sklears_kernel_approximation/

quantum_kernel_methods.rs

//! Quantum Kernel Methods and Quantum-Inspired Approximations
//!
//! This module implements quantum kernel approximations and quantum-inspired
//! classical algorithms for kernel methods. These methods simulate quantum feature
//! maps using classical computation while providing theoretical quantum advantage insights.
//!
//! # References
//! - Havlicek et al. (2019): "Supervised learning with quantum-enhanced feature spaces"
//! - Schuld & Killoran (2019): "Quantum Machine Learning in Feature Hilbert Spaces"
//! - Liu et al. (2021): "Rigorous Guarantees for Quantum Computational Advantage"
//! - Huang et al. (2021): "Power of data in quantum machine learning"

use scirs2_core::ndarray::{Array1, Array2};
use scirs2_core::random::essentials::Normal;
use scirs2_core::random::thread_rng;
use serde::{Deserialize, Serialize};
use sklears_core::{
    error::{Result, SklearsError},
    prelude::{Fit, Transform},
    traits::{Estimator, Trained, Untrained},
    types::Float,
};
use std::f64::consts::PI;
use std::marker::PhantomData;

/// Quantum feature map types
#[derive(Debug, Clone, Copy, Serialize, Deserialize, PartialEq)]
pub enum QuantumFeatureMap {
    /// Pauli-Z evolution feature map
    PauliZ,
    /// Pauli-ZZ entangling feature map
    PauliZZ,
    /// General Pauli feature map with X, Y, Z rotations
    GeneralPauli,
    /// Amplitude encoding feature map
    AmplitudeEncoding,
    /// Hamiltonian evolution feature map
    HamiltonianEvolution,
}

/// Configuration for quantum kernel approximation
#[derive(Debug, Clone, Serialize, Deserialize)]
pub struct QuantumKernelConfig {
    /// Quantum feature map to use
    pub feature_map: QuantumFeatureMap,
    /// Number of qubits (simulated)
    pub n_qubits: usize,
    /// Circuit depth (number of repetitions)
    pub circuit_depth: usize,
    /// Entangling configuration
    pub entanglement: EntanglementPattern,
    /// Number of classical samples for approximation
    pub n_samples: usize,
}

impl Default for QuantumKernelConfig {
    fn default() -> Self {
        Self {
            feature_map: QuantumFeatureMap::PauliZZ,
            n_qubits: 4,
            circuit_depth: 2,
            entanglement: EntanglementPattern::Linear,
            n_samples: 1000,
        }
    }
}

/// Entanglement patterns for quantum circuits
#[derive(Debug, Clone, Copy, Serialize, Deserialize, PartialEq)]
pub enum EntanglementPattern {
    /// No entanglement (product state)
    None,
    /// Linear nearest-neighbor entanglement
    Linear,
    /// Circular entanglement
    Circular,
    /// All-to-all entanglement
    AllToAll,
}

/// Quantum Kernel Approximation
///
/// Simulates quantum feature maps using classical computation to approximate
/// quantum kernels. The quantum kernel is defined as:
///
/// K(x, x') = |⟨φ(x)|φ(x')⟩|²
///
/// where |φ(x)⟩ is a quantum feature map encoding classical data x.
///
/// # Mathematical Background
///
/// Quantum feature maps encode classical data into quantum states:
/// - Pauli-Z: U(x) = exp(-i Σ_j x_j Z_j)
/// - Pauli-ZZ: U(x) = exp(-i Σ_{j,k} (π - x_j)(π - x_k) Z_j Z_k)
/// - Amplitude: |ψ⟩ = Σ_i √(x_i/||x||) |i⟩
///
/// The resulting kernel often provides exponential feature space dimension
/// advantages over classical kernels for certain problems.
///
/// # Examples
///
/// ```rust,ignore
/// use sklears_kernel_approximation::quantum_kernel_methods::{QuantumKernelApproximation, QuantumKernelConfig};
/// use scirs2_core::ndarray::array;
/// use sklears_core::traits::{Fit, Transform};
///
/// let config = QuantumKernelConfig::default();
/// let qkernel = QuantumKernelApproximation::new(config);
///
/// let X = array![[1.0, 2.0], [3.0, 4.0]];
/// let fitted = qkernel.fit(&X, &()).unwrap();
/// let features = fitted.transform(&X).unwrap();
/// ```
#[derive(Debug, Clone)]
pub struct QuantumKernelApproximation<State = Untrained> {
    config: QuantumKernelConfig,

    // Fitted attributes
    x_train: Option<Array2<Float>>,
    feature_basis: Option<Array2<Float>>,

    _state: PhantomData<State>,
}

// Common methods for all states
impl<State> QuantumKernelApproximation<State> {
    /// Simulate quantum feature map
    /// Returns a classical approximation of the quantum state's amplitude
    fn simulate_feature_map(&self, x: &Array1<Float>) -> Vec<Float> {
        match self.config.feature_map {
            QuantumFeatureMap::PauliZ => self.pauli_z_feature_map(x),
            QuantumFeatureMap::PauliZZ => self.pauli_zz_feature_map(x),
            QuantumFeatureMap::GeneralPauli => self.general_pauli_feature_map(x),
            QuantumFeatureMap::AmplitudeEncoding => self.amplitude_encoding_feature_map(x),
            QuantumFeatureMap::HamiltonianEvolution => self.hamiltonian_evolution_feature_map(x),
        }
    }

    /// Pauli-Z feature map: exp(-i Σ x_j Z_j)
    fn pauli_z_feature_map(&self, x: &Array1<Float>) -> Vec<Float> {
        let dim = 1 << self.config.n_qubits; // 2^n_qubits
        let mut amplitudes = vec![0.0; dim];

        // Start with |0...0⟩ state
        amplitudes[0] = 1.0;

        // Apply Pauli-Z rotations
        for depth in 0..self.config.circuit_depth {
            let mut new_amplitudes = amplitudes.clone();

            for qubit in 0..self.config.n_qubits.min(x.len()) {
                let feature_idx = (qubit + depth * self.config.n_qubits) % x.len();
                let angle = x[feature_idx];

                // Apply Z rotation: diagonal in computational basis
                for state in 0..dim {
                    if (state >> qubit) & 1 == 1 {
                        // Qubit is in |1⟩ state
                        new_amplitudes[state] *= (-angle).cos();
                    } else {
                        // Qubit is in |0⟩ state
                        new_amplitudes[state] *= angle.cos();
                    }
                }
            }

            amplitudes = new_amplitudes;
        }

        amplitudes
    }

    /// Pauli-ZZ feature map with entanglement
    fn pauli_zz_feature_map(&self, x: &Array1<Float>) -> Vec<Float> {
        let dim = 1 << self.config.n_qubits;
        let mut amplitudes = vec![0.0; dim];
        amplitudes[0] = 1.0;

        for _depth in 0..self.config.circuit_depth {
            let mut new_amplitudes = amplitudes.clone();

            // Apply entangling ZZ gates
            let pairs = self.get_entangling_pairs();
            for (q1, q2) in pairs {
                if q1 < x.len() && q2 < x.len() {
                    let angle = (PI - x[q1]) * (PI - x[q2]);

                    for state in 0..dim {
                        let bit1 = (state >> q1) & 1;
                        let bit2 = (state >> q2) & 1;

                        // ZZ interaction: phase depends on both qubits
                        let phase = if bit1 == bit2 { 1.0 } else { -1.0 };
                        new_amplitudes[state] *= phase * angle.cos();
                    }
                }
            }

            amplitudes = new_amplitudes;
        }

        // Normalize
        let norm: Float = amplitudes.iter().map(|a| a * a).sum::<Float>().sqrt();
        if norm > 1e-10 {
            amplitudes.iter_mut().for_each(|a| *a /= norm);
        }

        amplitudes
    }

    /// General Pauli feature map with X, Y, Z rotations
    fn general_pauli_feature_map(&self, x: &Array1<Float>) -> Vec<Float> {
        let dim = 1 << self.config.n_qubits;
        let mut amplitudes = vec![0.0; dim];
        amplitudes[0] = 1.0;

        for depth in 0..self.config.circuit_depth {
            for qubit in 0..self.config.n_qubits.min(x.len()) {
                let feature_idx = (qubit + depth * self.config.n_qubits) % x.len();
                let angle = x[feature_idx];

                // Apply Hadamard-like mixing (simplified)
                let cos_half = (angle / 2.0).cos();
                let sin_half = (angle / 2.0).sin();

                let mut new_amplitudes = vec![0.0; dim];
                for state in 0..dim {
                    let flipped_state = state ^ (1 << qubit);

                    new_amplitudes[state] += amplitudes[state] * cos_half;
                    new_amplitudes[state] += amplitudes[flipped_state] * sin_half;
                }

                amplitudes = new_amplitudes;
            }
        }

        // Normalize
        let norm: Float = amplitudes.iter().map(|a| a * a).sum::<Float>().sqrt();
        if norm > 1e-10 {
            amplitudes.iter_mut().for_each(|a| *a /= norm);
        }

        amplitudes
    }

    /// Amplitude encoding feature map
    fn amplitude_encoding_feature_map(&self, x: &Array1<Float>) -> Vec<Float> {
        let dim = 1 << self.config.n_qubits;
        let mut amplitudes = vec![0.0; dim];

        // Encode data as amplitudes (normalized)
        let norm: Float = x.iter().map(|v| v * v).sum::<Float>().sqrt().max(1e-10);

        for (i, &val) in x.iter().enumerate().take(dim) {
            amplitudes[i] = val / norm;
        }

        amplitudes
    }

    /// Hamiltonian evolution feature map
    fn hamiltonian_evolution_feature_map(&self, x: &Array1<Float>) -> Vec<Float> {
        // Simplified Hamiltonian evolution
        // H = Σ_i x_i Z_i + Σ_{i,j} x_i x_j Z_i Z_j
        let dim = 1 << self.config.n_qubits;
        let mut amplitudes = vec![0.0; dim];
        amplitudes[0] = 1.0;

        let evolution_time = 1.0;

        for state in 0..dim {
            let mut energy = 0.0;

            // Single-qubit terms
            for qubit in 0..self.config.n_qubits.min(x.len()) {
                let bit = (state >> qubit) & 1;
                let z_eigenvalue = if bit == 1 { -1.0 } else { 1.0 };
                energy += x[qubit] * z_eigenvalue;
            }

            // Two-qubit terms (simplified)
            let pairs = self.get_entangling_pairs();
            for (q1, q2) in pairs.iter().take(3) {
                // Limit for performance
                if *q1 < x.len() && *q2 < x.len() {
                    let bit1 = (state >> q1) & 1;
                    let bit2 = (state >> q2) & 1;
                    let zz_eigenvalue = if bit1 == bit2 { 1.0 } else { -1.0 };
                    energy += 0.1 * x[*q1] * x[*q2] * zz_eigenvalue;
                }
            }

            amplitudes[state] = (-energy * evolution_time).exp();
        }

        // Normalize
        let norm: Float = amplitudes.iter().map(|a| a * a).sum::<Float>().sqrt();
        if norm > 1e-10 {
            amplitudes.iter_mut().for_each(|a| *a /= norm);
        }

        amplitudes
    }

    /// Get entangling pairs based on pattern
    fn get_entangling_pairs(&self) -> Vec<(usize, usize)> {
        let n = self.config.n_qubits;
        match self.config.entanglement {
            EntanglementPattern::None => vec![],
            EntanglementPattern::Linear => (0..n.saturating_sub(1)).map(|i| (i, i + 1)).collect(),
            EntanglementPattern::Circular => {
                let mut pairs: Vec<_> = (0..n.saturating_sub(1)).map(|i| (i, i + 1)).collect();
                if n > 2 {
                    pairs.push((n - 1, 0));
                }
                pairs
            }
            EntanglementPattern::AllToAll => {
                let mut pairs = Vec::new();
                for i in 0..n {
                    for j in (i + 1)..n {
                        pairs.push((i, j));
                    }
                }
                pairs
            }
        }
    }
}

impl QuantumKernelApproximation<Untrained> {
    /// Create a new quantum kernel approximation
    pub fn new(config: QuantumKernelConfig) -> Self {
        Self {
            config,
            x_train: None,
            feature_basis: None,
            _state: PhantomData,
        }
    }

    /// Create with default configuration
    pub fn with_qubits(n_qubits: usize) -> Self {
        Self {
            config: QuantumKernelConfig {
                n_qubits,
                ..Default::default()
            },
            x_train: None,
            feature_basis: None,
            _state: PhantomData,
        }
    }

    /// Set feature map type
    pub fn feature_map(mut self, feature_map: QuantumFeatureMap) -> Self {
        self.config.feature_map = feature_map;
        self
    }

    /// Set circuit depth
    pub fn circuit_depth(mut self, depth: usize) -> Self {
        self.config.circuit_depth = depth;
        self
    }

    /// Set entanglement pattern
    pub fn entanglement(mut self, pattern: EntanglementPattern) -> Self {
        self.config.entanglement = pattern;
        self
    }
}

impl Estimator for QuantumKernelApproximation<Untrained> {
    type Config = QuantumKernelConfig;
    type Error = SklearsError;
    type Float = Float;

    fn config(&self) -> &Self::Config {
        &self.config
    }
}

impl Fit<Array2<Float>, ()> for QuantumKernelApproximation<Untrained> {
    type Fitted = QuantumKernelApproximation<Trained>;

    fn fit(self, x: &Array2<Float>, _y: &()) -> Result<Self::Fitted> {
        if x.nrows() == 0 || x.ncols() == 0 {
            return Err(SklearsError::InvalidInput(
                "Input array cannot be empty".to_string(),
            ));
        }

        let x_train = x.clone();

        // Generate random feature basis for quantum kernel approximation
        // This uses random projections similar to classical Random Fourier Features
        // but inspired by quantum sampling
        let mut rng = thread_rng();
        let normal = Normal::new(0.0, 1.0).unwrap();

        let feature_dim = (1 << self.config.n_qubits).min(self.config.n_samples);
        let feature_basis = Array2::from_shape_fn((x.ncols(), feature_dim), |_| rng.sample(normal));

        Ok(QuantumKernelApproximation {
            config: self.config,
            x_train: Some(x_train),
            feature_basis: Some(feature_basis),
            _state: PhantomData,
        })
    }
}

impl Transform<Array2<Float>, Array2<Float>> for QuantumKernelApproximation<Trained> {
    fn transform(&self, x: &Array2<Float>) -> Result<Array2<Float>> {
        let feature_basis = self.feature_basis.as_ref().unwrap();

        if x.ncols() != feature_basis.nrows() {
            return Err(SklearsError::InvalidInput(format!(
                "Feature dimension mismatch: expected {}, got {}",
                feature_basis.nrows(),
                x.ncols()
            )));
        }

        let n_samples = x.nrows();
        let n_features = feature_basis.ncols();
        let mut output = Array2::zeros((n_samples, n_features));

        // Apply quantum-inspired feature transformation
        for i in 0..n_samples {
            let sample = x.row(i).to_owned();

            // Simulate quantum feature map (simplified for performance)
            let quantum_features = self.simulate_feature_map(&sample);

            // Project onto random basis
            let projection = sample.dot(feature_basis);

            for j in 0..n_features {
                // Combine quantum simulation with classical projection
                let quantum_component = if j < quantum_features.len() {
                    quantum_features[j]
                } else {
                    0.0
                };

                output[[i, j]] =
                    (projection[j] + quantum_component).cos() / (n_features as Float).sqrt();
            }
        }

        Ok(output)
    }
}

impl QuantumKernelApproximation<Trained> {
    /// Get the training data
    pub fn x_train(&self) -> &Array2<Float> {
        self.x_train.as_ref().unwrap()
    }

    /// Get the feature basis
    pub fn feature_basis(&self) -> &Array2<Float> {
        self.feature_basis.as_ref().unwrap()
    }

    /// Compute quantum kernel matrix explicitly (slow, for small datasets)
    pub fn compute_kernel_matrix(&self, x: &Array2<Float>) -> Array2<Float> {
        let n = x.nrows();
        let mut kernel = Array2::zeros((n, n));

        for i in 0..n {
            for j in i..n {
                let features_i = self.simulate_feature_map(&x.row(i).to_owned());
                let features_j = self.simulate_feature_map(&x.row(j).to_owned());

                // Inner product of quantum states (overlap)
                let overlap: Float = features_i
                    .iter()
                    .zip(features_j.iter())
                    .map(|(a, b)| a * b)
                    .sum();

                kernel[[i, j]] = overlap.abs();
                kernel[[j, i]] = overlap.abs();
            }
        }

        kernel
    }
}

#[cfg(test)]
mod tests {
    use super::*;
    use scirs2_core::ndarray::array;

    #[test]
    fn test_quantum_kernel_basic() {
        let config = QuantumKernelConfig {
            n_qubits: 3,
            circuit_depth: 1,
            feature_map: QuantumFeatureMap::PauliZ,
            entanglement: EntanglementPattern::Linear,
            n_samples: 50,
        };

        let qkernel = QuantumKernelApproximation::new(config);
        let x = array![[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]];

        let fitted = qkernel.fit(&x, &()).unwrap();
        let features = fitted.transform(&x).unwrap();

        assert_eq!(features.nrows(), 2);
        assert!(features.ncols() > 0);
    }

    #[test]
    fn test_different_feature_maps() {
        let feature_maps = vec![
            QuantumFeatureMap::PauliZ,
            QuantumFeatureMap::PauliZZ,
            QuantumFeatureMap::GeneralPauli,
            QuantumFeatureMap::AmplitudeEncoding,
            QuantumFeatureMap::HamiltonianEvolution,
        ];

        let x = array![[1.0, 2.0], [3.0, 4.0]];

        for feature_map in feature_maps {
            let qkernel = QuantumKernelApproximation::with_qubits(2).feature_map(feature_map);

            let fitted = qkernel.fit(&x, &()).unwrap();
            let features = fitted.transform(&x).unwrap();

            assert_eq!(features.nrows(), 2);
            assert!(features.iter().all(|&v| v.is_finite()));
        }
    }

    #[test]
    fn test_different_entanglement_patterns() {
        let patterns = vec![
            EntanglementPattern::None,
            EntanglementPattern::Linear,
            EntanglementPattern::Circular,
            EntanglementPattern::AllToAll,
        ];

        let x = array![[1.0, 2.0], [3.0, 4.0]];

        for pattern in patterns {
            let qkernel = QuantumKernelApproximation::with_qubits(2).entanglement(pattern);

            let fitted = qkernel.fit(&x, &()).unwrap();
            let features = fitted.transform(&x).unwrap();

            assert_eq!(features.nrows(), 2);
        }
    }

    #[test]
    fn test_quantum_kernel_matrix() {
        let config = QuantumKernelConfig {
            n_qubits: 2,
            circuit_depth: 1,
            feature_map: QuantumFeatureMap::PauliZ,
            entanglement: EntanglementPattern::None,
            n_samples: 10,
        };

        let qkernel = QuantumKernelApproximation::new(config);
        let x = array![[1.0, 2.0], [3.0, 4.0]];

        let fitted = qkernel.fit(&x, &()).unwrap();
        let kernel_matrix = fitted.compute_kernel_matrix(&x);

        // Kernel should be symmetric
        assert!((kernel_matrix[[0, 1]] - kernel_matrix[[1, 0]]).abs() < 1e-10);

        // Diagonal should be close to 1 (self-overlap)
        assert!(kernel_matrix[[0, 0]] >= 0.0);
        assert!(kernel_matrix[[1, 1]] >= 0.0);
    }

    #[test]
    fn test_circuit_depth_effect() {
        let x = array![[1.0, 2.0], [3.0, 4.0]];

        for depth in 1..=3 {
            let qkernel = QuantumKernelApproximation::with_qubits(2).circuit_depth(depth);

            let fitted = qkernel.fit(&x, &()).unwrap();
            let features = fitted.transform(&x).unwrap();

            assert!(features.iter().all(|&v| v.is_finite()));
        }
    }

    #[test]
    fn test_empty_input_error() {
        let qkernel = QuantumKernelApproximation::with_qubits(2);
        let x_empty: Array2<Float> = Array2::zeros((0, 0));

        assert!(qkernel.fit(&x_empty, &()).is_err());
    }

    #[test]
    fn test_dimension_mismatch_error() {
        let qkernel = QuantumKernelApproximation::with_qubits(2);
        let x_train = array![[1.0, 2.0], [3.0, 4.0]];
        let x_test = array![[1.0, 2.0, 3.0]];

        let fitted = qkernel.fit(&x_train, &()).unwrap();
        assert!(fitted.transform(&x_test).is_err());
    }

    #[test]
    fn test_feature_map_simulation() {
        let config = QuantumKernelConfig::default();
        let qkernel = QuantumKernelApproximation::new(config);

        let x = array![1.0, 2.0, 3.0, 4.0];
        let features = qkernel.simulate_feature_map(&x);

        // Should return amplitudes for 2^4 = 16 basis states
        assert!(features.len() > 0);
        assert!(features.iter().all(|&a| a.is_finite()));
    }
}