Portfolio optimization is an essential use case in Finance, but its computational complexity forces financial institutions to resort to approximated solutions, which are still time consuming. Thus, the scientific community is looking at how Quantum Computing can be used for efficient and accurate portfolio optimization.
Portfolio optimization can be formulated as a quadratic program, with the cost function enforcing risk minimization for a targeted return. Of particular interest is the mean-variance portfolio optimization problem.
Using the method of Lagrange multipliers, the program can be converted into a system of linear equations and potentially benefit from the exponential speedup provided by the HHL quantum algorithm. However, multiple components in HHL are unsuitable for execution on Noisy Intermediate Scale Quantum (NISQ) hardware.
This paper introduces NISQ-HHL, the first hybrid formulation of HHL suitable for the end-to-end execution of small-scale portfolio-optimization problems on NISQ devices. NISQ-HHL extends the hybrid HHL variant with newly available quantum-hardware features: mid-circuit measurement, Quantum Conditional Logic (QCL), and qubit reset and reuse. To best of our knowledge, NISQ-HHL is the first algorithm incorporating a QCL-enhanced version of Phase Estimation that was executed on real hardware. In addition, NISQ-HHL includes a novel method for choosing the optimal evolution time for the Hamiltonian simulation. Although this paper focuses on portfolio optimization, the techniques it proposes to make HHL more scalable are generally applicable to any problem that can be solved via HHL in the NISQ era.
We empirically demonstrate the effectiveness of NISQ-HHL by presenting the experimental results we obtained on a real quantum device, the trapped-ion Honeywell System Model H1.
Portfolio optimization is an essential use case in Finance, but its computational complexity forces financial institutions to resort to approximated solutions, which are still time consuming. Thus, the scientific community is looking at how Quantum Computing can be used for efficient and accurate portfolio optimization.
Portfolio optimization can be formulated as a quadratic program, with the cost function enforcing risk minimization for a targeted return. Of particular interest is the mean-variance portfolio optimization problem.
Using the method of Lagrange multipliers, the program can be converted into a system of linear equations and potentially benefit from the exponential speedup provided by the HHL quantum algorithm. However, multiple components in HHL are unsuitable for execution on Noisy Intermediate Scale Quantum (NISQ) hardware.
This paper introduces NISQ-HHL, the first hybrid formulation of HHL suitable for the end-to-end execution of small-scale portfolio-optimization problems on NISQ devices. NISQ-HHL extends the hybrid HHL variant with newly available quantum-hardware features: mid-circuit measurement, Quantum Conditional Logic (QCL), and qubit reset and reuse. To best of our knowledge, NISQ-HHL is the first algorithm incorporating a QCL-enhanced version of Phase Estimation that was executed on real hardware. In addition, NISQ-HHL includes a novel method for choosing the optimal evolution time for the Hamiltonian simulation. Although this paper focuses on portfolio optimization, the techniques it proposes to make HHL more scalable are generally applicable to any problem that can be solved via HHL in the NISQ era.
We empirically demonstrate the effectiveness of NISQ-HHL by presenting the experimental results we obtained on a real quantum device, the trapped-ion Honeywell System Model H1.
The limited capabilities of current quantum hardware significantly constrain the scale of experimental demonstrations of most quantum algorithmic primitives. This makes it challenging to perform benchmarking of the current hardware using useful quantum algorithms, i.e., application-oriented benchmarking. In particular, the Harrow-Hassidim-Lloyd (HHL) algorithm is a critical quantum linear algebra primitive, but the majority of the components of HHL are far out of the reach of noisy intermediate-scale quantum devices, which has led to the proposal of hybrid classical-quantum variants. The goal of this work is to further bridge the gap between proposed near-term friendly implementations of HHL and the kinds of quantum circuits that can be executed on noisy hardware. Our proposal adds to the existing literature of hybrid quantum algorithms for linear algebra that are more compatible with the current scale of quantum devices. Specifically, we propose two modifications to the Hybrid HHL algorithm proposed by Lee et al., leading to our algorithm Hybrid HHL++ : (1) propose a novel algorithm for determining a scaling factor for the linear system matrix that maximizes the utility of the amount of ancillary qubits allocated to the phase estimation component of HHL, and (2) introduce a heuristic for compressing the HHL circuit. We demonstrate the efficacy of our work by running our modified Hybrid HHL on Quantinuum System Model H-series trapped-ion quantum computers to solve different problem instances of small-scale portfolio optimization problems, leading to the largest experimental demonstrations of HHL for an application to date.
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.
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.
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 under conditions of risk and market uncertainty remains a foundational challenge in financial economics and investment management. Classical optimization frameworks, including mean–variance models and stochastic programming approaches, often struggle with high-dimensional asset universes, nonlinear correlations, and dynamic market regimes. This paper examines quantum computing approaches to portfolio optimization under risk and market uncertainty, focusing on hybrid quantum–classical algorithms designed to enhance combinatorial optimization and probabilistic modeling. By integrating quantum annealing, variational quantum circuits, and quantum-enhanced sampling techniques with classical financial analytics, the study evaluates their potential to improve computational efficiency and scenario sensitivity. The analysis considers both theoretical advantages and practical limitations within the Noisy Intermediate-Scale Quantum (NISQ) era. The findings suggest that quantum-assisted optimization may complement traditional portfolio models by expanding feasible solution spaces and improving robustness under uncertainty, thereby contributing to more adaptive and resilient investment strategies.
Portfolio optimization under realistic constraints represents a computationally intensive problem for classical financial models, particularly in volatile and high-dimensional markets such as India.Conventional mean-variance approaches become NP-hard when integer, cardinality, and regulatory constraints are incorporated.This paper investigates the applicability of quantum computing techniques for constrained portfolio optimization in the Indian financial context.The study analyzes quantum annealing and gate-based variational algorithms, including the Quantum Approximate Optimization Algorithm (QAOA) and Variational Quantum Eigensolver (VQE), with emphasis on Quadratic Unconstrained Binary Optimization (QUBO) formulations.In addition, quantum amplitude estimation methods are examined for accelerating financial risk metrics such as Value-at-Risk and Conditional Value-at-Risk.Empirical results derived from thematic and regression analyses indicate that perceived computational advantage and organizational readiness significantly influence adoption intent, while quantum-related security risks act as a moderating factor.The findings suggest that hybrid classical-quantum architectures provide a practical pathway during the Noisy Intermediate-Scale Quantum era.The paper concludes that early-stage quantum adoption, combined with post-quantum cryptographic preparedness, can enhance portfolio efficiency and risk governance for Indian institutional investors