Constrained Portfolio Optimization via Quantum Approximate Optimization Algorithm (QAOA) with XY-Mixers and Trotterized Initialization: A Hybrid Approach for Direct Indexing

TL;DR

This paper introduces a constraint-preserving QAOA approach with XY-mixers and Trotterized initialization for portfolio optimization, demonstrating superior backtested performance over classical methods in direct indexing scenarios.

Contribution

It develops a novel QAOA ansatz with XY-mixers and Dicke state initialization that strictly enforces portfolio size constraints, improving optimization in quantum finance applications.

Findings

QAOA achieves a Sharpe Ratio of 1.81 in backtesting.

The approach outperforms classical methods like SA and HRP.

High turnover of 76.8% highlights operational trade-offs.

Abstract

Portfolio optimization under strict cardinality constraints is a combinatorial challenge that defies classical convex optimization techniques, particularly in the context of "Direct Indexing" and ESG-constrained mandates. In the Noisy Intermediate-Scale Quantum (NISQ) era, the Quantum Approximate Optimization Algorithm (QAOA) offers a promising hybrid approach. However, standard QAOA implementations utilizing transverse field mixers often fail to strictly enforce hard constraints, necessitating soft penalties that distort the energy landscape. This paper presents a comprehensive analysis of a constraint-preserving QAOA formulation against Simulated Annealing (SA) and Hierarchical Risk Parity (HRP). We implement a specific QAOA ansatz utilizing a Dicke state initialization and an XY-mixer Hamiltonian that strictly preserves the Hamming weight of the solution, ensuring only valid portfolios of size K are explored. Furthermore, we introduce a Trotterized parameter initialization schedule inspired by adiabatic quantum computing to mitigate the "Barren Plateau" problem. Backtesting on a basket of 10 US equities over 2025 reveals that our QAOA approach achieves a Sharpe Ratio of 1.81, significantly outperforming Simulated Annealing (1.31) and HRP (0.98). We further analyze the operational implications of the algorithm's high turnover (76.8%), discussing the trade-offs between theoretical optimality and implementation costs in institutional settings.