| 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
| Method | Sharpe Ratio | Turnover |
|---|
| QAOA (XY-mixer) | 1.81 | 76.8% |
| Simulated Annealing | 1.31 | - |
| Hierarchical Risk Parity | 0.98 | - |
Backtested on 10 US equities over 2025.
Pattern: Constraint-Preserving Ansatz Design
When quantum algorithms must respect hard constraints:
- Identify the conserved quantity (e.g., Hamming weight = K)
- Design a mixer Hamiltonian that preserves it
- Initialize in a state that satisfies the constraint
- Verify constraint preservation throughout the circuit
Comparison: XY-Mixer vs Transverse Field
| Property | XY-Mixer | Transverse Field |
|---|
| Hamming weight | Preserved | Not preserved |
| Search space | C(n,K) valid states | 2^n all states |
| Penalty terms | Not needed | Required (distorts landscape) |
| Circuit complexity | O(n²) gates | O(n) gates |
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
- Hardware gap (2026-07-07 NEW): On real hardware (ibm_kingston, Heron r2), XY-mixer simulator-hardware energy gap stays within |0.8| across all depths — but penalty-based variants diverge up to |53.9|. The structural guarantee partially breaks under gate-level noise, so honest hardware accounting is essential.
Real-Hardware Validation (COMET, arXiv: 2607.02622)
First real-hardware comparison of XY-mixer vs penalty QAOA on 12-qubit CRISPR gene editing instance:
- XY-mixer: >95% optimum probability at p=3; hardware gap |ΔE| ≤ 0.8
- Penalty variants: <6% optimum at every depth; hardware gap up to |ΔE| = 53.9
- No penalty coefficient tuning needed with XY-mixer — enforced structurally
- Related umbrella skill:
comet-constraint-preserving-qaoa
References
- arXiv: 2602.14827 - "Constrained Portfolio Optimization via QAOA with XY-Mixers"
