quantrs2_sim/holographic_quantum_error_correction/

encoding.rs

//! Holographic encoding methods for quantum error correction.
//!
//! This module contains methods for creating various holographic encoding matrices
//! used in the holographic quantum error correction framework.

use scirs2_core::ndarray::Array2;
use scirs2_core::Complex64;
use std::f64::consts::PI;

use crate::error::Result;

use super::config::HolographicCodeType;
use super::simulator::HolographicQECSimulator;

impl HolographicQECSimulator {
    /// Create holographic encoding matrix using tensor network structure
    pub fn create_holographic_encoding_matrix(
        &self,
        boundary_dim: usize,
        bulk_dim: usize,
    ) -> Result<Array2<Complex64>> {
        let mut encoding_matrix = Array2::zeros((bulk_dim, boundary_dim));

        match self.config.error_correction_code {
            HolographicCodeType::AdSRindler => {
                self.create_ads_rindler_encoding(&mut encoding_matrix)?;
            }
            HolographicCodeType::HolographicStabilizer => {
                self.create_holographic_stabilizer_encoding(&mut encoding_matrix)?;
            }
            HolographicCodeType::BulkGeometry => {
                self.create_bulk_geometry_encoding(&mut encoding_matrix)?;
            }
            HolographicCodeType::TensorNetwork => {
                self.create_tensor_network_encoding(&mut encoding_matrix)?;
            }
            HolographicCodeType::HolographicSurface => {
                self.create_holographic_surface_encoding(&mut encoding_matrix)?;
            }
            HolographicCodeType::PerfectTensor => {
                self.create_perfect_tensor_encoding(&mut encoding_matrix)?;
            }
            HolographicCodeType::EntanglementEntropy => {
                self.create_entanglement_entropy_encoding(&mut encoding_matrix)?;
            }
            HolographicCodeType::AdSCFTCode => {
                self.create_ads_cft_encoding(&mut encoding_matrix)?;
            }
        }

        Ok(encoding_matrix)
    }

    /// Create AdS-Rindler holographic encoding
    pub fn create_ads_rindler_encoding(
        &self,
        encoding_matrix: &mut Array2<Complex64>,
    ) -> Result<()> {
        let (bulk_dim, boundary_dim) = encoding_matrix.dim();

        // AdS-Rindler encoding based on Rindler coordinates
        for i in 0..bulk_dim {
            for j in 0..boundary_dim {
                let rindler_factor = self.calculate_rindler_factor(i, j);
                let entanglement_factor = self.calculate_entanglement_factor(i, j);

                encoding_matrix[[i, j]] = Complex64::new(rindler_factor * entanglement_factor, 0.0);
            }
        }

        // Normalize the encoding matrix
        Self::normalize_encoding_matrix(encoding_matrix)?;
        Ok(())
    }

    /// Calculate Rindler factor for AdS-Rindler encoding
    #[must_use]
    pub fn calculate_rindler_factor(&self, bulk_index: usize, boundary_index: usize) -> f64 {
        let rindler_horizon = self.config.ads_radius;
        let bulk_position = (bulk_index as f64) / f64::from(1 << self.config.bulk_qubits);
        let boundary_position =
            (boundary_index as f64) / f64::from(1 << self.config.boundary_qubits);

        // Rindler transformation factor with phase shift to avoid zeros
        let factor = (rindler_horizon * bulk_position).cosh()
            * (2.0 * PI).mul_add(boundary_position, PI / 4.0).cos();

        factor.abs().max(1e-10)
    }

    /// Calculate entanglement factor for holographic encoding
    #[must_use]
    pub fn calculate_entanglement_factor(&self, bulk_index: usize, boundary_index: usize) -> f64 {
        let mutual_information = self.calculate_mutual_information(bulk_index, boundary_index);
        let entanglement_entropy = self.calculate_entanglement_entropy(bulk_index, boundary_index);

        (mutual_information * entanglement_entropy).sqrt()
    }

    /// Calculate mutual information between bulk and boundary regions
    pub(crate) fn calculate_mutual_information(
        &self,
        bulk_index: usize,
        boundary_index: usize,
    ) -> f64 {
        let bulk_entropy = self.calculate_region_entropy(bulk_index, true);
        let boundary_entropy = self.calculate_region_entropy(boundary_index, false);
        let joint_entropy = self.calculate_joint_entropy(bulk_index, boundary_index);

        bulk_entropy + boundary_entropy - joint_entropy
    }

    /// Calculate entanglement entropy for region
    pub(crate) fn calculate_entanglement_entropy(
        &self,
        bulk_index: usize,
        boundary_index: usize,
    ) -> f64 {
        // Use Ryu-Takayanagi prescription: S = Area/(4G)
        let area = self.calculate_rt_surface_area(bulk_index, boundary_index);
        let gravitational_constant = 1.0; // Natural units

        area / (4.0 * gravitational_constant)
    }

    /// Calculate Ryu-Takayanagi surface area
    pub(crate) fn calculate_rt_surface_area(
        &self,
        bulk_index: usize,
        boundary_index: usize,
    ) -> f64 {
        let bulk_position = (bulk_index as f64) / f64::from(1 << self.config.bulk_qubits);
        let boundary_position =
            (boundary_index as f64) / f64::from(1 << self.config.boundary_qubits);

        // Minimal surface area calculation
        let radial_distance = (bulk_position - boundary_position).abs();
        let ads_factor = self.config.ads_radius / radial_distance.mul_add(radial_distance, 1.0);

        ads_factor * self.config.central_charge
    }

    /// Calculate region entropy
    pub(crate) fn calculate_region_entropy(&self, region_index: usize, is_bulk: bool) -> f64 {
        let max_index = if is_bulk {
            Self::safe_dimension(self.config.bulk_qubits).unwrap_or(8)
        } else {
            Self::safe_dimension(self.config.boundary_qubits).unwrap_or(4)
        };

        // Ensure we have at least a reasonable minimum for computation
        let max_index = max_index.max(2);
        let region_size = ((region_index % max_index) as f64 + 0.1) / (max_index as f64 + 0.2);

        // Von Neumann entropy approximation with improved bounds
        if region_size > 0.01 && region_size < 0.99 {
            (-region_size).mul_add(
                region_size.ln(),
                -((1.0 - region_size) * (1.0 - region_size).ln()),
            )
        } else {
            // Return a small positive entropy instead of zero
            0.1
        }
    }

    /// Calculate joint entropy
    fn calculate_joint_entropy(&self, bulk_index: usize, boundary_index: usize) -> f64 {
        let combined_entropy = self.calculate_region_entropy(bulk_index, true)
            + self.calculate_region_entropy(boundary_index, false);

        // Add quantum correlations
        let correlation_factor = self.calculate_correlation_factor(bulk_index, boundary_index);
        combined_entropy * (1.0 - correlation_factor)
    }

    /// Calculate correlation factor between bulk and boundary
    pub(crate) fn calculate_correlation_factor(
        &self,
        bulk_index: usize,
        boundary_index: usize,
    ) -> f64 {
        let bulk_position = (bulk_index as f64) / f64::from(1 << self.config.bulk_qubits);
        let boundary_position =
            (boundary_index as f64) / f64::from(1 << self.config.boundary_qubits);

        // Correlation based on holographic correspondence
        let distance = (bulk_position - boundary_position).abs();
        (-distance / self.config.ads_radius).exp()
    }

    /// Create holographic stabilizer encoding
    pub(crate) fn create_holographic_stabilizer_encoding(
        &self,
        encoding_matrix: &mut Array2<Complex64>,
    ) -> Result<()> {
        let (bulk_dim, boundary_dim) = encoding_matrix.dim();

        // Create stabilizer-based holographic encoding
        for i in 0..bulk_dim {
            for j in 0..boundary_dim {
                let stabilizer_factor = Self::calculate_stabilizer_factor(i, j);
                let holographic_factor = self.calculate_holographic_factor(i, j);

                encoding_matrix[[i, j]] =
                    Complex64::new(stabilizer_factor * holographic_factor, 0.0);
            }
        }

        Self::normalize_encoding_matrix(encoding_matrix)?;
        Ok(())
    }

    /// Calculate stabilizer factor for encoding
    fn calculate_stabilizer_factor(bulk_index: usize, boundary_index: usize) -> f64 {
        let bulk_parity = f64::from(bulk_index.count_ones() % 2);
        let boundary_parity = f64::from(boundary_index.count_ones() % 2);

        // Stabilizer correlation
        if bulk_parity == boundary_parity {
            1.0 / (1.0 + bulk_index as f64).sqrt()
        } else {
            -1.0 / (1.0 + bulk_index as f64).sqrt()
        }
    }

    /// Calculate holographic factor for encoding
    pub(crate) fn calculate_holographic_factor(
        &self,
        bulk_index: usize,
        boundary_index: usize,
    ) -> f64 {
        let bulk_weight = f64::from(bulk_index.count_ones());
        let boundary_weight = f64::from(boundary_index.count_ones());

        // Holographic weight correlation
        let weight_correlation = (bulk_weight - boundary_weight).abs();
        (-weight_correlation / self.config.central_charge).exp()
    }

    /// Create bulk geometry encoding
    pub(crate) fn create_bulk_geometry_encoding(
        &self,
        encoding_matrix: &mut Array2<Complex64>,
    ) -> Result<()> {
        let (bulk_dim, boundary_dim) = encoding_matrix.dim();

        // Encoding based on bulk geometry and geodesics
        for i in 0..bulk_dim {
            for j in 0..boundary_dim {
                let geodesic_length = self.calculate_geodesic_length(i, j);
                let geometric_factor = self.calculate_geometric_factor(i, j);

                encoding_matrix[[i, j]] = Complex64::new(
                    (-geodesic_length / self.config.ads_radius).exp() * geometric_factor,
                    0.0,
                );
            }
        }

        Self::normalize_encoding_matrix(encoding_matrix)?;
        Ok(())
    }

    /// Calculate geodesic length in `AdS` space
    pub(crate) fn calculate_geodesic_length(
        &self,
        bulk_index: usize,
        boundary_index: usize,
    ) -> f64 {
        let bulk_position = (bulk_index as f64) / f64::from(1 << self.config.bulk_qubits);
        let boundary_position =
            (boundary_index as f64) / f64::from(1 << self.config.boundary_qubits);

        // AdS geodesic length calculation
        let radial_bulk = 1.0 / (1.0 - bulk_position);
        let radial_boundary = 1.0 / (1.0 - boundary_position);

        self.config.ads_radius * (radial_bulk / radial_boundary).ln().abs()
    }

    /// Calculate geometric factor
    pub(crate) fn calculate_geometric_factor(
        &self,
        bulk_index: usize,
        boundary_index: usize,
    ) -> f64 {
        let bulk_curvature = self.calculate_bulk_curvature(bulk_index);
        let boundary_curvature = self.calculate_boundary_curvature(boundary_index);

        (bulk_curvature.abs() / boundary_curvature).sqrt()
    }

    /// Calculate bulk curvature
    fn calculate_bulk_curvature(&self, bulk_index: usize) -> f64 {
        let position = (bulk_index as f64) / f64::from(1 << self.config.bulk_qubits);
        let ads_curvature = -1.0 / (self.config.ads_radius * self.config.ads_radius);

        ads_curvature * (1.0 - position).powi(2)
    }

    /// Calculate boundary curvature
    fn calculate_boundary_curvature(&self, boundary_index: usize) -> f64 {
        let position = (boundary_index as f64) / f64::from(1 << self.config.boundary_qubits);

        // Boundary is typically flat, but can have induced curvature
        // Ensure positive curvature to avoid division by zero
        0.1f64
            .mul_add((2.0 * PI * position).sin(), 1.0)
            .abs()
            .max(0.1)
    }

    /// Create tensor network encoding
    pub(crate) fn create_tensor_network_encoding(
        &self,
        encoding_matrix: &mut Array2<Complex64>,
    ) -> Result<()> {
        let (bulk_dim, boundary_dim) = encoding_matrix.dim();

        // Tensor network based holographic encoding
        for i in 0..bulk_dim {
            for j in 0..boundary_dim {
                let tensor_element = self.calculate_tensor_network_element(i, j);
                encoding_matrix[[i, j]] = tensor_element;
            }
        }

        Self::normalize_encoding_matrix(encoding_matrix)?;
        Ok(())
    }

    /// Calculate tensor network element
    pub(crate) fn calculate_tensor_network_element(
        &self,
        bulk_index: usize,
        boundary_index: usize,
    ) -> Complex64 {
        let bulk_legs = Self::get_tensor_legs(bulk_index, true);
        let boundary_legs = Self::get_tensor_legs(boundary_index, false);

        // Contract tensor legs between bulk and boundary
        let contraction_value = Self::contract_tensor_legs(&bulk_legs, &boundary_legs);

        Complex64::new(contraction_value, 0.0)
    }

    /// Get tensor legs for given index
    fn get_tensor_legs(index: usize, is_bulk: bool) -> Vec<f64> {
        let mut legs = Vec::new();
        let num_legs = if is_bulk { 4 } else { 2 }; // Bulk tensors have more legs

        for i in 0..num_legs {
            let leg_value = (((index >> i) & 1) as f64).mul_add(2.0, -1.0); // Convert to {-1, 1}
            legs.push(leg_value);
        }

        legs
    }

    /// Contract tensor legs
    fn contract_tensor_legs(bulk_legs: &[f64], boundary_legs: &[f64]) -> f64 {
        let mut contraction = 1.0;

        // Contract matching legs
        let min_legs = bulk_legs.len().min(boundary_legs.len());
        for i in 0..min_legs {
            contraction *= bulk_legs[i] * boundary_legs[i];
        }

        // Add remaining bulk leg contributions
        for leg in &bulk_legs[min_legs..] {
            contraction *= leg;
        }

        contraction / (bulk_legs.len() as f64).sqrt()
    }

    /// Create holographic surface encoding
    pub(crate) fn create_holographic_surface_encoding(
        &self,
        encoding_matrix: &mut Array2<Complex64>,
    ) -> Result<()> {
        let (bulk_dim, boundary_dim) = encoding_matrix.dim();

        // Surface code based holographic encoding
        for i in 0..bulk_dim {
            for j in 0..boundary_dim {
                let surface_element = self.calculate_surface_code_element(i, j);
                encoding_matrix[[i, j]] = surface_element;
            }
        }

        Self::normalize_encoding_matrix(encoding_matrix)?;
        Ok(())
    }

    /// Calculate surface code element
    fn calculate_surface_code_element(
        &self,
        bulk_index: usize,
        boundary_index: usize,
    ) -> Complex64 {
        let bulk_x = bulk_index % (1 << (self.config.bulk_qubits / 2));
        let bulk_y = bulk_index / (1 << (self.config.bulk_qubits / 2));
        let boundary_x = boundary_index % (1 << (self.config.boundary_qubits / 2));
        let boundary_y = boundary_index / (1 << (self.config.boundary_qubits / 2));

        // Surface code connectivity
        let x_parity = (bulk_x ^ boundary_x).count_ones() % 2;
        let y_parity = (bulk_y ^ boundary_y).count_ones() % 2;

        let amplitude = if x_parity == y_parity {
            1.0 / (1.0 + (bulk_x + bulk_y) as f64).sqrt()
        } else {
            // Use suppressed but non-zero value for off-parity connections
            1e-8 / (2.0 + (bulk_x + bulk_y) as f64).sqrt()
        };

        Complex64::new(amplitude, 0.0)
    }

    /// Create perfect tensor encoding
    pub(crate) fn create_perfect_tensor_encoding(
        &self,
        encoding_matrix: &mut Array2<Complex64>,
    ) -> Result<()> {
        let (bulk_dim, boundary_dim) = encoding_matrix.dim();

        // Perfect tensor network encoding
        for i in 0..bulk_dim {
            for j in 0..boundary_dim {
                let perfect_element = self.calculate_perfect_tensor_element(i, j);
                encoding_matrix[[i, j]] = perfect_element;
            }
        }

        Self::normalize_encoding_matrix(encoding_matrix)?;
        Ok(())
    }

    /// Calculate perfect tensor element
    fn calculate_perfect_tensor_element(
        &self,
        bulk_index: usize,
        boundary_index: usize,
    ) -> Complex64 {
        let bulk_state = Self::index_to_state_vector(bulk_index, self.config.bulk_qubits);
        let boundary_state =
            Self::index_to_state_vector(boundary_index, self.config.boundary_qubits);

        // Perfect tensor conditions
        let overlap = Self::calculate_state_overlap(&bulk_state, &boundary_state);
        let perfect_factor = Self::calculate_perfect_tensor_factor(bulk_index, boundary_index);

        Complex64::new(overlap * perfect_factor, 0.0)
    }

    /// Convert index to state vector
    fn index_to_state_vector(index: usize, num_qubits: usize) -> Vec<f64> {
        let mut state = vec![0.0; num_qubits];
        for (i, elem) in state.iter_mut().enumerate() {
            *elem = if (index >> i) & 1 == 1 { 1.0 } else { 0.0 };
        }
        state
    }

    /// Calculate state overlap
    fn calculate_state_overlap(state1: &[f64], state2: &[f64]) -> f64 {
        let min_len = state1.len().min(state2.len());
        let mut overlap = 0.0;

        for i in 0..min_len {
            overlap += state1[i] * state2[i];
        }

        overlap / (min_len as f64).sqrt()
    }

    /// Calculate perfect tensor factor
    fn calculate_perfect_tensor_factor(bulk_index: usize, boundary_index: usize) -> f64 {
        let bulk_weight = f64::from(bulk_index.count_ones());
        let boundary_weight = f64::from(boundary_index.count_ones());

        // Perfect tensor satisfies specific weight conditions
        if (bulk_weight - boundary_weight).abs() <= 1.0 {
            1.0 / (1.0 + bulk_weight).sqrt()
        } else {
            // Use exponentially suppressed but non-zero value
            1e-6 / (1.0 + (bulk_weight - boundary_weight).abs()).sqrt()
        }
    }

    /// Create entanglement entropy encoding
    pub(crate) fn create_entanglement_entropy_encoding(
        &self,
        encoding_matrix: &mut Array2<Complex64>,
    ) -> Result<()> {
        let (bulk_dim, boundary_dim) = encoding_matrix.dim();

        // Encoding based on entanglement entropy structure
        for i in 0..bulk_dim {
            for j in 0..boundary_dim {
                let entropy_element = self.calculate_entanglement_entropy_element(i, j);
                encoding_matrix[[i, j]] = entropy_element;
            }
        }

        Self::normalize_encoding_matrix(encoding_matrix)?;
        Ok(())
    }

    /// Calculate entanglement entropy element
    fn calculate_entanglement_entropy_element(
        &self,
        bulk_index: usize,
        boundary_index: usize,
    ) -> Complex64 {
        let bulk_entropy = self.calculate_region_entropy(bulk_index, true);
        let boundary_entropy = self.calculate_region_entropy(boundary_index, false);
        let mutual_information = self.calculate_mutual_information(bulk_index, boundary_index);

        // Entanglement entropy based amplitude
        let amplitude = (mutual_information / (bulk_entropy + boundary_entropy + 1e-10)).sqrt();

        Complex64::new(amplitude, 0.0)
    }

    /// Create AdS/CFT encoding
    pub(crate) fn create_ads_cft_encoding(
        &self,
        encoding_matrix: &mut Array2<Complex64>,
    ) -> Result<()> {
        let (bulk_dim, boundary_dim) = encoding_matrix.dim();

        // AdS/CFT correspondence encoding
        for i in 0..bulk_dim {
            for j in 0..boundary_dim {
                let ads_cft_element = self.calculate_ads_cft_element(i, j);
                encoding_matrix[[i, j]] = ads_cft_element;
            }
        }

        Self::normalize_encoding_matrix(encoding_matrix)?;
        Ok(())
    }

    /// Calculate AdS/CFT element
    pub(crate) fn calculate_ads_cft_element(
        &self,
        bulk_index: usize,
        boundary_index: usize,
    ) -> Complex64 {
        let bulk_field = self.calculate_bulk_field_value(bulk_index);
        let boundary_field = self.calculate_boundary_field_value(boundary_index);
        let correlation_function = self.calculate_correlation_function(bulk_index, boundary_index);

        // AdS/CFT dictionary
        let amplitude = bulk_field * boundary_field * correlation_function;

        Complex64::new(amplitude, 0.0)
    }

    /// Calculate bulk field value
    pub(crate) fn calculate_bulk_field_value(&self, bulk_index: usize) -> f64 {
        let position = (bulk_index as f64) / f64::from(1 << self.config.bulk_qubits);
        let radial_coordinate = 1.0 / (1.0 - position);

        // Bulk field in AdS space
        (radial_coordinate / self.config.ads_radius).powf(self.calculate_conformal_dimension())
    }

    /// Calculate boundary field value
    pub(crate) fn calculate_boundary_field_value(&self, boundary_index: usize) -> f64 {
        let position = (boundary_index as f64) / f64::from(1 << self.config.boundary_qubits);

        // Boundary CFT field
        (2.0 * PI * position).sin() / (1.0 + position).sqrt()
    }

    /// Calculate conformal dimension
    pub(crate) fn calculate_conformal_dimension(&self) -> f64 {
        // Conformal dimension based on central charge
        (self.config.central_charge / 12.0).sqrt()
    }

    /// Calculate correlation function
    pub(crate) fn calculate_correlation_function(
        &self,
        bulk_index: usize,
        boundary_index: usize,
    ) -> f64 {
        let bulk_position = (bulk_index as f64) / f64::from(1 << self.config.bulk_qubits);
        let boundary_position =
            (boundary_index as f64) / f64::from(1 << self.config.boundary_qubits);

        // Two-point correlation function
        let distance = (bulk_position - boundary_position).abs();
        let conformal_dimension = self.calculate_conformal_dimension();

        1.0 / (1.0 + distance).powf(2.0 * conformal_dimension)
    }

    /// Normalize encoding matrix
    pub(crate) fn normalize_encoding_matrix(encoding_matrix: &mut Array2<Complex64>) -> Result<()> {
        let (rows, cols) = encoding_matrix.dim();

        // Normalize each column
        for j in 0..cols {
            let mut column_norm = 0.0;
            for i in 0..rows {
                column_norm += encoding_matrix[[i, j]].norm_sqr();
            }

            if column_norm > 1e-10 {
                let norm_factor = 1.0 / column_norm.sqrt();
                for i in 0..rows {
                    encoding_matrix[[i, j]] *= norm_factor;
                }
            } else {
                // If column is all zeros, add small diagonal elements
                if j < rows {
                    encoding_matrix[[j, j]] = Complex64::new(1e-6, 0.0);
                } else {
                    encoding_matrix[[0, j]] = Complex64::new(1e-6, 0.0);
                }
            }
        }

        Ok(())
    }
}