Quantum Growth Modeling

---
name: quantum-growth-modeling
description: "Quantum growth modeling methodology using parameterized quantum circuits, EWL quantum games, and Dirac-Hamiltonian economic simulation. Applies quantum computing to economic growth, innovation dynamics, capital accumulation, and policy recommendation systems. Use when: analyzing economic growth with quantum methods, modeling innovation ecosystems as quantum systems, implementing quantum game theory for strategic decision-making, simulating capital trajectories with quantum Hamiltonians, building quantum-enhanced recommender systems for policy. Keywords: quantum growth model, Dirac Hamiltonian economics, Solow-Swan quantum, EWL quantum game, innovation ecosystem, capital accumulation quantum, policy recommender quantum, quantum strategic decision, 量子经济增长, 量子哈密顿经济学."
---

# Quantum Growth Modeling

Methodology for modeling economic growth, innovation dynamics, and capital accumulation using quantum computing techniques.

## Core Framework

### Quantum Game Theory for Economic Modeling

Use Eisert-Wilkens-Lewenstein (EWL) quantum game circuits to model strategic interactions in multi-agent economic systems:

1. **Entangle agents**: Apply multi-qubit EWL entangler to create superposition of strategies
2. **Parameterize local rotations**: Each agent's strategy operator is tuned by normalized dominance weights from real data
3. **Apply inverse entangler**: Collapse entangled state
4. **Measure**: Measurement probabilities become recommender scores for outcomes

### Dirac-Solow-Swan Hamiltonian Integration

Map quantum game outcomes to economic growth simulation:

1. **Extract game probabilities**: Measurement outcomes from EWL circuit
2. **Construct Dirac potential**: Map probabilities to diagonal of Dirac Hamiltonian
3. **Integrate Solow-Swan dynamics**: Combine quantum game outcomes with classical growth model
4. **Time-evolve**: Simulate capital accumulation and bifurcation dynamics

## Implementation Pattern

```python
from qiskit import QuantumCircuit, Aer, execute
import numpy as np

def ewl_quantum_game(weights, n_rounds=1):
    """EWL quantum game circuit parameterized by real economic weights.
    
    Args:
        weights: Normalized dominance weights from real data
        n_rounds: Number of entanglement-measurement rounds
    
    Returns:
        Measurement probabilities as recommender scores
    """
    n = len(weights)
    qc = QuantumCircuit(n)
    
    # EWL Entangler (multi-qubit)
    for i in range(n - 1):
        qc.cx(i, i + 1)
    for i in range(n):
        qc.h(i)
    
    # Parameterized local rotations
    for i, w in enumerate(weights):
        theta = w * np.pi
        qc.ry(theta, i)
    
    # Inverse entangler
    for i in range(n):
        qc.h(i)
    for i in range(n - 2, -1, -1):
        qc.cx(i, i + 1)
    
    # Measure
    qc.measure_all()
    
    backend = Aer.get_backend('qasm_simulator')
    result = execute(qc, backend, shots=1024).result()
    return result.get_counts()
```

### Dirac-Solow-Swan Hamiltonian

```python
def dirac_solow_swan(game_probs, K0, s, delta, n, T=10):
    """Time-evolution of capital under quantum game-influenced dynamics.
    
    Args:
        game_probs: Measurement probabilities from EWL circuit
        K0: Initial capital
        s: Savings rate
        delta: Depreciation rate
        n: Population growth
        T: Time steps
    """
    # Map game probs to Dirac potential (diagonal)
    V = np.diag(np.array(list(game_probs.values())))
    
    # Dirac Hamiltonian: H = alpha.p + beta.m + V
    H = np.zeros((len(V), len(V)))
    for t in range(T):
        K_t = K0 * ((1 - delta) + s * game_probs.get('disruptive', 0.5)) ** t
        # Bifurcation detection
        if abs(K_t - K0 * (1 + n) ** t) > threshold:
            return 'bifurcation', K_t
    return 'stable', K_T
```

## Key Principles

- **NISQ-compatible**: Circuits with <25 gates and depth <15 are executable on current hardware
- **Real data integration**: Strategy weights from empirical funding/participation data
- **Scaling**: Circuit scales as O(n) for n-round helix communications
- **Interpretability**: Measurement probabilities directly map to policy recommender scores

## Activation

Keywords: quantum growth model, Dirac Hamiltonian economics, Solow-Swan quantum, EWL quantum game, innovation ecosystem, capital accumulation, quantum recommender, policy simulation quantum, quadruple helix innovation.