Quantum Portfolio Optimizer

---
name: quantum-portfolio-optimizer
description: "Quantum computing portfolio optimization skill. Uses QAOA, quantum annealing, and hybrid quantum-classical methods for financial portfolio optimization with higher-order moments (skewness, kurtosis) and real-world constraints (cardinality, turnover limits). Activation: quantum portfolio, quantum optimization, QAOA portfolio, 量子组合优化, quantum finance optimization, portfolio optimization quantum."
---

# Quantum Portfolio Optimizer

Quantum computing approach to portfolio optimization using QAOA, quantum annealing, and hybrid methods.

## Quick Start

```python
# Basic quantum portfolio optimization
from qiskit_optimization import QuadraticProgram
from qiskit_optimization.algorithms import QAOA

# 1. Define portfolio problem as QUBO
qp = QuadraticProgram("portfolio")
qp.binary_var_list(assets)  # n assets -> n binary variables

# 2. Set objective: maximize return - risk penalty
qp.minimize(linear=-expected_returns, quadratic=risk_covariance)

# 3. Add constraints (optional)
qp.linear_constraint(cardinality_constraint)  # max k assets

# 4. Solve with QAOA
qaoa = QAOA(quantum_instance, reps=3)
result = qaoa.solve(qp)
selected_assets = result.x
```

## Core Workflow

### Step 1: Problem Formulation

Convert portfolio optimization to QUBO (Quadratic Unconstrained Binary Optimization):

```
Objective: min Σ_i(-μ_i x_i) + λ Σ_iΣ_j(σ_ij x_i x_j)

Where:
- x_i ∈ {0,1}: binary selection variable for asset i
- μ_i: expected return of asset i
- σ_ij: covariance between assets i and j
- λ: risk penalty coefficient
```

**Higher-order moments extension** (from arxiv:2509.01496):
```
Add skewness S and kurtosis K terms:
min Σ_i(-μ_i x_i) + λ Σ_ij(σ_ij x_i x_j) - γ Σ_ijk(S_ijk x_i x_j x_k) + δ Σ_ijkl(K_ijkl x_i x_j x_k x_l)
```

### Step 2: Algorithm Selection

| Algorithm | Best For | Implementation |
|-----------|----------|----------------|
| **QAOA** | Gate-based QC, general problems | `qiskit_optimization.algorithms.QAOA` |
| **Quantum Annealing** | D-Wave, large sparse problems | `dwave.system.samplers.DWaveSampler` |
| **VQE** | Variational approach, NISQ | `qiskit.algorithms.VQE` |
| **Hybrid** | Practical applications | Classical optimizer + quantum sampling |

### Step 3: Constraint Handling

Real-world constraints from arxiv:2504.08843:

```python
# Cardinality constraint: select exactly k assets
qp.linear_constraint(
    constraint={'x_1': 1, 'x_2': 1, ...},
    sense='==',
    rhs=k,
    name='cardinality'
)

# Turnover constraint: limit change from current portfolio
qp.linear_constraint(
    constraint={f'delta_{i}': 1 for i in assets},
    sense='<=',
    rhs=max_turnover,
    name='turnover'
)

# Budget constraint
qp.linear_constraint(
    constraint={f'w_{i}': price[i] for i in assets},
    sense='<=',
    rhs=budget,
    name='budget'
)
```

### Step 4: Hybrid Quantum-Classical

For practical NISQ-era implementation:

```python
# Hybrid approach from arxiv:2407.19857
def hybrid_portfolio_optimization(returns, covariance, params):
    """
    1. Classical preprocessing: filter candidates, compute statistics
    2. Quantum optimization: solve reduced QUBO
    3. Classical postprocessing: refine allocation, validate constraints
    """
    # Pre-classical: reduce problem size
    top_assets = classical_filter(returns, top_n=20)
    
    # Quantum: solve reduced problem
    qp = build_qubo(top_assets, returns, covariance)
    quantum_solution = qaoa.solve(qp)
    
    # Post-classical: continuous allocation
    weights = refine_allocation(quantum_solution, constraints)
    
    return weights
```

## Tools Used

- `exec`: Run Python quantum optimization scripts
- `write`: Create optimization code and results
- `read`: Load existing configurations and data

## Key Papers (References)

| Paper | Contribution | arXiv ID |
|-------|--------------|----------|
| Free Energy-Entropy Duality for Risk-Sensitive RL Investment | LQG stochastic game reformulation of benchmarked allocation | 2606.20903 |
| Heuristic Portfolio Optimization (HPO) | Information-restricted projection of Markowitz solution | 2606.12612 |
| Higher-Order Portfolio Optimization with QAOA | Skewness, kurtosis terms | 2509.01496 |
| Variational and Annealing-Based Quantum Combinatorial Optimization | QAOA vs QA comparative survey | 2603.19117 |
| End-to-End Portfolio Optimization with Quantum Annealing | Practical hybrid approach | 2504.08843 |
| PO-QA Framework | Quantum algorithm framework | 2407.19857 |
| Quantum Computing for Finance: State of the Art | Comprehensive review | 2006.14510 |
| Reverse Quantum Annealing | Alternative annealing strategy | 1810.08584 |

## Activation Keywords

- quantum portfolio
- quantum optimization
- QAOA portfolio
- 量子组合优化
- quantum finance
- portfolio quantum
- quantum annealing portfolio
- higher-order portfolio
- HUBO portfolio optimization

## Error Handling

### QUBO Conversion Issues
- Check covariance matrix is positive semi-definite
- Scale parameters appropriately (λ, γ, δ)
- Validate binary variable encoding

### Quantum Hardware Limitations
- For NISQ devices: reduce problem size first
- Use noise mitigation techniques
- Consider hybrid classical fallback

### Constraint Violation
- **Penalty methods**: Add penalty terms to objective function (traditional approach — see PITFALL below)
- **Dicke state ansatz** (arxiv:2606.08504): Structural constraint encoding — mixed Dicke states eliminate penalty terms entirely by encoding Hamming weight constraints into circuit structure. Pure Dicke for equality, mixed for inequality, tensor products for multiple groups. Validated on IBM NISQ with CMA-ES. See `dicke-state-portfolio-qaoa` skill.
- **CQM with hard constraints** (D-Wave): Use `LeapHybridCQMSampler` with `ConstrainedQuadraticModel()` for penalty-free constraint handling on annealers

## Examples

### Example 1: Basic Portfolio Selection

**User**: "用量子计算优化一个5资产组合"

**Agent**: Creates QUBO formulation, runs QAOA simulation, returns optimal asset selection with expected return and risk metrics.

### Example 2: Higher-Order Optimization

**User**: "考虑偏度和峰度的量子组合优化"

**Agent**: Extends QUBO with skewness (3rd order) and kurtosis (4th order) terms, solves with deeper QAOA circuit, reports improved risk-adjusted returns.

### Example 3: Constrained Optimization

**User**: "量子组合优化加上基数约束（最多选3个资产）"

**Agent**: Adds cardinality constraint to QUBO, uses penalty method or constraint encoding, validates final selection meets constraint.

## Related Skills
- `free-energy-rl-investment` - Risk-sensitive RL via free energy-entropy duality for continuous-time benchmarked allocation
- `variational-annealing-quantum-combinatorial` - QAOA vs quantum annealing comparative methodology
- `heuristic-portfolio-optimization` - Heuristic portfolio optimization (economics/)
- `qrl-dynamic-portfolio` - Quantum RL for dynamic portfolio optimization
- `quantum-portfolio-optimization` - QAOA with constraint-preserving mixers
- `dicke-state-portfolio-qaoa` - Dicke state ansatz for constraint-preserving QAOA
- `quantum-resilient-decentralized-ai-economy` - Decentralized AI economy with post-quantum security
- `quantum-market-entanglement` - Quantum game theory for market stabilization

## Resources

- `references/algorithms.md`: Detailed algorithm comparison
- See also: `quantum-optimization/higher-order-quantum-optimization-finance` — HOBO for legally-constrained financial problems (CSA, margin, concentration limits) without QUBO reduction overhead
- `references/qubo_formulation.md`: Mathematical formulation guide
- `scripts/portfolio_qaoa.py`: Ready-to-run QAOA implementation
- `scripts/hybrid_optimizer.py`: Hybrid quantum-classical workflow