qaoa-xy-mixers-portfolio

name: qaoa-xy-mixers-portfolio description: "Constraint-preserving QAOA formulation using Dicke state initialization and XY-mixer Hamiltonian for direct indexing portfolio optimization. Use when: implementing QAOA with hard cardinality constraints; designing constraint-preserving quantum ansatzes; mitigating barren plateaus via Trotterized initialization; comparing quantum vs classical portfolio optimization (SA, HRP); Direct Indexing with ESG constraints. Keywords: qaoa, xy-mixer, dicke state, portfolio optimization, constraint-preserving, direct indexing, barren plateau, hamming weight"

QAOA with XY-Mixers for Constrained Portfolio Optimization

Core Concept

Standard QAOA with transverse field mixers fails to enforce hard cardinality constraints, requiring soft penalties that distort the energy landscape. This formulation uses XY-mixer Hamiltonians that strictly preserve Hamming weight, ensuring only valid portfolios of size K are explored.

Key Innovation: Constraint Preservation

Standard QAOA (fails for constraints)

H_mixer = Σ X_i  # Transverse field

Mixes all bitstrings → violates cardinality constraint K

XY-Mixer QAOA (preserves constraints)

H_mixer = Σ (X_i X_j + Y_i Y_j)  # For adjacent pairs

Conserves total magnetization → preserves Hamming weight K

Algorithm Steps

Step 1: Dicke State Initialization

Initialize in a Dicke state |D(n,K)⟩ with exactly K qubits in |1⟩:

• Encodes the cardinality constraint from the start
• Inspired by adiabatic quantum computing
• Avoids the "all-zeros" barren plateau

Step 2: Trotterized Parameter Schedule

Use adiabatic-inspired parameter initialization:

γ_p = (p/P) * γ_max   # Problem Hamiltonian
β_p = (1 - p/P) * β_max  # Mixer Hamiltonian

This mimics adiabatic evolution, reducing barren plateaus.

Step 3: QAOA Circuit Construction

|ψ⟩ = ∏_p [exp(-i β_p H_XY) · exp(-i γ_p H_cost)] |D(n,K)⟩

Step 4: Measurement and Portfolio Selection

• Measure the final state
• Select the bitstring with minimum energy
• The K assets with bit=1 form the portfolio

Performance Benchmarks

Backtested on 10 US equities over 2025.

Pattern: Constraint-Preserving Ansatz Design

When quantum algorithms must respect hard constraints:

1. Identify the conserved quantity (e.g., Hamming weight = K)
2. Design a mixer Hamiltonian that preserves it
3. Initialize in a state that satisfies the constraint
4. Verify constraint preservation throughout the circuit

Comparison: XY-Mixer vs Transverse Field

Operational Considerations

• High turnover (76.8%): Trade-off between theoretical optimality and implementation costs
• Transaction costs: Must factor into backtesting
• Institutional constraints: ESG mandates, sector limits
• Rebalancing frequency: Weekly vs monthly impacts costs

Pitfalls

• Barren plateaus: Without Trotterized initialization, gradients vanish
• Gate overhead: XY-mixer requires O(n²) gates vs O(n) for transverse field
• NISQ limitations: Deep circuits suffer from noise
• Classical comparison: Ensure fair comparison with tuned classical baselines

References

• arXiv: 2602.14827 - "Constrained Portfolio Optimization via QAOA with XY-Mixers"