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

Paper: arXiv 2602.14827

Authors: Javier Mancilla, Theodoros D. Bouloumis, Frederic Goguikian

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.

Complexity vs Empirical Score

• Math Complexity: 6.5/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents an innovative hybrid quantum-classical pipeline using constraint-preserving QAOA with Dicke states and XY-mixers for direct indexing. It balances mathematical formulation with practical financial implementation, evaluating realistic frictions like transaction costs and turnover on out-of-sample equities.

Research Flowchart

  flowchart TD
    A[Goal: Solve Constrained Portfolio Optimization via QAOA for Direct Indexing] --> B[Input: Basket of 10 US Equities Market Data]
    B --> C[Methodology: Dicke State Prep & XY-Mixer QAOA with Trotterized Schedule]
    C --> D[Execution & Benchmarking: QAOA vs. Simulated Annealing vs. HRP]
    D --> E[Key Outcomes: QAOA Achieves Top Sharpe Ratio 1.81 vs SA 1.31 & HRP 0.98]

Cited References (25)

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2. Computational study of a family of mixed-integer quadratic programming problems (1996)
3. Efficient Asset Management: A Practical Guide to Stock Portfolio Optimization and Asset Allocation (2008)
4. The Sharpe Ratio Efficient Frontier (2012)
5. Honey, I Shrunk the Sample Covariance Matrix (2004)
6. Robust Portfolio Optimization and Management (2007)
7. Building Diversified Portfolios that Outperform Out of Sample (2016)
8. Algorithm for Cardinality Constrained Quadratic Optimization (2009)

0 of 25 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.