# Optical/Photonic Computing for Ultra-Fast Linear System Solving

## Executive Summary

Optical computing leverages the speed of light and massive parallelism of photonic systems to achieve matrix operations at the speed of light propagation. With zero-energy computation (passive optical elements) and inherent parallelism, photonic solvers can achieve 1000× speedups with 100× lower energy consumption.

## Core Innovation: Computing with Light

Light naturally performs linear operations:
1. **Interference** = Addition
2. **Diffraction** = Convolution
3. **Refraction** = Matrix multiplication
4. **Polarization** = Complex arithmetic
5. **Speed** = 10ps operations (100GHz)

## Photonic Linear Algebra Primitives

### 1. Optical Matrix Multiplication

```python
class PhotonicMatrixMultiplier:
    """
    Silicon photonic mesh for matrix-vector multiplication
    Based on Mach-Zehnder interferometer (MZI) arrays
    """
    def __init__(self, size=64):
        self.size = size
        self.mzi_mesh = self.create_universal_mesh(size)
        self.phase_shifters = np.zeros((size, size, 2))  # θ and φ per MZI

    def create_universal_mesh(self, n):
        """
        Reck or Clements mesh topology
        Universal linear optical network
        """
        mesh = []
        # Triangular arrangement of MZIs
        for layer in range(n):
            for pos in range(n - layer - 1):
                mzi = MachZehnderInterferometer(
                    input_ports=(pos, pos + 1),
                    layer=layer
                )
                mesh.append(mzi)
        return mesh

    def decompose_matrix(self, matrix):
        """
        Decompose arbitrary matrix into MZI settings
        Using Reck decomposition algorithm
        """
        U, S, V = np.linalg.svd(matrix)

        # Convert to MZI phase settings
        phases = []
        for layer in self.mzi_mesh:
            theta, phi = self.extract_phases(U, layer)
            phases.append((theta, phi))

        return phases

    def compute(self, input_vector):
        """
        Propagate light through mesh
        Speed: O(1) after setup!
        """
        # Encode input as light intensities/phases
        optical_input = self.encode_optical(input_vector)

        # Single propagation through mesh
        optical_output = self.propagate(optical_input)

        # Decode output
        return self.decode_optical(optical_output)

    def propagate(self, light):
        """
        Physics simulation or actual hardware
        """
        for mzi in self.mzi_mesh:
            light = mzi.transform(light, self.phase_shifters[mzi.id])
        return light
```

### 2. Coherent Ising Machine for Optimization

```cpp
class CoherentIsingMachine {
    // Solves quadratic optimization via optical parametric oscillators
private:
    int num_spins;
    std::vector<OpticalOscillator> oscillators;
    FeedbackNetwork feedback;

public:
    Vector solve_linear_system(const Matrix& A, const Vector& b) {
        // Transform Ax=b to Ising problem
        // Minimize: x^T A^T A x - 2b^T A x

        auto ising_couplings = transform_to_ising(A, b);

        // Initialize oscillators
        for (int i = 0; i < num_spins; i++) {
            oscillators[i].set_coupling(ising_couplings[i]);
            oscillators[i].inject_pump_light();
        }

        // Let system evolve (microseconds)
        while (!reached_steady_state()) {
            // Optical feedback implements matrix multiplication
            for (int i = 0; i < num_spins; i++) {
                complex<double> feedback = 0;
                for (int j = 0; j < num_spins; j++) {
                    feedback += ising_couplings[i][j] *
                               oscillators[j].get_amplitude();
                }
                oscillators[i].apply_feedback(feedback);
            }

            // Natural evolution toward minimum
            propagate_time_step();
        }

        // Read out solution
        return decode_spin_configuration();
    }

private:
    bool reached_steady_state() {
        // Check if oscillator phases locked
        return calculate_phase_variance() < threshold;
    }
};
```

### 3. Reservoir Computing with Photonics

```python
class PhotonicReservoir:
    """
    Random photonic network for solving via physical computation
    No training needed - uses natural dynamics
    """
    def __init__(self, nodes=1000):
        self.reservoir = self.create_random_photonic_network(nodes)
        self.readout_weights = None

    def create_random_photonic_network(self, n):
        """
        Silicon photonic chip with random connections
        """
        network = {
            'waveguides': self.random_waveguide_mesh(n),
            'couplers': self.random_directional_couplers(n),
            'delays': np.random.exponential(1.0, n),  # ps
            'nonlinearities': self.kerr_nonlinearities(n)
        }
        return network

    def solve(self, A, b):
        """
        Inject problem, let light evolve, read solution
        """
        # Encode input
        input_light = self.encode_problem(A, b)

        # Inject into reservoir
        self.inject_light(input_light)

        # Let photonic network evolve (100ps)
        time_series = self.record_evolution(duration_ps=100)

        # Linear readout gives solution!
        if self.readout_weights is None:
            self.train_readout(time_series, expected_solution)

        return self.readout_weights @ time_series[-1]

    def record_evolution(self, duration_ps):
        """
        Record optical intensities at each node
        """
        samples = int(duration_ps / 0.01)  # 100GHz sampling
        evolution = np.zeros((samples, len(self.reservoir['waveguides'])))

        for t in range(samples):
            evolution[t] = self.measure_intensities()
            self.propagate_timestep(0.01)  # 10fs

        return evolution
```

## Breakthrough Architectures

### 1. Neuromorphic Photonic Processor

```python
class NeuromorphicPhotonicSolver:
    """
    Spiking photonic neural network for iterative solving
    """
    def __init__(self):
        self.photonic_neurons = self.create_laser_neuron_array()
        self.optical_synapses = self.create_weight_bank()

    def create_laser_neuron_array(self, n=256):
        """
        Semiconductor lasers with saturable absorbers
        Natural spiking dynamics at 10GHz
        """
        neurons = []
        for i in range(n):
            neuron = GrapheneLaserNeuron(
                threshold_current=1.5,  # mA
                refractory_period=10,   # ps
                wavelength=1550 + i * 0.1  # nm (WDM)
            )
            neurons.append(neuron)
        return neurons

    def solve_iteratively(self, A, b):
        """
        Map linear system to spiking dynamics
        """
        # Configure synaptic weights from matrix A
        self.configure_optical_weights(A)

        # Set input currents from vector b
        self.set_bias_currents(b)

        # Run until convergence (microseconds)
        spike_trains = self.run_dynamics(duration_us=10)

        # Decode solution from spike rates
        return self.decode_spike_rates(spike_trains)

    def configure_optical_weights(self, matrix):
        """
        Program microring resonator weight banks
        """
        for i, row in enumerate(matrix):
            for j, weight in enumerate(row):
                # Thermal tuning of microring resonance
                self.optical_synapses[i][j].set_transmission(weight)
```

### 2. Quantum-Classical Hybrid Photonic

```rust
struct HybridPhotonicSolver {
    classical_mesh: PhotonicMesh,
    quantum_unit: BosonSampler,
    interface: ClassicalQuantumInterface,
}

impl HybridPhotonicSolver {
    fn solve_with_quantum_speedup(&self, A: &Matrix, b: &Vector) -> Vector {
        // Decompose problem
        let (classical_part, quantum_part) = self.decompose_problem(A, b);

        // Classical photonic for bulk computation
        let classical_result = self.classical_mesh.compute(classical_part);

        // Quantum photonic for hard kernel
        let quantum_result = self.quantum_unit.sample_solution(quantum_part);

        // Combine results
        self.interface.merge_solutions(classical_result, quantum_result)
    }

    fn quantum_unit_sample(&self, problem: &QuantumProblem) -> Sample {
        // Boson sampling for #P-hard problems
        // Exponential speedup for certain matrices

        // Prepare Fock state input
        let input_state = self.prepare_fock_state(problem);

        // Linear optical network
        let unitary = self.program_unitary(problem.matrix);

        // Detection (photon counting)
        let output_distribution = self.measure_photons();

        // Post-select for solution
        self.postselect_solution(output_distribution)
    }
}
```

### 3. Free-Space Optical Processor

```python
class FreeSpaceOpticalComputer:
    """
    Lens-based computing - matrix ops at speed of light
    Based on 4f optical system
    """
    def __init__(self):
        self.spatial_light_modulator = SLM(resolution=(4096, 4096))
        self.fourier_lens = FourierTransformLens(focal_length=100)  # mm
        self.detector = CCDArray(resolution=(4096, 4096))

    def optical_matrix_multiply(self, matrix, vector):
        """
        Single-pass optical computation
        Time: Speed of light through 4f system (~1ns)
        """
        # Encode matrix as hologram
        hologram = self.encode_matrix_hologram(matrix)
        self.spatial_light_modulator.display(hologram)

        # Encode vector as light pattern
        input_light = self.encode_vector_amplitude(vector)

        # Propagate through 4f system
        # Fourier -> Multiply -> Inverse Fourier
        light = input_light
        light = self.fourier_lens.transform(light)  # F
        light = light * hologram                     # Multiply
        light = self.fourier_lens.transform(light)  # F^-1

        # Detect result
        result = self.detector.measure_intensity(light)
        return self.decode_result(result)

    def solve_via_fourier(self, A, b):
        """
        Solve in Fourier domain where convolution = multiplication
        """
        # Transform to Fourier space
        A_fourier = self.optical_fourier_transform(A)
        b_fourier = self.optical_fourier_transform(b)

        # Division in Fourier space (via interference)
        x_fourier = self.optical_divide(b_fourier, A_fourier)

        # Inverse transform
        return self.optical_inverse_fourier(x_fourier)
```

## Performance Metrics

### Speed Comparison

| Operation | Electronic | Photonic | Speedup |
|-----------|------------|----------|---------|
| Matrix-Vector (1000×1000) | 1μs | 10ps | 100,000× |
| FFT (1M points) | 100μs | 100ps | 1,000,000× |
| Convolution | 10ms | 10ps | 1,000,000,000× |
| Neural Network Layer | 10μs | 10ps | 1,000,000× |

### Energy Efficiency

```python
def energy_comparison():
    """
    Energy per operation comparison
    """
    # Electronic (45nm CMOS)
    electronic_energy = {
        'add_32bit': 0.1e-12,      # 0.1 pJ
        'multiply_32bit': 3.0e-12,  # 3 pJ
        'memory_access': 10e-12,    # 10 pJ
        'matrix_mult_1k': 3e-6,     # 3 μJ
    }

    # Photonic
    photonic_energy = {
        'add_32bit': 0,            # Passive interference
        'multiply_32bit': 1e-15,   # 1 fJ (modulation only)
        'memory_access': 0,         # Optical delay lines
        'matrix_mult_1k': 1e-9,     # 1 nJ (detection only)
    }

    # 3000× more efficient!
    return electronic_energy['matrix_mult_1k'] / photonic_energy['matrix_mult_1k']
```

## Cutting-Edge Research

### Recent Breakthroughs

1. **Shen et al. (2017)**: "Deep Learning with Coherent Nanophotonic Circuits"
   - First optical neural network
   - Nature Photonics

2. **Wetzstein et al. (2020)**: "Inference in Artificial Intelligence with Deep Optics"
   - Stanford optical AI processor
   - Nature

3. **Xu et al. (2021)**: "11 TOPS Photonic Convolutional Accelerator"
   - Record performance
   - Nature

4. **Lightmatter (2021)**: "Envise: Commercial Photonic AI Chip"
   - 10× faster than A100
   - Commercial product

5. **Hamerly et al. (2019)**: "Large-Scale Optical Neural Networks"
   - Scaling to millions of neurons
   - Physical Review X

### Companies & Labs

- **Lightmatter**: AI inference chips
- **Lightelligence**: Optical AI computing
- **Optalysys**: Optical correlators
- **Xanadu**: Photonic quantum computing
- **MIT Photonic Computing Lab**
- **Stanford Nanophotonics Lab**

## Implementation Example

```python
import numpy as np
from photonic_sim import PhotonicCircuit, MZI, Detector

class SublinearPhotonicSolver:
    """
    Combines sublinear algorithms with photonic hardware
    """
    def __init__(self):
        self.circuit = PhotonicCircuit()
        self.build_universal_processor()

    def build_universal_processor(self, size=64):
        """
        Construct reconfigurable photonic processor
        """
        # Input couplers
        for i in range(size):
            self.circuit.add_input_coupler(port=i)

        # MZI mesh (Clements architecture)
        for layer in range(size):
            for i in range(0, size - layer - 1, 2):
                mzi = MZI(
                    ports=(i + layer % 2, i + 1 + layer % 2),
                    layer=layer
                )
                self.circuit.add_component(mzi)

        # Output detectors
        for i in range(size):
            self.circuit.add_detector(port=i, type='homodyne')

    def solve_sublinear(self, A_sparse, b, epsilon=1e-6):
        """
        Sublinear solving with photonic acceleration
        """
        n = len(b)
        x = np.zeros(n)

        # Configure photonic processor for sparse ops
        sampled_rows = self.importance_sample(A_sparse)

        for batch in self.batch_rows(sampled_rows, batch_size=64):
            # Extract submatrix
            A_batch = A_sparse[batch, :]

            # Configure photonic circuit
            self.circuit.configure_matrix(A_batch)

            # Optical computation (single shot)
            x_batch = self.circuit.compute(b[batch])

            # Update solution
            x[batch] = x_batch

        return x

    def importance_sample(self, A_sparse):
        """
        Sample rows based on leverage scores
        """
        leverage = np.sum(A_sparse**2, axis=1)
        probs = leverage / np.sum(leverage)
        n_samples = int(np.log(len(A_sparse)) * 100)
        return np.random.choice(len(A_sparse), n_samples, p=probs)
```

## Future Directions

### Integration Challenges
1. **Photonic-Electronic Interface**: High-speed DACs/ADCs
2. **Thermal Stability**: Phase drift compensation
3. **Packaging**: 3D photonic-electronic integration
4. **Programmability**: Universal photonic gates

### Emerging Technologies
- **Plasmonics**: Sub-wavelength computation
- **Metamaterials**: Engineered optical response
- **Topological Photonics**: Robust light propagation
- **Nonlinear Optics**: All-optical logic

## Conclusion

Optical computing offers the ultimate speed for linear algebra—literally at the speed of light. Combined with sublinear algorithms, photonic processors can solve massive systems in nanoseconds with minimal energy. The future of high-performance computing is photonic.