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

Pablo Thomassin, Guillaume Guerard, Sonia Djebali, Vincent Marc Lambert

TL;DR

The paper addresses the challenge of encoding inequality constraints in quantum portfolio optimization and proposes embedding slack variables as ancilla qubits within the problem Hamiltonian to yield a QUBO form for QAOA. It demonstrates, via simulation, that the slack-ancilla QAOA consistently finds the optimal portfolio where standard penalty-based QAOA fails, and it discusses a fundamental quantum bound on joint risk and return. The contributions include an architectural Hamiltonian redesign that internalizes constraints, a reduced need for high penalty weights, and a feasible demonstration on a small three-asset test. The work highlights a practical route to applying quantum optimization to constrained financial problems and suggests a quantum-limited trade-off between risk and return.

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.

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

TL;DR

The paper addresses the challenge of encoding inequality constraints in quantum portfolio optimization and proposes embedding slack variables as ancilla qubits within the problem Hamiltonian to yield a QUBO form for QAOA. It demonstrates, via simulation, that the slack-ancilla QAOA consistently finds the optimal portfolio where standard penalty-based QAOA fails, and it discusses a fundamental quantum bound on joint risk and return. The contributions include an architectural Hamiltonian redesign that internalizes constraints, a reduced need for high penalty weights, and a feasible demonstration on a small three-asset test. The work highlights a practical route to applying quantum optimization to constrained financial problems and suggests a quantum-limited trade-off between risk and return.

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.
Paper Structure (22 sections, 3 theorems, 26 equations, 3 figures, 2 tables)

This paper contains 22 sections, 3 theorems, 26 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Literature Review and Context
  3. Background on Quantum Computing in Finance
  4. Markowitz Portfolio Theory
  5. Integration of Quantum Computing with Markowitz Portfolio Theory
  6. Context of the proposed method
  7. Methodology
  8. Classical Model
  9. Quantum Model
  10. Hamiltonian Computing
  11. Conjecture
  12. Motivation and Scope of the Risk–Return Variance Bound
  13. Methodology Summary and Novel Contributions
  14. Experimentation
  15. Test-Instance Specification
  16. ...and 7 more sections

Key Result

Theorem 1

Let $\omega \in \mathbb{R}^N_+$, where $N \in \mathbb{N}^*$, represent the portfolio vector where $\forall i \in \mathbb{N}^*$, $\omega_i \in \mathbb{R}_+$ denotes the allocation for asset $i$. Let $\Sigma$ be the covariance matrix and $\mu$ the mean vector of returns for each asset. Introduce $\lam subject to:

Figures (3)

  • Figure 1: The variational circuit structure configured for a depth of $p=2$ and 3 qubits. The Rz gates encode the problem Hamiltonian for the phase-separation step of QAOA, while the Rx gates serve as mixer operations.
  • Figure 2: Mathematical exponentiation of the Hamiltonian terms. This operation is a core component for constructing the unitary operators used in the QAOA algorithm.
  • Figure 3: Illustrative convergence behavior of the standard penalty-based QAOA. The consistent selection of the infeasible ‘111’ state demonstrates a common failure mode where the optimizer does not satisfy the problem's hard constraints.

Theorems & Definitions (6)

  • Theorem 1
  • Definition 1
  • Definition 2
  • Lemma 1
  • Proposition 2
  • Conjecture 1