quantrs2_device/topological/

braiding.rs

//! Braiding operations for topological quantum computing
//!
//! This module implements braiding operations, braid group representations,
//! and the calculation of braiding matrices for topological quantum computation.

use super::{
    Anyon, BraidingDirection, BraidingOperation, BraidingResult, NonAbelianAnyonType,
    TopologicalCharge, TopologicalError, TopologicalResult,
};
use serde::{Deserialize, Serialize};
use std::collections::HashMap;
use std::f64::consts::PI;

/// Braid group generator for topological quantum computation
#[derive(Debug, Clone, Serialize, Deserialize)]
pub struct BraidGenerator {
    /// Generator index
    pub index: usize,
    /// Anyon indices involved in this generator
    pub anyon_indices: (usize, usize),
    /// Braiding direction
    pub direction: BraidingDirection,
}

/// Braid group element representing a sequence of generators
#[derive(Debug, Clone, Serialize, Deserialize)]
pub struct BraidGroupElement {
    /// Sequence of generators
    pub generators: Vec<BraidGenerator>,
    /// Total number of anyons
    pub anyon_count: usize,
}

impl BraidGroupElement {
    /// Create a new braid group element
    pub const fn new(anyon_count: usize) -> Self {
        Self {
            generators: Vec::new(),
            anyon_count,
        }
    }

    /// Add a generator to the braid
    pub fn add_generator(&mut self, generator: BraidGenerator) -> TopologicalResult<()> {
        if generator.anyon_indices.0 >= self.anyon_count
            || generator.anyon_indices.1 >= self.anyon_count
        {
            return Err(TopologicalError::InvalidBraiding(
                "Anyon indices out of bounds for braid".to_string(),
            ));
        }
        self.generators.push(generator);
        Ok(())
    }

    /// Compose with another braid element
    pub fn compose(&self, other: &Self) -> TopologicalResult<Self> {
        if self.anyon_count != other.anyon_count {
            return Err(TopologicalError::InvalidBraiding(
                "Cannot compose braids with different anyon counts".to_string(),
            ));
        }

        let mut result = self.clone();
        result.generators.extend(other.generators.clone());
        Ok(result)
    }

    /// Calculate the inverse braid
    #[must_use]
    pub fn inverse(&self) -> Self {
        let mut inverse_generators = Vec::new();

        for generator in self.generators.iter().rev() {
            let inverse_direction = match generator.direction {
                BraidingDirection::Clockwise => BraidingDirection::Counterclockwise,
                BraidingDirection::Counterclockwise => BraidingDirection::Clockwise,
            };

            inverse_generators.push(BraidGenerator {
                index: generator.index,
                anyon_indices: generator.anyon_indices,
                direction: inverse_direction,
            });
        }

        Self {
            generators: inverse_generators,
            anyon_count: self.anyon_count,
        }
    }
}

/// Braiding matrix calculator for different anyon types
pub struct BraidingMatrixCalculator {
    anyon_type: NonAbelianAnyonType,
}

impl BraidingMatrixCalculator {
    /// Create a new braiding matrix calculator
    pub const fn new(anyon_type: NonAbelianAnyonType) -> Self {
        Self { anyon_type }
    }

    /// Calculate R-matrix for braiding two charges
    pub fn calculate_r_matrix(
        &self,
        charge1: &TopologicalCharge,
        charge2: &TopologicalCharge,
        fusion_channel: &str,
    ) -> TopologicalResult<Vec<Vec<f64>>> {
        match self.anyon_type {
            NonAbelianAnyonType::Fibonacci => {
                self.fibonacci_r_matrix(charge1, charge2, fusion_channel)
            }
            NonAbelianAnyonType::Ising => self.ising_r_matrix(charge1, charge2, fusion_channel),
            _ => {
                // Default identity matrix for unknown types
                Ok(vec![vec![1.0, 0.0], vec![0.0, 1.0]])
            }
        }
    }

    /// Calculate Fibonacci R-matrix
    fn fibonacci_r_matrix(
        &self,
        charge1: &TopologicalCharge,
        charge2: &TopologicalCharge,
        fusion_channel: &str,
    ) -> TopologicalResult<Vec<Vec<f64>>> {
        let phi = f64::midpoint(1.0, 5.0_f64.sqrt()); // Golden ratio

        match (
            charge1.label.as_str(),
            charge2.label.as_str(),
            fusion_channel,
        ) {
            ("τ", "τ", "I") => {
                // R-matrix for τ × τ → I channel
                let phase = (-4.0 * PI / 5.0).exp();
                Ok(vec![vec![phase, 0.0], vec![0.0, phase]])
            }
            ("τ", "τ", "τ") => {
                // R-matrix for τ × τ → τ channel
                let r11 = (-4.0 * PI / 5.0).exp();
                let r22 = (3.0 * PI / 5.0).exp();
                let off_diagonal = (1.0 / phi).sqrt();

                Ok(vec![vec![r11, off_diagonal], vec![off_diagonal, r22]])
            }
            ("I", _, _) | (_, "I", _) => {
                // Identity braiding
                Ok(vec![vec![1.0, 0.0], vec![0.0, 1.0]])
            }
            _ => Err(TopologicalError::InvalidBraiding(format!(
                "Unknown Fibonacci braiding: {} × {} → {}",
                charge1.label, charge2.label, fusion_channel
            ))),
        }
    }

    /// Calculate Ising R-matrix
    fn ising_r_matrix(
        &self,
        charge1: &TopologicalCharge,
        charge2: &TopologicalCharge,
        fusion_channel: &str,
    ) -> TopologicalResult<Vec<Vec<f64>>> {
        match (
            charge1.label.as_str(),
            charge2.label.as_str(),
            fusion_channel,
        ) {
            ("σ", "σ", "I") => {
                // R-matrix for σ × σ → I channel
                let phase = (PI / 8.0).exp();
                Ok(vec![vec![phase, 0.0], vec![0.0, phase]])
            }
            ("σ", "σ", "ψ") => {
                // R-matrix for σ × σ → ψ channel
                let phase = (PI / 8.0).exp();
                Ok(vec![vec![phase, 0.0], vec![0.0, -phase]])
            }
            ("σ", "ψ", "σ") | ("ψ", "σ", "σ") => {
                // R-matrix for σ × ψ → σ channel
                let phase = (-PI / 2.0).exp();
                Ok(vec![vec![phase, 0.0], vec![0.0, phase]])
            }
            ("I", _, _) | (_, "I", _) => {
                // Identity braiding
                Ok(vec![vec![1.0, 0.0], vec![0.0, 1.0]])
            }
            ("ψ", "ψ", "I") => {
                // Fermion anticommutation
                Ok(vec![vec![-1.0, 0.0], vec![0.0, -1.0]])
            }
            _ => Err(TopologicalError::InvalidBraiding(format!(
                "Unknown Ising braiding: {} × {} → {}",
                charge1.label, charge2.label, fusion_channel
            ))),
        }
    }
}

/// Advanced braiding operations manager
pub struct BraidingOperationManager {
    anyon_type: NonAbelianAnyonType,
    matrix_calculator: BraidingMatrixCalculator,
    operation_history: Vec<BraidingOperation>,
}

impl BraidingOperationManager {
    /// Create a new braiding operation manager
    pub fn new(anyon_type: NonAbelianAnyonType) -> Self {
        Self {
            anyon_type: anyon_type.clone(),
            matrix_calculator: BraidingMatrixCalculator::new(anyon_type),
            operation_history: Vec::new(),
        }
    }

    /// Perform a braiding operation with detailed tracking
    pub fn perform_braiding(
        &mut self,
        anyon1: &Anyon,
        anyon2: &Anyon,
        direction: BraidingDirection,
        braid_count: usize,
        fusion_channel: Option<&str>,
    ) -> TopologicalResult<BraidingResult> {
        // Calculate the braiding result
        let result = self.calculate_braiding_result(
            &anyon1.charge,
            &anyon2.charge,
            &direction,
            braid_count,
            fusion_channel,
        )?;

        // Record the operation
        let operation = BraidingOperation {
            operation_id: self.operation_history.len(),
            anyon1: anyon1.anyon_id,
            anyon2: anyon2.anyon_id,
            direction,
            braid_count,
            result: result.clone(),
            timestamp: 0.0, // Would be set to current time in practice
        };

        self.operation_history.push(operation);
        Ok(result)
    }

    /// Calculate braiding result with matrix computation
    fn calculate_braiding_result(
        &self,
        charge1: &TopologicalCharge,
        charge2: &TopologicalCharge,
        direction: &BraidingDirection,
        braid_count: usize,
        fusion_channel: Option<&str>,
    ) -> TopologicalResult<BraidingResult> {
        let channel = fusion_channel.unwrap_or("I");

        // Get the R-matrix for single braid
        let r_matrix = self
            .matrix_calculator
            .calculate_r_matrix(charge1, charge2, channel)?;

        // For multiple braids, raise the matrix to the power of braid_count
        let final_matrix = self.matrix_power(&r_matrix, braid_count)?;

        // Adjust for braiding direction
        let result_matrix = match direction {
            BraidingDirection::Clockwise => final_matrix,
            BraidingDirection::Counterclockwise => self.matrix_inverse(&final_matrix)?,
        };

        // Extract the braiding phase or return the full matrix
        if result_matrix.len() == 2 && result_matrix[0].len() == 2 {
            // For 2x2 matrices, extract phase from diagonal
            if (result_matrix[0][0] - result_matrix[1][1]).abs() < 1e-10 {
                let phase = 0.0; // For real matrices, imaginary part is 0
                Ok(BraidingResult::Phase(phase))
            } else {
                Ok(BraidingResult::UnitaryMatrix(result_matrix))
            }
        } else {
            Ok(BraidingResult::UnitaryMatrix(result_matrix))
        }
    }

    /// Calculate matrix power
    fn matrix_power(&self, matrix: &[Vec<f64>], power: usize) -> TopologicalResult<Vec<Vec<f64>>> {
        if power == 0 {
            // Return identity matrix
            let size = matrix.len();
            let mut identity = vec![vec![0.0; size]; size];
            for i in 0..size {
                identity[i][i] = 1.0;
            }
            return Ok(identity);
        }

        let mut result = matrix.to_vec();
        for _ in 1..power {
            result = self.matrix_multiply(&result, matrix)?;
        }
        Ok(result)
    }

    /// Matrix multiplication
    fn matrix_multiply(&self, a: &[Vec<f64>], b: &[Vec<f64>]) -> TopologicalResult<Vec<Vec<f64>>> {
        if a[0].len() != b.len() {
            return Err(TopologicalError::InvalidBraiding(
                "Matrix dimensions incompatible for multiplication".to_string(),
            ));
        }

        let rows = a.len();
        let cols = b[0].len();
        let inner = a[0].len();

        let mut result = vec![vec![0.0; cols]; rows];
        for i in 0..rows {
            for j in 0..cols {
                for k in 0..inner {
                    result[i][j] += a[i][k] * b[k][j];
                }
            }
        }
        Ok(result)
    }

    /// Matrix inverse (simplified for 2x2 matrices)
    fn matrix_inverse(&self, matrix: &[Vec<f64>]) -> TopologicalResult<Vec<Vec<f64>>> {
        if matrix.len() != 2 || matrix[0].len() != 2 {
            return Err(TopologicalError::InvalidBraiding(
                "Matrix inverse only implemented for 2x2 matrices".to_string(),
            ));
        }

        let det = matrix[0][0].mul_add(matrix[1][1], -(matrix[0][1] * matrix[1][0]));
        if det.abs() < 1e-10 {
            return Err(TopologicalError::InvalidBraiding(
                "Matrix is singular and cannot be inverted".to_string(),
            ));
        }

        let inv_det = 1.0 / det;
        Ok(vec![
            vec![matrix[1][1] * inv_det, -matrix[0][1] * inv_det],
            vec![-matrix[1][0] * inv_det, matrix[0][0] * inv_det],
        ])
    }

    /// Get braiding operation history
    pub fn get_operation_history(&self) -> &[BraidingOperation] {
        &self.operation_history
    }

    /// Calculate total accumulated phase from all operations
    pub fn calculate_total_phase(&self) -> f64 {
        self.operation_history
            .iter()
            .map(|op| match &op.result {
                BraidingResult::Phase(phase) => *phase,
                _ => 0.0,
            })
            .sum()
    }
}

/// Braid word optimizer for reducing braid complexity
pub struct BraidWordOptimizer {
    anyon_count: usize,
}

impl BraidWordOptimizer {
    /// Create a new braid word optimizer
    pub const fn new(anyon_count: usize) -> Self {
        Self { anyon_count }
    }

    /// Optimize a braid word by removing redundant operations
    pub fn optimize(&self, braid: &BraidGroupElement) -> BraidGroupElement {
        let mut optimized = braid.clone();

        // Remove inverse pairs
        self.remove_inverse_pairs(&mut optimized);

        // Apply braid relations
        self.apply_braid_relations(&mut optimized);

        optimized
    }

    /// Remove adjacent inverse pairs from the braid word
    fn remove_inverse_pairs(&self, braid: &mut BraidGroupElement) {
        let mut i = 0;
        while i < braid.generators.len().saturating_sub(1) {
            let current = &braid.generators[i];
            let next = &braid.generators[i + 1];

            if current.anyon_indices == next.anyon_indices && current.direction != next.direction {
                // Remove the inverse pair
                braid.generators.remove(i);
                braid.generators.remove(i);
                i = i.saturating_sub(1);
            } else {
                i += 1;
            }
        }
    }

    /// Apply braid group relations to simplify the word
    fn apply_braid_relations(&self, braid: &mut BraidGroupElement) {
        // Apply Yang-Baxter relation: σᵢσᵢ₊₁σᵢ = σᵢ₊₁σᵢσᵢ₊₁
        // This is a simplified implementation
        let mut changed = true;
        while changed {
            changed = false;

            for i in 0..braid.generators.len().saturating_sub(2) {
                if self.is_yang_baxter_pattern(&braid.generators[i..i + 3]) {
                    // Apply the relation
                    self.apply_yang_baxter_relation(braid, i);
                    changed = true;
                    break;
                }
            }
        }
    }

    /// Check if a sequence matches Yang-Baxter pattern
    fn is_yang_baxter_pattern(&self, generators: &[BraidGenerator]) -> bool {
        if generators.len() < 3 {
            return false;
        }

        // Check for σᵢσᵢ₊₁σᵢ pattern
        let g1 = &generators[0];
        let g2 = &generators[1];
        let g3 = &generators[2];

        // Adjacent generators and proper ordering
        g1.anyon_indices == g3.anyon_indices
            && g1.direction == g3.direction
            && ((g2.anyon_indices.0 == g1.anyon_indices.1
                && g2.anyon_indices.1 == g1.anyon_indices.1 + 1)
                || (g2.anyon_indices.1 == g1.anyon_indices.0
                    && g2.anyon_indices.0 == g1.anyon_indices.0 - 1))
    }

    /// Apply Yang-Baxter relation transformation
    fn apply_yang_baxter_relation(&self, braid: &mut BraidGroupElement, start_index: usize) {
        // This is a placeholder for the actual Yang-Baxter transformation
        // In practice, this would swap the order according to the relation
        // σᵢσᵢ₊₁σᵢ → σᵢ₊₁σᵢσᵢ₊₁

        if start_index + 2 < braid.generators.len() {
            let g1 = braid.generators[start_index].clone();
            let g2 = braid.generators[start_index + 1].clone();
            let g3 = braid.generators[start_index + 2].clone();

            // Apply the transformation (simplified)
            braid.generators[start_index] = g2.clone();
            braid.generators[start_index + 1] = g1;
            braid.generators[start_index + 2] = g2;
        }
    }
}

#[cfg(test)]
mod tests {
    use super::*;

    #[test]
    fn test_braid_group_element() {
        let mut braid = BraidGroupElement::new(4);

        let generator = BraidGenerator {
            index: 0,
            anyon_indices: (0, 1),
            direction: BraidingDirection::Clockwise,
        };

        assert!(braid.add_generator(generator).is_ok());
        assert_eq!(braid.generators.len(), 1);
    }

    #[test]
    fn test_braid_inverse() {
        let mut braid = BraidGroupElement::new(3);

        let generator = BraidGenerator {
            index: 0,
            anyon_indices: (0, 1),
            direction: BraidingDirection::Clockwise,
        };

        braid
            .add_generator(generator)
            .expect("Generator should be valid for 3-anyon braid");
        let inverse = braid.inverse();

        assert_eq!(inverse.generators.len(), 1);
        assert_eq!(
            inverse.generators[0].direction,
            BraidingDirection::Counterclockwise
        );
    }

    #[test]
    fn test_fibonacci_r_matrix() {
        let calculator = BraidingMatrixCalculator::new(NonAbelianAnyonType::Fibonacci);
        let tau_charge = TopologicalCharge::fibonacci_tau();

        let r_matrix = calculator
            .calculate_r_matrix(&tau_charge, &tau_charge, "I")
            .expect("Fibonacci tau x tau -> I R-matrix should be valid");
        assert_eq!(r_matrix.len(), 2);
        assert_eq!(r_matrix[0].len(), 2);
    }

    #[test]
    fn test_braiding_operation_manager() {
        let mut manager = BraidingOperationManager::new(NonAbelianAnyonType::Fibonacci);

        let anyon1 = Anyon {
            anyon_id: 0,
            charge: TopologicalCharge::fibonacci_tau(),
            position: (0.0, 0.0),
            is_qubit_part: false,
            qubit_id: None,
            creation_time: 0.0,
        };

        let anyon2 = Anyon {
            anyon_id: 1,
            charge: TopologicalCharge::fibonacci_tau(),
            position: (1.0, 0.0),
            is_qubit_part: false,
            qubit_id: None,
            creation_time: 0.0,
        };

        let result =
            manager.perform_braiding(&anyon1, &anyon2, BraidingDirection::Clockwise, 1, Some("I"));

        assert!(result.is_ok());
        assert_eq!(manager.get_operation_history().len(), 1);
    }
}