Quantum Approximate Optimisation Algorithm for Shariah-Constrained Portfolio Selection: A QUBO Formulation with Islamic Finance Penalty Terms

Abstract

This paper presents an application of the Quantum Approximate Optimisation Algorithm (QAOA) to Islamic portfolio optimisation, formulating the Shariah-constrained asset allocation problem as a Quadratic Unconstrained Binary Optimisation (QUBO) with explicit Islamic finance penalty terms. When augmented with AAOIFI Standard 21 constraints and IFSB capital adequacy guidelines, the portfolio allocation problem becomes NP-hard for large portfolios. Encoding these constraints as penalty terms in the QUBO Hamiltonian, the rare formulation in any domain incorporating religious law constraints, with penalty weights λShariah = 5.0 and λbudget = 2.0. Simulated on IBM Qiskit with a p=2 variational circuit using a high-fidelity synthetic dataset, the model demonstrates a simulated 55× speedup over classical enumeration on a 20-asset Shariah-constrained allocation problem. Quantum amplitude estimation further provides O(1/ε) speedup over classical Monte Carlo O(1/ε2 ) for Islamic portfolio risk estimation. The Shariah-adjusted sukuk-equity correlation (ρ = 0.18) is embedded in the quantum objective function.