quantrs2_tytan/

topological_optimization.rs

//! Topological optimization for quantum computing.
//!
//! This module provides topological approaches to optimization including
//! topological quantum computing concepts, anyonic computation, and
//! topological data analysis for optimization.

#![allow(dead_code)]

use scirs2_core::ndarray::{Array2, Array3};
use scirs2_core::random::prelude::*;
use scirs2_core::random::prelude::*;
use scirs2_core::Complex64;
use std::collections::HashMap;
use std::f64::consts::PI;

/// Topological Quantum Optimizer using anyonic braiding
pub struct TopologicalOptimizer {
    /// Number of anyons
    n_anyons: usize,
    /// Anyon type
    anyon_type: AnyonType,
    /// Braiding sequence depth
    braid_depth: usize,
    /// Temperature for thermal anyons
    temperature: f64,
    /// Use error correction
    use_error_correction: bool,
    /// Fusion rules
    fusion_rules: FusionRules,
}

#[derive(Debug, Clone)]
pub enum AnyonType {
    /// Abelian anyons (simple phase)
    Abelian { phase: f64 },
    /// Fibonacci anyons (universal for quantum computation)
    Fibonacci,
    /// Ising anyons (Majorana zero modes)
    Ising,
    /// SU(2)_k anyons
    SU2 { level: usize },
    /// Custom non-abelian anyons
    Custom { name: String, dimension: usize },
}

#[derive(Debug, Clone)]
pub struct FusionRules {
    /// Fusion tensor F^k_{ij}
    fusion_tensor: Array3<f64>,
    /// R-matrix for braiding
    r_matrix: Array2<Complex64>,
    /// Quantum dimensions
    quantum_dimensions: Vec<f64>,
}

impl TopologicalOptimizer {
    /// Create new topological optimizer
    pub fn new(n_anyons: usize, anyon_type: AnyonType) -> Self {
        let fusion_rules = match &anyon_type {
            AnyonType::Fibonacci => FusionRules::fibonacci_rules(),
            AnyonType::Ising => FusionRules::ising_rules(),
            _ => FusionRules::default_rules(),
        };

        Self {
            n_anyons,
            anyon_type,
            braid_depth: 10,
            temperature: 0.1,
            use_error_correction: true,
            fusion_rules,
        }
    }

    /// Set braiding depth
    pub const fn with_braid_depth(mut self, depth: usize) -> Self {
        self.braid_depth = depth;
        self
    }

    /// Set temperature
    pub const fn with_temperature(mut self, temp: f64) -> Self {
        self.temperature = temp;
        self
    }

    /// Optimize using topological braiding
    pub fn optimize(
        &self,
        cost_function: &dyn Fn(&[bool]) -> f64,
    ) -> Result<TopologicalResult, String> {
        let mut best_state = vec![false; self.n_anyons];
        let mut best_cost = f64::INFINITY;
        let mut braid_history = Vec::new();

        // Initialize anyonic state
        let mut anyon_state = self.initialize_anyons()?;

        // Perform braiding optimization
        for iteration in 0..self.braid_depth {
            // Generate braiding sequence
            let braid_sequence = self.generate_braid_sequence(iteration);

            // Apply braiding
            anyon_state = self.apply_braiding(&anyon_state, &braid_sequence)?;

            // Measure and evaluate
            let measured_state = self.measure_anyons(&anyon_state)?;
            let cost = cost_function(&measured_state);

            if cost < best_cost {
                best_cost = cost;
                best_state = measured_state.clone();
            }

            braid_history.push(BraidStep {
                sequence: braid_sequence,
                cost,
                state: measured_state,
            });

            // Apply error correction if enabled
            if self.use_error_correction {
                anyon_state = self.error_correct_anyons(anyon_state)?;
            }
        }

        Ok(TopologicalResult {
            best_state,
            best_cost,
            braid_history,
            topological_invariant: self.compute_topological_invariant(&anyon_state),
        })
    }

    /// Initialize anyonic state
    fn initialize_anyons(&self) -> Result<AnyonState, String> {
        match &self.anyon_type {
            AnyonType::Fibonacci => Ok(AnyonState::Fibonacci(FibonacciState::new(self.n_anyons))),
            AnyonType::Ising => Ok(AnyonState::Ising(IsingAnyonState::new(self.n_anyons))),
            _ => Ok(AnyonState::Generic(GenericAnyonState::new(self.n_anyons))),
        }
    }

    /// Generate braiding sequence
    fn generate_braid_sequence(&self, iteration: usize) -> Vec<BraidOperation> {
        let mut rng = thread_rng();
        let mut sequence = Vec::new();

        // Deterministic part based on iteration
        for i in 0..self.n_anyons - 1 {
            if (iteration + i) % 3 == 0 {
                sequence.push(BraidOperation::Exchange(i, i + 1));
            }
        }

        // Random exploration
        for _ in 0..3 {
            let i = rng.gen_range(0..self.n_anyons - 1);
            sequence.push(BraidOperation::Exchange(i, i + 1));
        }

        // Fusion operations
        if self.n_anyons > 2 && rng.gen_bool(0.3) {
            let i = rng.gen_range(0..self.n_anyons - 1);
            sequence.push(BraidOperation::Fusion(i, i + 1));
        }

        sequence
    }

    /// Apply braiding operations
    fn apply_braiding(
        &self,
        state: &AnyonState,
        sequence: &[BraidOperation],
    ) -> Result<AnyonState, String> {
        let mut current_state = state.clone();

        for operation in sequence {
            current_state = match operation {
                BraidOperation::Exchange(i, j) => self.apply_exchange(&current_state, *i, *j)?,
                BraidOperation::Fusion(i, j) => self.apply_fusion(&current_state, *i, *j)?,
                BraidOperation::Creation(i) => self.apply_creation(&current_state, *i)?,
            };
        }

        Ok(current_state)
    }

    /// Apply exchange (braiding) operation
    fn apply_exchange(&self, state: &AnyonState, i: usize, j: usize) -> Result<AnyonState, String> {
        match state {
            AnyonState::Fibonacci(fib_state) => Ok(AnyonState::Fibonacci(fib_state.braid(i, j)?)),
            AnyonState::Ising(ising_state) => Ok(AnyonState::Ising(ising_state.braid(i, j)?)),
            AnyonState::Generic(gen_state) => Ok(AnyonState::Generic(gen_state.braid(
                i,
                j,
                &self.fusion_rules,
            )?)),
        }
    }

    /// Apply fusion operation
    fn apply_fusion(&self, state: &AnyonState, _i: usize, _j: usize) -> Result<AnyonState, String> {
        // Simplified fusion
        Ok(state.clone())
    }

    /// Apply creation operation
    fn apply_creation(&self, state: &AnyonState, _i: usize) -> Result<AnyonState, String> {
        // Simplified creation
        Ok(state.clone())
    }

    /// Measure anyonic state
    fn measure_anyons(&self, state: &AnyonState) -> Result<Vec<bool>, String> {
        match state {
            AnyonState::Fibonacci(fib_state) => fib_state.measure(),
            AnyonState::Ising(ising_state) => ising_state.measure(),
            AnyonState::Generic(gen_state) => gen_state.measure(),
        }
    }

    /// Error correction for anyonic states
    fn error_correct_anyons(&self, state: AnyonState) -> Result<AnyonState, String> {
        // Topological error correction using stabilizer measurements
        match state {
            AnyonState::Ising(ising_state) => {
                // Majorana parity checks
                Ok(AnyonState::Ising(ising_state.parity_correct()?))
            }
            _ => Ok(state), // Other anyon types don't have error correction yet
        }
    }

    /// Compute topological invariant
    const fn compute_topological_invariant(&self, state: &AnyonState) -> f64 {
        match state {
            AnyonState::Fibonacci(_) => 1.618, // Golden ratio
            AnyonState::Ising(_) => 1.414,     // sqrt(2)
            AnyonState::Generic(_) => 1.0,
        }
    }
}

/// Anyonic state representations
#[derive(Debug, Clone)]
pub enum AnyonState {
    Fibonacci(FibonacciState),
    Ising(IsingAnyonState),
    Generic(GenericAnyonState),
}

/// Fibonacci anyon state
#[derive(Debug, Clone)]
pub struct FibonacciState {
    /// Number of anyons
    n: usize,
    /// Fusion tree amplitudes
    amplitudes: Vec<Complex64>,
    /// Fusion basis labels
    basis_labels: Vec<Vec<usize>>,
}

impl FibonacciState {
    fn new(n: usize) -> Self {
        // Initialize in computational basis
        let num_basis_states = fibonacci(n + 1);
        let amplitudes =
            vec![Complex64::new(1.0 / (num_basis_states as f64).sqrt(), 0.0); num_basis_states];
        let basis_labels = Self::generate_fusion_basis(n);

        Self {
            n,
            amplitudes,
            basis_labels,
        }
    }

    fn generate_fusion_basis(n: usize) -> Vec<Vec<usize>> {
        // Generate valid fusion sequences
        let mut basis = Vec::new();

        // Simplified: just enumerate some valid configurations
        for i in 0..(1 << n) {
            let config: Vec<usize> = (0..n).map(|j| usize::from(i & (1 << j) != 0)).collect();
            basis.push(config);
        }

        basis
    }

    fn braid(&self, i: usize, j: usize) -> Result<Self, String> {
        let mut new_state = self.clone();

        // Apply R-matrix for Fibonacci anyons
        let theta = 2.0 * PI / 5.0; // Pentagon equation solution
        let r_matrix = Complex64::new(theta.cos(), theta.sin());

        for (idx, amplitude) in new_state.amplitudes.iter_mut().enumerate() {
            if self.basis_labels[idx][i] != self.basis_labels[idx][j] {
                *amplitude *= r_matrix;
            }
        }

        Ok(new_state)
    }

    fn measure(&self) -> Result<Vec<bool>, String> {
        let mut rng = thread_rng();

        // Sample from amplitude distribution
        let probabilities: Vec<f64> = self.amplitudes.iter().map(|a| a.norm_sqr()).collect();

        let total_prob: f64 = probabilities.iter().sum();
        let normalized: Vec<f64> = probabilities.iter().map(|p| p / total_prob).collect();

        // Sample basis state
        let mut cumsum = 0.0;
        let r = rng.gen::<f64>();

        for (idx, &prob) in normalized.iter().enumerate() {
            cumsum += prob;
            if r < cumsum {
                return Ok(self.basis_labels[idx].iter().map(|&x| x != 0).collect());
            }
        }

        Ok(vec![false; self.n])
    }
}

/// Ising anyon state (Majorana fermions)
#[derive(Debug, Clone)]
pub struct IsingAnyonState {
    /// Number of Majorana modes
    n: usize,
    /// Majorana operators (antisymmetric matrix)
    majorana_matrix: Array2<f64>,
    /// Parity sectors
    parity_sectors: Vec<bool>,
}

impl IsingAnyonState {
    fn new(n: usize) -> Self {
        let mut majorana_matrix = Array2::zeros((n, n));

        // Initialize with random couplings
        let mut rng = thread_rng();
        for i in 0..n {
            for j in i + 1..n {
                let coupling = rng.gen_range(-1.0..1.0);
                majorana_matrix[[i, j]] = coupling;
                majorana_matrix[[j, i]] = -coupling;
            }
        }

        Self {
            n,
            majorana_matrix,
            parity_sectors: vec![false; n / 2],
        }
    }

    fn braid(&self, i: usize, j: usize) -> Result<Self, String> {
        let mut new_state = self.clone();

        // Braiding Majoranas: γ_i → γ_j, γ_j → -γ_i
        let _phase = Complex64::new(0.0, PI / 4.0).exp();

        // Update Majorana couplings
        for k in 0..self.n {
            if k != i && k != j {
                let temp = new_state.majorana_matrix[[i, k]];
                new_state.majorana_matrix[[i, k]] = new_state.majorana_matrix[[j, k]];
                new_state.majorana_matrix[[j, k]] = -temp;

                new_state.majorana_matrix[[k, i]] = -new_state.majorana_matrix[[i, k]];
                new_state.majorana_matrix[[k, j]] = -new_state.majorana_matrix[[j, k]];
            }
        }

        Ok(new_state)
    }

    fn measure(&self) -> Result<Vec<bool>, String> {
        // Measure fermion parity
        let mut result = Vec::new();

        for i in 0..self.n / 2 {
            result.push(self.parity_sectors[i]);
        }

        // Pad if necessary
        while result.len() < self.n {
            result.push(false);
        }

        Ok(result)
    }

    fn parity_correct(&self) -> Result<Self, String> {
        let mut corrected = self.clone();

        // Check parity conservation
        let total_parity = self.parity_sectors.iter().filter(|&&p| p).count() % 2;

        if total_parity != 0 {
            // Flip a random parity sector to restore conservation
            let mut rng = thread_rng();
            let idx = rng.gen_range(0..self.parity_sectors.len());
            corrected.parity_sectors[idx] = !corrected.parity_sectors[idx];
        }

        Ok(corrected)
    }
}

/// Generic anyon state
#[derive(Debug, Clone)]
pub struct GenericAnyonState {
    n: usize,
    state_vector: Vec<Complex64>,
}

impl GenericAnyonState {
    fn new(n: usize) -> Self {
        let dim = 1 << n;
        let state_vector = vec![Complex64::new(1.0 / (dim as f64).sqrt(), 0.0); dim];

        Self { n, state_vector }
    }

    fn braid(&self, i: usize, j: usize, rules: &FusionRules) -> Result<Self, String> {
        let mut new_state = self.clone();

        // Apply R-matrix from fusion rules
        let r_element = rules.r_matrix[[i.min(j), i.max(j)]];

        for amplitude in &mut new_state.state_vector {
            *amplitude *= r_element;
        }

        Ok(new_state)
    }

    fn measure(&self) -> Result<Vec<bool>, String> {
        let mut rng = StdRng::seed_from_u64(42);
        let probabilities: Vec<f64> = self.state_vector.iter().map(|a| a.norm_sqr()).collect();

        let idx = weighted_sample(&probabilities, &mut rng);

        Ok((0..self.n).map(|i| idx & (1 << i) != 0).collect())
    }
}

impl FusionRules {
    fn fibonacci_rules() -> Self {
        // Fibonacci anyon fusion rules: 1 × 1 = 0 + 1
        let mut fusion_tensor = Array3::zeros((2, 2, 2));
        fusion_tensor[[0, 0, 0]] = 1.0; // 0 × 0 = 0
        fusion_tensor[[0, 1, 1]] = 1.0; // 0 × 1 = 1
        fusion_tensor[[1, 0, 1]] = 1.0; // 1 × 0 = 1
        fusion_tensor[[1, 1, 0]] = 1.0; // 1 × 1 = 0
        fusion_tensor[[1, 1, 1]] = 1.0; // 1 × 1 = 1

        let phi = f64::midpoint(1.0, 5.0_f64.sqrt()); // Golden ratio
        let r_matrix = Array2::from_shape_fn((2, 2), |(i, j)| {
            if i == 0 && j == 0 {
                Complex64::new(1.0, 0.0)
            } else if i == 1 && j == 1 {
                Complex64::new((-1.0 / phi).cos(), (-1.0 / phi).sin())
            } else {
                Complex64::new(0.0, 0.0)
            }
        });

        Self {
            fusion_tensor,
            r_matrix,
            quantum_dimensions: vec![1.0, phi],
        }
    }

    fn ising_rules() -> Self {
        // Ising anyon fusion rules
        let mut fusion_tensor = Array3::zeros((3, 3, 3));
        fusion_tensor[[0, 0, 0]] = 1.0; // 1 × 1 = 1
        fusion_tensor[[0, 1, 1]] = 1.0; // 1 × σ = σ
        fusion_tensor[[0, 2, 2]] = 1.0; // 1 × ψ = ψ
        fusion_tensor[[1, 0, 1]] = 1.0; // σ × 1 = σ
        fusion_tensor[[1, 1, 0]] = 1.0; // σ × σ = 1
        fusion_tensor[[1, 1, 2]] = 1.0; // σ × σ = ψ
        fusion_tensor[[2, 0, 2]] = 1.0; // ψ × 1 = ψ
        fusion_tensor[[2, 1, 1]] = 1.0; // ψ × σ = σ
        fusion_tensor[[2, 2, 0]] = 1.0; // ψ × ψ = 1

        let sqrt2 = 2.0_f64.sqrt();
        let r_matrix = Array2::from_shape_fn((3, 3), |(i, j)| match (i, j) {
            (0, 0) => Complex64::new(1.0, 0.0),
            (1, 1) => Complex64::new(0.0, 1.0) / sqrt2,
            (2, 2) => Complex64::new(-1.0, 0.0),
            _ => Complex64::new(0.0, 0.0),
        });

        Self {
            fusion_tensor,
            r_matrix,
            quantum_dimensions: vec![1.0, sqrt2, 1.0],
        }
    }

    fn default_rules() -> Self {
        // Create a 2x2x2 identity-like fusion tensor
        let mut fusion_tensor = Array3::zeros((2, 2, 2));
        fusion_tensor[[0, 0, 0]] = 1.0; // 1 × 1 = 1
        fusion_tensor[[1, 1, 0]] = 1.0; // σ × σ = 1

        let r_matrix = Array2::<f64>::eye(2).mapv(Complex64::from);

        Self {
            fusion_tensor,
            r_matrix,
            quantum_dimensions: vec![1.0; 2],
        }
    }
}

#[derive(Debug, Clone)]
pub enum BraidOperation {
    Exchange(usize, usize),
    Fusion(usize, usize),
    Creation(usize),
}

#[derive(Debug, Clone)]
pub struct BraidStep {
    sequence: Vec<BraidOperation>,
    cost: f64,
    state: Vec<bool>,
}

#[derive(Debug, Clone)]
pub struct TopologicalResult {
    pub best_state: Vec<bool>,
    pub best_cost: f64,
    pub braid_history: Vec<BraidStep>,
    pub topological_invariant: f64,
}

/// Persistent homology for optimization landscape analysis
pub struct PersistentHomology {
    /// Maximum dimension to compute
    max_dimension: usize,
    /// Filtration type
    filtration: FiltrationType,
    /// Resolution for discretization
    resolution: f64,
}

#[derive(Debug, Clone)]
pub enum FiltrationType {
    /// Sublevel set filtration
    Sublevel,
    /// Vietoris-Rips filtration
    VietorisRips,
    /// Alpha complex
    AlphaComplex,
    /// Cubical complex
    Cubical,
}

impl PersistentHomology {
    /// Create new persistent homology analyzer
    pub const fn new(max_dimension: usize) -> Self {
        Self {
            max_dimension,
            filtration: FiltrationType::Sublevel,
            resolution: 0.01,
        }
    }

    /// Set filtration type
    pub const fn with_filtration(mut self, filtration: FiltrationType) -> Self {
        self.filtration = filtration;
        self
    }

    /// Analyze optimization landscape
    pub fn analyze_landscape(
        &self,
        samples: &[(Vec<bool>, f64)],
    ) -> Result<HomologyResult, String> {
        // Build filtration
        let filtration = self.build_filtration(samples)?;

        // Compute persistent homology
        let persistence_pairs = self.compute_persistence(&filtration)?;

        // Extract features
        let features = self.extract_topological_features(&persistence_pairs);

        let betti_numbers = self.compute_betti_numbers(&persistence_pairs);
        let optimal_regions = self.identify_optimal_regions(&persistence_pairs, samples);

        Ok(HomologyResult {
            persistence_pairs,
            betti_numbers,
            features,
            optimal_regions,
        })
    }

    /// Build filtration from samples
    fn build_filtration(&self, samples: &[(Vec<bool>, f64)]) -> Result<Filtration, String> {
        match self.filtration {
            FiltrationType::Sublevel => self.build_sublevel_filtration(samples),
            FiltrationType::VietorisRips => self.build_vietoris_rips(samples),
            _ => Err("Filtration type not implemented".to_string()),
        }
    }

    /// Build sublevel set filtration
    fn build_sublevel_filtration(
        &self,
        samples: &[(Vec<bool>, f64)],
    ) -> Result<Filtration, String> {
        let mut simplices = Vec::new();

        // Sort samples by cost
        let mut sorted_samples = samples.to_vec();
        sorted_samples.sort_by(|a, b| a.1.partial_cmp(&b.1).unwrap_or(std::cmp::Ordering::Equal));

        // Add vertices
        for (idx, (_, cost)) in sorted_samples.iter().enumerate() {
            simplices.push(Simplex {
                vertices: vec![idx],
                filtration_value: *cost,
                dimension: 0,
            });
        }

        // Add edges between nearby points
        for i in 0..sorted_samples.len() {
            for j in i + 1..sorted_samples.len() {
                let dist = hamming_distance(&sorted_samples[i].0, &sorted_samples[j].0);
                if dist == 1 {
                    simplices.push(Simplex {
                        vertices: vec![i, j],
                        filtration_value: sorted_samples[j].1.max(sorted_samples[i].1),
                        dimension: 1,
                    });
                }
            }
        }

        Ok(Filtration { simplices })
    }

    /// Build Vietoris-Rips complex
    fn build_vietoris_rips(&self, samples: &[(Vec<bool>, f64)]) -> Result<Filtration, String> {
        let mut simplices = Vec::new();
        let n = samples.len();

        // Compute pairwise distances
        let mut distances = Array2::zeros((n, n));
        for i in 0..n {
            for j in i + 1..n {
                let dist = hamming_distance(&samples[i].0, &samples[j].0) as f64;
                distances[[i, j]] = dist;
                distances[[j, i]] = dist;
            }
        }

        // Add vertices
        for i in 0..n {
            simplices.push(Simplex {
                vertices: vec![i],
                filtration_value: 0.0,
                dimension: 0,
            });
        }

        // Add higher dimensional simplices
        for dim in 1..=self.max_dimension {
            self.add_simplices_of_dimension(&mut simplices, &distances, dim);
        }

        Ok(Filtration { simplices })
    }

    /// Add simplices of given dimension
    fn add_simplices_of_dimension(
        &self,
        simplices: &mut Vec<Simplex>,
        distances: &Array2<f64>,
        dim: usize,
    ) {
        // Generate all possible simplices of dimension dim
        // This is simplified - proper implementation would use combinatorial enumeration
        let n = distances.shape()[0];

        if dim == 1 {
            // Add edges
            for i in 0..n {
                for j in i + 1..n {
                    simplices.push(Simplex {
                        vertices: vec![i, j],
                        filtration_value: distances[[i, j]],
                        dimension: 1,
                    });
                }
            }
        }
    }

    /// Compute persistent homology
    fn compute_persistence(&self, filtration: &Filtration) -> Result<Vec<PersistencePair>, String> {
        // Simplified persistence computation
        let mut pairs = Vec::new();

        // Group simplices by dimension
        let mut by_dimension: HashMap<usize, Vec<&Simplex>> = HashMap::new();
        for simplex in &filtration.simplices {
            by_dimension
                .entry(simplex.dimension)
                .or_default()
                .push(simplex);
        }

        // Compute persistence for each dimension
        for dim in 0..=self.max_dimension {
            if let Some(simplices) = by_dimension.get(&dim) {
                // Simplified: create persistence pairs
                for (i, simplex) in simplices.iter().enumerate() {
                    if i % 2 == 0 && i + 1 < simplices.len() {
                        pairs.push(PersistencePair {
                            dimension: dim,
                            birth: simplex.filtration_value,
                            death: simplices[i + 1].filtration_value,
                        });
                    }
                }
            }
        }

        Ok(pairs)
    }

    /// Extract topological features
    fn extract_topological_features(&self, pairs: &[PersistencePair]) -> TopologicalFeatures {
        let mut features = TopologicalFeatures {
            total_persistence: 0.0,
            max_persistence: vec![0.0; self.max_dimension + 1],
            persistence_entropy: 0.0,
            landscape: Vec::new(),
        };

        // Compute total persistence
        for pair in pairs {
            let persistence = pair.death - pair.birth;
            features.total_persistence += persistence;
            features.max_persistence[pair.dimension] =
                features.max_persistence[pair.dimension].max(persistence);
        }

        // Compute persistence entropy
        if !pairs.is_empty() {
            let total = features.total_persistence;
            features.persistence_entropy = pairs
                .iter()
                .map(|p| {
                    let persistence = p.death - p.birth;
                    let prob = persistence / total;
                    if prob > 0.0 {
                        -prob * prob.ln()
                    } else {
                        0.0
                    }
                })
                .sum();
        }

        // Compute persistence landscape
        features.landscape = self.compute_persistence_landscape(pairs);

        features
    }

    /// Compute persistence landscape
    fn compute_persistence_landscape(&self, pairs: &[PersistencePair]) -> Vec<Vec<f64>> {
        // Simplified persistence landscape
        let resolution = 100;
        let mut landscape = vec![vec![0.0; resolution]; self.max_dimension + 1];

        for pair in pairs {
            let start = (pair.birth * resolution as f64) as usize;
            let end = (pair.death * resolution as f64) as usize;

            for i in start..end.min(resolution) {
                landscape[pair.dimension][i] = f64::max(
                    landscape[pair.dimension][i],
                    1.0 - (i as f64 / resolution as f64 - pair.birth).abs(),
                );
            }
        }

        landscape
    }

    /// Compute Betti numbers
    fn compute_betti_numbers(&self, pairs: &[PersistencePair]) -> Vec<usize> {
        let mut betti = vec![0; self.max_dimension + 1];

        for pair in pairs {
            if pair.death - pair.birth > self.resolution {
                betti[pair.dimension] += 1;
            }
        }

        betti
    }

    /// Identify optimal regions
    fn identify_optimal_regions(
        &self,
        pairs: &[PersistencePair],
        samples: &[(Vec<bool>, f64)],
    ) -> Vec<OptimalRegion> {
        let mut regions = Vec::new();

        // Find persistent 0-dimensional features (connected components)
        let persistent_components: Vec<_> = pairs
            .iter()
            .filter(|p| p.dimension == 0 && p.death - p.birth > self.resolution)
            .collect();

        for component in persistent_components {
            // Find samples in this component
            let component_samples: Vec<_> = samples
                .iter()
                .filter(|(_, cost)| *cost >= component.birth && *cost < component.death)
                .collect();

            if !component_samples.is_empty() {
                let min_cost = component_samples
                    .iter()
                    .map(|(_, cost)| *cost)
                    .fold(f64::INFINITY, f64::min);

                regions.push(OptimalRegion {
                    persistence: component.death - component.birth,
                    min_cost,
                    size: component_samples.len(),
                    representative: component_samples[0].0.clone(),
                });
            }
        }

        regions.sort_by(|a, b| {
            a.min_cost
                .partial_cmp(&b.min_cost)
                .unwrap_or(std::cmp::Ordering::Equal)
        });
        regions
    }
}

#[derive(Debug, Clone)]
struct Filtration {
    simplices: Vec<Simplex>,
}

#[derive(Debug, Clone)]
struct Simplex {
    vertices: Vec<usize>,
    filtration_value: f64,
    dimension: usize,
}

#[derive(Debug, Clone)]
pub struct PersistencePair {
    dimension: usize,
    birth: f64,
    death: f64,
}

#[derive(Debug, Clone)]
pub struct TopologicalFeatures {
    pub total_persistence: f64,
    pub max_persistence: Vec<f64>,
    pub persistence_entropy: f64,
    pub landscape: Vec<Vec<f64>>,
}

#[derive(Debug, Clone)]
pub struct OptimalRegion {
    pub persistence: f64,
    pub min_cost: f64,
    pub size: usize,
    pub representative: Vec<bool>,
}

#[derive(Debug, Clone)]
pub struct HomologyResult {
    pub persistence_pairs: Vec<PersistencePair>,
    pub betti_numbers: Vec<usize>,
    pub features: TopologicalFeatures,
    pub optimal_regions: Vec<OptimalRegion>,
}

/// Helper functions
fn fibonacci(n: usize) -> usize {
    if n <= 1 {
        n
    } else {
        fibonacci(n - 1) + fibonacci(n - 2)
    }
}

fn hamming_distance(a: &[bool], b: &[bool]) -> usize {
    a.iter().zip(b.iter()).filter(|(&x, &y)| x != y).count()
}

fn weighted_sample(weights: &[f64], rng: &mut StdRng) -> usize {
    let total: f64 = weights.iter().sum();
    let mut cumsum = 0.0;
    let r = rng.gen::<f64>() * total;

    for (idx, &weight) in weights.iter().enumerate() {
        cumsum += weight;
        if r < cumsum {
            return idx;
        }
    }

    weights.len() - 1
}

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

    #[test]
    fn test_topological_optimizer() {
        let optimizer = TopologicalOptimizer::new(4, AnyonType::Fibonacci);

        let cost_fn = |state: &[bool]| state.iter().filter(|&&x| x).count() as f64;

        let mut result = optimizer.optimize(&cost_fn);
        assert!(result.is_ok());
    }

    #[test]
    fn test_fibonacci_state() {
        let mut state = FibonacciState::new(3);
        assert_eq!(state.n, 3);

        let braided = state.braid(0, 1);
        assert!(braided.is_ok());
    }

    #[test]
    fn test_persistent_homology() {
        let ph = PersistentHomology::new(2);

        let mut samples = vec![
            (vec![false, false], 0.0),
            (vec![true, false], 1.0),
            (vec![false, true], 1.0),
            (vec![true, true], 2.0),
        ];

        let mut result = ph.analyze_landscape(&samples);
        assert!(result.is_ok());

        let homology = result.expect("Homology analysis should succeed");
        assert!(!homology.persistence_pairs.is_empty());
    }
}