Merge commit 'd803bfe2b1fe7f5e219e50ac20d6801a0a58ac75' as 'vendor/ruvector'

vendor/ruvector/examples/prime-radiant/docs/adr/ADR-006-quantum-topology.md

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+# ADR-006: Quantum Topology for Representation Analysis
+
+**Status**: Accepted
+**Date**: 2024-12-15
+**Authors**: RuVector Team
+**Supersedes**: None
+
+---
+
+## Context
+
+High-dimensional neural network representations exhibit complex geometric and topological structure that classical methods struggle to capture. We need tools that can:
+
+1. **Handle superpositions**: Representations often encode multiple concepts simultaneously
+2. **Measure entanglement**: Detect non-local correlations between features
+3. **Track topological invariants**: Identify persistent structural properties
+4. **Model uncertainty**: Represent distributional properties of activations
+
+Quantum-inspired methods offer advantages because:
+- Superposition naturally models polysemy and context-dependence
+- Entanglement captures feature interactions beyond correlation
+- Density matrices provide natural uncertainty representation
+- Topological quantum invariants are robust to noise
+
+---
+
+## Decision
+
+We implement **quantum topology** methods for advanced representation analysis, including density matrix representations, entanglement measures, and topological invariants.
+
+### Core Structures
+
+#### 1. Quantum State Representation
+
+```rust
+use num_complex::Complex64;
+
+/// A quantum state representing neural activations
+pub struct QuantumState {
+    /// Amplitudes in computational basis
+    amplitudes: Vec<Complex64>,
+    /// Number of qubits (log2 of dimension)
+    num_qubits: usize,
+}
+
+impl QuantumState {
+    /// Create from real activation vector (amplitude encoding)
+    pub fn from_activations(activations: &[f64]) -> Self {
+        let n = activations.len();
+        let num_qubits = (n as f64).log2().ceil() as usize;
+        let dim = 1 << num_qubits;
+
+        // Normalize
+        let norm: f64 = activations.iter().map(|x| x * x).sum::<f64>().sqrt();
+
+        let mut amplitudes = vec![Complex64::new(0.0, 0.0); dim];
+        for (i, &a) in activations.iter().enumerate() {
+            amplitudes[i] = Complex64::new(a / norm, 0.0);
+        }
+
+        Self { amplitudes, num_qubits }
+    }
+
+    /// Inner product (fidelity for pure states)
+    pub fn fidelity(&self, other: &Self) -> f64 {
+        let inner: Complex64 = self.amplitudes.iter()
+            .zip(other.amplitudes.iter())
+            .map(|(a, b)| a.conj() * b)
+            .sum();
+        inner.norm_sqr()
+    }
+
+    /// Convert to density matrix
+    pub fn to_density_matrix(&self) -> DensityMatrix {
+        let dim = self.amplitudes.len();
+        let mut rho = vec![vec![Complex64::new(0.0, 0.0); dim]; dim];
+
+        for i in 0..dim {
+            for j in 0..dim {
+                rho[i][j] = self.amplitudes[i] * self.amplitudes[j].conj();
+            }
+        }
+
+        DensityMatrix { matrix: rho, dim }
+    }
+}
+```
+
+#### 2. Density Matrix
+
+```rust
+/// Density matrix for mixed state representation
+pub struct DensityMatrix {
+    /// The density matrix elements
+    matrix: Vec<Vec<Complex64>>,
+    /// Dimension
+    dim: usize,
+}
+
+impl DensityMatrix {
+    /// Create maximally mixed state
+    pub fn maximally_mixed(dim: usize) -> Self {
+        let mut matrix = vec![vec![Complex64::new(0.0, 0.0); dim]; dim];
+        let val = Complex64::new(1.0 / dim as f64, 0.0);
+        for i in 0..dim {
+            matrix[i][i] = val;
+        }
+        Self { matrix, dim }
+    }
+
+    /// From ensemble of pure states
+    pub fn from_ensemble(states: &[(f64, QuantumState)]) -> Self {
+        let dim = states[0].1.amplitudes.len();
+        let mut matrix = vec![vec![Complex64::new(0.0, 0.0); dim]; dim];
+
+        for (prob, state) in states {
+            let rho = state.to_density_matrix();
+            for i in 0..dim {
+                for j in 0..dim {
+                    matrix[i][j] += Complex64::new(*prob, 0.0) * rho.matrix[i][j];
+                }
+            }
+        }
+
+        Self { matrix, dim }
+    }
+
+    /// Von Neumann entropy: S(rho) = -Tr(rho log rho)
+    pub fn entropy(&self) -> f64 {
+        let eigenvalues = self.eigenvalues();
+        -eigenvalues.iter()
+            .filter(|&e| *e > 1e-10)
+            .map(|e| e * e.ln())
+            .sum::<f64>()
+    }
+
+    /// Purity: Tr(rho^2)
+    pub fn purity(&self) -> f64 {
+        let mut trace = Complex64::new(0.0, 0.0);
+        for i in 0..self.dim {
+            for k in 0..self.dim {
+                trace += self.matrix[i][k] * self.matrix[k][i];
+            }
+        }
+        trace.re
+    }
+
+    /// Eigenvalues of density matrix
+    pub fn eigenvalues(&self) -> Vec<f64> {
+        // Convert to nalgebra matrix and compute eigenvalues
+        let mut m = DMatrix::zeros(self.dim, self.dim);
+        for i in 0..self.dim {
+            for j in 0..self.dim {
+                m[(i, j)] = self.matrix[i][j].re; // Hermitian, so real eigenvalues
+            }
+        }
+        let eigen = m.symmetric_eigenvalues();
+        eigen.iter().cloned().collect()
+    }
+}
+```
+
+#### 3. Entanglement Measures
+
+```rust
+/// Entanglement analysis for bipartite systems
+pub struct EntanglementAnalysis {
+    /// Subsystem A
+    subsystem_a: Vec<usize>,
+    /// Subsystem B
+    subsystem_b: Vec<usize>,
+}
+
+impl EntanglementAnalysis {
+    /// Compute partial trace over subsystem B
+    pub fn partial_trace_b(&self, rho: &DensityMatrix) -> DensityMatrix {
+        let dim_a = 1 << self.subsystem_a.len();
+        let dim_b = 1 << self.subsystem_b.len();
+
+        let mut rho_a = vec![vec![Complex64::new(0.0, 0.0); dim_a]; dim_a];
+
+        for i in 0..dim_a {
+            for j in 0..dim_a {
+                for k in 0..dim_b {
+                    let row = i * dim_b + k;
+                    let col = j * dim_b + k;
+                    rho_a[i][j] += rho.matrix[row][col];
+                }
+            }
+        }
+
+        DensityMatrix { matrix: rho_a, dim: dim_a }
+    }
+
+    /// Entanglement entropy: S(rho_A)
+    pub fn entanglement_entropy(&self, rho: &DensityMatrix) -> f64 {
+        let rho_a = self.partial_trace_b(rho);
+        rho_a.entropy()
+    }
+
+    /// Mutual information: I(A:B) = S(A) + S(B) - S(AB)
+    pub fn mutual_information(&self, rho: &DensityMatrix) -> f64 {
+        let rho_a = self.partial_trace_b(rho);
+        let rho_b = self.partial_trace_a(rho);
+
+        rho_a.entropy() + rho_b.entropy() - rho.entropy()
+    }
+
+    /// Concurrence (for 2-qubit systems)
+    pub fn concurrence(&self, rho: &DensityMatrix) -> f64 {
+        if rho.dim != 4 {
+            return 0.0; // Only defined for 2 qubits
+        }
+
+        // Spin-flip matrix
+        let sigma_y = [[Complex64::new(0.0, 0.0), Complex64::new(0.0, -1.0)],
+                       [Complex64::new(0.0, 1.0), Complex64::new(0.0, 0.0)]];
+
+        // rho_tilde = (sigma_y x sigma_y) rho* (sigma_y x sigma_y)
+        let rho_tilde = self.spin_flip_transform(rho, &sigma_y);
+
+        // R = rho * rho_tilde
+        let r = self.matrix_multiply(rho, &rho_tilde);
+
+        // Eigenvalues of R
+        let eigenvalues = r.eigenvalues();
+        let mut lambdas: Vec<f64> = eigenvalues.iter()
+            .map(|e| e.sqrt())
+            .collect();
+        lambdas.sort_by(|a, b| b.partial_cmp(a).unwrap());
+
+        // C = max(0, lambda_1 - lambda_2 - lambda_3 - lambda_4)
+        (lambdas[0] - lambdas[1] - lambdas[2] - lambdas[3]).max(0.0)
+    }
+}
+```
+
+#### 4. Topological Invariants
+
+```rust
+/// Topological invariants for representation spaces
+pub struct TopologicalInvariant {
+    /// Type of invariant
+    pub kind: InvariantKind,
+    /// Computed value
+    pub value: f64,
+    /// Confidence/precision
+    pub precision: f64,
+}
+
+pub enum InvariantKind {
+    /// Euler characteristic
+    EulerCharacteristic,
+    /// Betti numbers
+    BettiNumber(usize),
+    /// Chern number (for complex bundles)
+    ChernNumber,
+    /// Berry phase
+    BerryPhase,
+    /// Winding number
+    WindingNumber,
+}
+
+impl TopologicalInvariant {
+    /// Compute Berry phase around a loop in parameter space
+    pub fn berry_phase(states: &[QuantumState]) -> Self {
+        let n = states.len();
+        let mut phase = Complex64::new(1.0, 0.0);
+
+        for i in 0..n {
+            let next = (i + 1) % n;
+            let overlap: Complex64 = states[i].amplitudes.iter()
+                .zip(states[next].amplitudes.iter())
+                .map(|(a, b)| a.conj() * b)
+                .sum();
+            phase *= overlap;
+        }
+
+        Self {
+            kind: InvariantKind::BerryPhase,
+            value: phase.arg(),
+            precision: 1e-10,
+        }
+    }
+
+    /// Compute winding number from phase function
+    pub fn winding_number(phases: &[f64]) -> Self {
+        let mut total_winding = 0.0;
+        for i in 0..phases.len() {
+            let next = (i + 1) % phases.len();
+            let mut delta = phases[next] - phases[i];
+
+            // Wrap to [-pi, pi]
+            while delta > std::f64::consts::PI { delta -= 2.0 * std::f64::consts::PI; }
+            while delta < -std::f64::consts::PI { delta += 2.0 * std::f64::consts::PI; }
+
+            total_winding += delta;
+        }
+
+        Self {
+            kind: InvariantKind::WindingNumber,
+            value: (total_winding / (2.0 * std::f64::consts::PI)).round(),
+            precision: 1e-6,
+        }
+    }
+}
+```
+
+### Simplicial Complex for TDA
+
+```rust
+/// Simplicial complex for topological data analysis
+pub struct SimplicialComplex {
+    /// Vertices
+    vertices: Vec<usize>,
+    /// Simplices by dimension
+    simplices: Vec<HashSet<Vec<usize>>>,
+    /// Boundary matrices
+    boundary_maps: Vec<DMatrix<f64>>,
+}
+
+impl SimplicialComplex {
+    /// Build Vietoris-Rips complex from point cloud
+    pub fn vietoris_rips(points: &[DVector<f64>], epsilon: f64, max_dim: usize) -> Self {
+        let n = points.len();
+        let vertices: Vec<usize> = (0..n).collect();
+
+        let mut simplices = vec![HashSet::new(); max_dim + 1];
+
+        // 0-simplices (vertices)
+        for i in 0..n {
+            simplices[0].insert(vec![i]);
+        }
+
+        // 1-simplices (edges)
+        for i in 0..n {
+            for j in (i+1)..n {
+                if (&points[i] - &points[j]).norm() <= epsilon {
+                    simplices[1].insert(vec![i, j]);
+                }
+            }
+        }
+
+        // Higher simplices (clique detection)
+        for dim in 2..=max_dim {
+            for simplex in &simplices[dim - 1] {
+                for v in 0..n {
+                    if simplex.contains(&v) { continue; }
+
+                    // Check if v is connected to all vertices in simplex
+                    let all_connected = simplex.iter().all(|&u| {
+                        simplices[1].contains(&vec![u.min(v), u.max(v)])
+                    });
+
+                    if all_connected {
+                        let mut new_simplex = simplex.clone();
+                        new_simplex.push(v);
+                        new_simplex.sort();
+                        simplices[dim].insert(new_simplex);
+                    }
+                }
+            }
+        }
+
+        Self { vertices, simplices, boundary_maps: vec![] }
+    }
+
+    /// Compute Betti numbers
+    pub fn betti_numbers(&self) -> Vec<usize> {
+        self.compute_boundary_maps();
+
+        let mut betti = Vec::new();
+        for k in 0..self.simplices.len() {
+            let kernel_dim = if k < self.boundary_maps.len() {
+                self.kernel_dimension(&self.boundary_maps[k])
+            } else {
+                self.simplices[k].len()
+            };
+
+            let image_dim = if k > 0 && k <= self.boundary_maps.len() {
+                self.image_dimension(&self.boundary_maps[k - 1])
+            } else {
+                0
+            };
+
+            betti.push(kernel_dim.saturating_sub(image_dim));
+        }
+
+        betti
+    }
+}
+```
+
+---
+
+## Consequences
+
+### Positive
+
+1. **Rich representation**: Density matrices capture distributional information
+2. **Entanglement detection**: Identifies non-local feature correlations
+3. **Topological robustness**: Invariants stable under continuous deformation
+4. **Quantum advantage**: Some computations exponentially faster
+5. **Uncertainty modeling**: Natural probabilistic interpretation
+
+### Negative
+
+1. **Computational cost**: Density matrices are O(d^2) in memory
+2. **Classical simulation**: Full quantum benefits require quantum hardware
+3. **Interpretation complexity**: Quantum concepts less intuitive
+4. **Limited applicability**: Not all problems benefit from quantum formalism
+
+### Mitigations
+
+1. **Low-rank approximations**: Use matrix product states for large systems
+2. **Tensor networks**: Efficient classical simulation of structured states
+3. **Hybrid classical-quantum**: Use quantum-inspired methods on classical hardware
+4. **Domain-specific applications**: Focus on problems with natural quantum structure
+
+---
+
+## Integration with Prime-Radiant
+
+### Connection to Sheaf Cohomology
+
+Quantum states form a sheaf:
+- Open sets: Subsystems
+- Sections: Quantum states
+- Restriction: Partial trace
+- Cohomology: Entanglement obstructions
+
+### Connection to Category Theory
+
+Quantum mechanics as a dagger category:
+- Objects: Hilbert spaces
+- Morphisms: Completely positive maps
+- Dagger: Adjoint
+
+---
+
+## References
+
+1. Nielsen, M.A., & Chuang, I.L. (2010). "Quantum Computation and Quantum Information." Cambridge.
+
+2. Carlsson, G. (2009). "Topology and Data." Bulletin of the AMS.
+
+3. Coecke, B., & Kissinger, A. (2017). "Picturing Quantum Processes." Cambridge.
+
+4. Schuld, M., & Petruccione, F. (2021). "Machine Learning with Quantum Computers." Springer.
+
+5. Edelsbrunner, H., & Harer, J. (2010). "Computational Topology." AMS.