A Quantum Model for Constrained Markowitz Modern Portfolio Using Slack Variables to Process Mixed-Binary Optimization under QAOA

TL;DR

This paper introduces a quantum approach for portfolio optimization that embeds constraints directly into the Hamiltonian using slack variables, improving solution accuracy over traditional penalty methods.

Contribution

It presents a novel slack-ancilla scheme for encoding inequality constraints in quantum optimization, enabling more effective portfolio solutions with QAOA.

Findings

The slack-ancilla method outperforms penalty-based QAOA in simulations.

The approach successfully finds optimal portfolios under constraints.

A quantum limit on precision for risk and return is proposed.

Abstract

Effectively encoding inequality constraints is a primary obstacle in applying quantum algorithms to financial optimization. A quantum model for Markowitz portfolio optimization is presented that resolves this by embedding slack variables directly into the problem Hamiltonian. The method maps each slack variable to a dedicated ancilla qubit, transforming the problem into a Quadratic Unconstrained Binary Optimization (QUBO) formulation suitable for the Quantum Approximate Optimization Algorithm (QAOA). This process internalizes the constraints within the quantum state, altering the problem's energy landscape to facilitate optimization. The model is empirically validated through simulation, showing it consistently finds the optimal portfolio where a standard penalty-based QAOA fails. This work demonstrates that modifying the Hamiltonian architecture via a slack-ancilla scheme provides a robust and effective pathway for solving constrained optimization problems on quantum computers. A fundamental quantum limit on the simultaneous precision of portfolio risk and return is also posited.