimization (QUBO) model through a finite binary encoding of asset weights, and subsequently embedded into a constrained quadratic model (CQM) solved by a hybrid quantum-classical optimization workflow. On the classical side, benchmark allocation strategies are introduced to provide an economically meaningful reference for performance assessment. The proposed framework jointly evaluates portfolio efficiency from two complementary perspectives: business performance, measured through return, volatility, and a measure of the trade-off between them, the Sharpe ratio; and computational performance, assessed through solver structure, constraint handling, and hardware execution characteristics. The resulting comparison provides a rigorous basis for understanding the practical role of quantum annealing in constrained portfolio optimization, while clarifying both its modeling advantages and its current implementation limitations relative to established classical approaches.