# Multiclass portfolio optimization via variational quantum Eigensolver with Dicke state ansatz

## TL;DR

This paper introduces a quantum computing method for portfolio optimization that better handles diversification constraints.

## Contribution

A novel quantum framework for multiclass portfolio optimization using Dicke states in a variational quantum eigensolver.

## Key findings

- The Dicke state ansatz initializes only feasible states, reducing search space and eliminating penalty terms.
- Combining the Dicke state ansatz with the CMA-ES optimizer improves convergence rate and approximation ratio.
- The method shows potential for practical diversification-aware portfolio optimization in finance.

## 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.