Multiclass Portfolio Optimization via Variational Quantum Eigensolver with Dicke State Ansatz

TL;DR

The paper addresses diversification-aware multiclass portfolio optimization by embedding diversification constraints directly into a parameterized Dicke-state ansatz within the Variational Quantum Eigensolver, eliminating penalty terms and constraining the search to the feasible subspace. By tensoring multiple Dicke states across asset classes, the method reduces the effective search space from $2^n$ to $igotimes_i inom{n_i}{k_i}$, enabling more stable convergence. Empirical results across several scenarios show the Dicke-state ansatz outperforms standard ansatzes in approximation ratio and ground-state probability, with CMA-ES consistently providing the strongest classical optimization signal. The work demonstrates a practical, hardware-agnostic framework for diversification-aware quantum portfolio optimization, while acknowledging that near-term hardware noise and the absence of proven quantum speedups limit immediate real-world deployment; it points to future directions including QAOA variants with Dicke states and CVaR-based objectives.

Abstract

Combinatorial optimization is a fundamental challenge in various domains, with portfolio optimization standing out as a key application in finance. Despite numerous quantum algorithmic approaches proposed for this problem, most overlook a critical feature of realistic portfolios: diversification. In this work, we introduce a novel quantum framework for multiclass portfolio optimization that explicitly incorporates diversification by leveraging multiple parametrized Dicke states, simultaneously initialized to encode the diversification constraints , as an ansatz of the Variational Quantum Eigensolver. A key strength of this ansatz is that it initializes the quantum system in a superposition of only feasible states, inherently satisfying the constraints. This significantly reduces the search space and eliminates the need for penalty terms. In addition, we also analyze the impact of different classical optimizers in this hybrid quantum-classical approach. Our findings demonstrate that, when combined with the CMA-ES optimizer, the Dicke state ansatz achieves superior performance in terms of convergence rate, approximation ratio, and measurement probability. These results underscore the potential of this method to solve practical, diversification-aware portfolio optimization problems relevant to the financial sector.

Figures (11)

• Figure 1: Illustrative example of a multiclass portfolio optimization problem with a predefined asset class allocation to ensure diversification aiming the reduction of market risk. The total number of assets considered is $820$, distributed as follows: $500$ stocks, $200$ cryptocurrencies, $30$ commodities, $10$ ETFs, $60$ REITs and $20$ Bonds. For each class there are $C_{n,k}$ possible portfolios, where $n$ represents the total number of assets in the class and $k$ is the predefined number of assets to be selected. The goal of this portfolio optimization is to find the best set of assets that will produce a portfolio that satisfies the constraints maximizing the return and minimizing the risk.
• Figure 2: An illustrative example of a VQE routine, which is defined by a quantum state preparation conducted on a quantum device or simulator, followed by an optimization process in a classical computer. In the quantum routine, we start with a quantum state where all qubits are in state $\vert0\rangle$, this initial state evolves according to the unitary $U(\vec{\theta_i})$, which defines ansatz structure and receives a set of parameters that will define the states probability distribution extracted from a certain number of measurements. The classical routine is focused on updating the set of parameters $\vec{\theta_i}$, in order to minimize the expectation value of the Hamiltonian that encodes the optimization problem. This process is repeated until the maximum number of iterations or when other stopping criteria are achieved.
• Figure 3: Schematic representation of the ansatzes explored in this work: $(a)$ and $(b)$ depict the Dicke State, while $(c)$ and $(d)$ illustrate the $R_y$ and Two Local ansatz, respectively. In this example, the portfolio consists of $5$ assets categorized into $2$ classes—$3$ bonds and $2$ stocks. The optimization goal is to select $2$ bonds and $1$ stock to maximize returns and minimize risks. For a consistent comparison, the different ansatzes are configured to have a comparable number of parameters, with the parameterized Dicke state serving as the reference.
• Figure 4: Ansatz comparison results for Scenario I taking into account all optimizers. The histogram shows the number of trials where each ansatz found the optimal solution as its most common output. Notably, out of the $20$ ansatzes tested, the Dicke state demonstrated the best performance, independently of the classical optimizer. Out of $500$ runs, in more than $50\%$, the VQE-Dicke state found the optimal result as the one with the highest probability. The inside table shows the absolute error $(\vert\vert Err\vert\vert)$ and the standard deviation $(\sigma)$ between the expected value of the quantum solution and the target. Again, the VQE-Dicke state presented the best metrics.
• Figure 5: Plots $(a)$, $(b)$, and $(c)$ compare the approximation ratio ($a_r$) distributions before (gray) and after (colored) optimization for each scenario (columns) and optimizers (rows). The gray represent $a_r$ distributions with the initial parameters, while the colored ones reflect optimized parameters, based on $100$ experiments per optimizer. A vertical black dashed line marks $a_r = 0.9$. Across all optimizers and scenarios, the initial distributions shift toward $a_r \geq 0.9$, indicating the positive impact of the hybrid approach and optimizers.In Scenario I (blue) all the optimizers guide the solution to the optimal region.