Quantum Systems Control Simulation

---
name: quantum-systems-control-simulation
description: "Quantum systems control theory and simulation framework. Covers coherent feedback control (H∞), physics-informed discrete-event simulation for quantum networks, and high-dimensional quantum photonics encoding. Use when: (1) designing control systems for quantum linear systems, (2) simulating polarization-encoded quantum networks, (3) implementing H∞ disturbance attenuation, (4) encoding quantum states in high-dimensional photonic modes, (5) analyzing quantum network stability and performance."
---

# Quantum Systems Control and Simulation

Framework for designing, analyzing, and simulating quantum control systems with physics-informed models.

## Core Concepts

### 1. Coherent Feedback H∞ Control

Design methodology for linear quantum systems with guaranteed stability and disturbance attenuation.

**Key Principles:**
- Closed-loop stability guarantee
- Prescribed disturbance attenuation level
- Simplified design for general linear quantum systems
- Riccati equation-based synthesis

**Design Steps:**
1. Define quantum linear system model (G)
2. Specify disturbance attenuation level (γ)
3. Solve H∞ Riccati equation
4. Construct coherent feedback controller (K)
5. Validate closed-loop stability

### 2. Physics-Informed Discrete-Event Simulation

Simulation framework integrating physical models with event-driven quantum network simulation.

**Components:**
- Jones calculus optical components
- SPDC Bell-state source models
- Wave plates and polarizing beam splitters
- Multi-section fiber models
- Quantum protocol timing

**Implementation:**
```python
# Extend SeQUeNCe simulator with physics models
from sequence import QuantumNetworkSimulator

class PhysicsInformedQuantumSimulator(QuantumNetworkSimulator):
    def __init__(self):
        super().__init__()
        self.add_jones_calculus_components()
        self.add_spdc_source()
        self.add_polarization_components()
    
    def simulate_bell_state_distribution(self, topology):
        # Discrete-event simulation with physics models
        events = self.generate_events(topology)
        return self.run_simulation(events)
```

### 3. High-Dimensional Quantum Photonics

Encoding multi-level quantum states using photonic degrees-of-freedom.

**Encoding Modes:**
- **Spatial modes**: Path encoding, orbital angular momentum
- **Temporal modes**: Time-bin encoding, pulse shaping
- **Spectral modes**: Frequency encoding, wavelength channels

**Workflow:**
1. Select encoding dimension (d)
2. Design generation scheme (SPDC, waveguides)
3. Define manipulation operations (unitary transformations)
4. Implement detection scheme (mode projection)
5. Characterize encoding fidelity

## Tools Used

- **exec**: Run simulation scripts, solve control equations
- **read**: Load reference materials, configuration files
- **write**: Save simulation results, controller designs
- **python**: Numerical computation (numpy, scipy, qutip)

## Usage Patterns

### Pattern 1: Design H∞ Quantum Controller

```
Design H∞ controller for quantum linear system with γ=0.5 attenuation
```

**Process:**
1. Parse system matrices (A, B, C, D)
2. Compute H∞ Riccati solution
3. Extract controller gains
4. Validate stability margin
5. Output controller transfer function

### Pattern 2: Simulate Quantum Network

```
Simulate polarization-encoded quantum network with Bell-state distribution
```

**Process:**
1. Define network topology
2. Configure optical components (Jones matrices)
3. Set timing parameters (discrete events)
4. Run physics-informed simulation
5. Analyze fidelity and timing statistics

### Pattern 3: Design High-Dimensional Encoding

```
Design 4-dimensional quantum encoding using temporal modes
```

**Process:**
1. Choose encoding scheme (time-bin)
2. Define generation parameters
3. Specify manipulation operations
4. Design detection protocol
5. Calculate information capacity

## Instructions for Agents

### Step 1: Identify Problem Type

Determine which quantum systems problem:
- **Control**: Stability, disturbance attenuation, feedback design
- **Simulation**: Network behavior, protocol timing, component modeling
- **Encoding**: State dimensionality, photonic modes, fidelity

### Step 2: Gather System Parameters

For control problems:
- System matrices (A, B, C, D)
- Disturbance characteristics
- Performance requirements (γ level)

For simulation problems:
- Network topology
- Component specifications
- Timing constraints

For encoding problems:
- Target dimensionality
- Available photonic modes
- Fidelity requirements

### Step 3: Apply Methodology

Execute appropriate design/analysis method:
- Solve Riccati equations for H∞ control
- Run discrete-event simulation with physics models
- Design encoding scheme with unitary transformations

### Step 4: Validate Results

Check:
- Stability conditions satisfied
- Simulation timing and fidelity meet specs
- Encoding scheme achieves target dimension

### Step 5: Document Output

Generate report with:
- Controller transfer function (for control)
- Simulation statistics (for simulation)
- Encoding protocol specification (for encoding)

## References

### H∞ Control Theory

See references/h-infinity-control.md for:
- Riccati equation derivation
- Stability conditions
- Disturbance attenuation proofs

### Physics-Informed Simulation

See references/discrete-event-simulation.md for:
- Jones calculus fundamentals
- SPDC source modeling
- Component timing parameters

### High-Dimensional Encoding

See references/high-dimensional-encoding.md for:
- Spatial/temporal/spectral mode theory
- Generation/manipulation/detection schemes
- Fidelity characterization methods

## Examples

### Example 1: H∞ Controller Design

**User:** "Design H∞ controller for quantum harmonic oscillator with γ=0.8"

**Agent:**
1. Parse oscillator model: A = [ω, 0; 0, -ω]
2. Solve H∞ Riccati: P = solve_riccati(A, B, γ)
3. Compute controller: K = -B^T P
4. Validate: ||Tzw||∞ < γ ✓
5. Output: K(s) = transfer_function(K)

### Example 2: Quantum Network Simulation

**User:** "Simulate Bell-state distribution over 3-node quantum network"

**Agent:**
1. Define topology: Node_A → Fiber → Node_B
2. Configure: SPDC source, PBS, wave plates
3. Set events: Generate, transmit, detect
4. Run simulation: 10000 trials
5. Results: Fidelity = 0.92, Latency = 15μs

## Related Skills

- **quantum-algorithm-framework-designer**: Algorithm design
- **quantum-error-correction-gauge-theory**: Error handling
- **distributed-quantum-computing**: Network architectures
- **complex-valued-kuramoto-control**: Oscillator control

## Dependencies

```bash
pip install qutip numpy scipy matplotlib
```

## Notes

- H∞ control requires linear system model
- Simulation accuracy depends on component models
- High-dimensional encoding limited by mode orthogonality
- Consider decoherence effects in all designs