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9425aed | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 | // Complex Hermitian eigensolver via nalgebra real-block reduction.
// Fixes spe_encoder.f90 sov_zheev stub.
//
// Method: for H Hermitian, decompose as H = A + iB where A = Re(H) symmetric,
// B = Im(H) antisymmetric. Build real block matrix M = [[A, -B],[B, A]] (2n×2n).
// M is real symmetric. Its eigenvalues are the doubled eigenvalues of H.
// Recover eigenvectors by decoding the real pairs back to complex.
use nalgebra::{DMatrix, DVector};
use num_complex::Complex64;
use thiserror::Error;
#[derive(Error, Debug)]
pub enum ZheevError {
#[error("Matrix is not square: {rows}x{cols}")]
NotSquare { rows: usize, cols: usize },
#[error("Matrix is not Hermitian (max off-symmetry: {max_err})")]
NotHermitian { max_err: f64 },
}
pub struct ZheevResult {
pub eigenvalues: DVector<f64>,
pub eigenvectors: DMatrix<Complex64>,
}
pub fn zheev(h: &DMatrix<Complex64>) -> Result<ZheevResult, ZheevError> {
let (rows, cols) = h.shape();
if rows != cols {
return Err(ZheevError::NotSquare { rows, cols });
}
let n = rows;
let max_err = hermitian_error(h);
if max_err > 1e-10 {
return Err(ZheevError::NotHermitian { max_err });
}
// Build real block matrix [[Re(H), -Im(H)], [Im(H), Re(H)]]
let mut m = DMatrix::<f64>::zeros(2 * n, 2 * n);
for i in 0..n {
for j in 0..n {
let h_ij = h[(i, j)];
m[(i, j)] = h_ij.re;
m[(i, n + j)] = -h_ij.im;
m[(n + i, j)] = h_ij.im;
m[(n + i, n + j)] = h_ij.re;
}
}
let eig = m.symmetric_eigen();
// eigenvalues come in pairs; take every other one (first n)
let mut pairs: Vec<(f64, usize)> = (0..2 * n)
.map(|i| (eig.eigenvalues[i], i))
.collect();
pairs.sort_by(|a, b| a.0.partial_cmp(&b.0).unwrap());
let mut eigenvalues = DVector::<f64>::zeros(n);
let mut eigenvectors = DMatrix::<Complex64>::zeros(n, n);
// eigenvalues are doubled (each appears twice); take one from each pair: indices 0,2,4,...
for (k, &(eval, col)) in pairs.iter().step_by(2).take(n).enumerate() {
eigenvalues[k] = eval;
for i in 0..n {
let re = eig.eigenvectors[(i, col)];
let im = eig.eigenvectors[(n + i, col)];
eigenvectors[(i, k)] = Complex64::new(re, im);
}
// normalize
let norm: f64 = (0..n).map(|i| eigenvectors[(i, k)].norm_sqr()).sum::<f64>().sqrt();
if norm > 1e-14 {
for i in 0..n {
eigenvectors[(i, k)] /= norm;
}
}
}
Ok(ZheevResult { eigenvalues, eigenvectors })
}
/// exp(-i * dt * H) for Hermitian H via eigendecomposition.
/// This is the unitary evolution operator U = exp(-i H dt).
pub fn exp_hermitian(h: &DMatrix<Complex64>, dt: f64) -> DMatrix<Complex64> {
let res = zheev(h).expect("exp_hermitian requires Hermitian input");
let n = res.eigenvalues.len();
let mut diag = DMatrix::<Complex64>::zeros(n, n);
for i in 0..n {
let phase = Complex64::new(0.0, -dt * res.eigenvalues[i]);
diag[(i, i)] = phase.exp();
}
&res.eigenvectors * &diag * res.eigenvectors.adjoint()
}
fn hermitian_error(h: &DMatrix<Complex64>) -> f64 {
let n = h.nrows();
let mut max = 0.0_f64;
for i in 0..n {
for j in 0..n {
let err = (h[(i, j)] - h[(j, i)].conj()).norm();
if err > max {
max = err;
}
}
}
max
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn test_real_diagonal() {
// H = diag(1, 2, 3) — trivially Hermitian
let mut h = DMatrix::<Complex64>::zeros(3, 3);
h[(0, 0)] = Complex64::new(1.0, 0.0);
h[(1, 1)] = Complex64::new(2.0, 0.0);
h[(2, 2)] = Complex64::new(3.0, 0.0);
let res = zheev(&h).unwrap();
let mut evals: Vec<f64> = res.eigenvalues.iter().copied().collect();
evals.sort_by(|a, b| a.partial_cmp(b).unwrap());
assert!((evals[0] - 1.0).abs() < 1e-10);
assert!((evals[1] - 2.0).abs() < 1e-10);
assert!((evals[2] - 3.0).abs() < 1e-10);
}
#[test]
fn test_exp_unitary() {
// H = [[1, 0],[0, -1]], dt = pi/2 → U = diag(e^{-i pi/2}, e^{i pi/2})
let mut h = DMatrix::<Complex64>::zeros(2, 2);
h[(0, 0)] = Complex64::new(1.0, 0.0);
h[(1, 1)] = Complex64::new(-1.0, 0.0);
let u = exp_hermitian(&h, std::f64::consts::PI / 2.0);
// U†U should be identity
let prod = u.adjoint() * &u;
let eye = DMatrix::<Complex64>::identity(2, 2);
let residual = (&prod - &eye).norm();
assert!(residual < 1e-12, "residual={}", residual);
}
}
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