# givens_decomposition_slater

### givens_decomposition_slater(orbital_coeffs)

Decomposes the occupied orbitals of a Slater determinant into Givens rotations.

This is the rectangular counterpart of [`givens_decomposition()`](qiskit_fermions.linalg.givens_decomposition.html.md#qiskit_fermions.linalg.givens_decomposition), specialized for Slater
determinant state preparation. Given the coefficient matrix of the occupied orbitals of a
Slater determinant, it returns a sequence of Givens rotations that, applied to the reference
configuration $\lvert 1 \cdots 1 0 \cdots 0 \rangle$ (the first $m$ orbitals
occupied), prepares the Slater determinant. Here `orbital_coeffs` is an $m \times n$
matrix whose rows are the $m$ occupied orbitals expressed in a basis of $n$ spatial
orbitals ($m \le n$); its rows are assumed to be orthonormal.

Unlike [`givens_decomposition()`](qiskit_fermions.linalg.givens_decomposition.html.md#qiskit_fermions.linalg.givens_decomposition), this decomposition only needs to realize the $m$
occupied orbitals rather than a full $n \times n$ orbital rotation, so it uses at most
$m (n - m)$ Givens rotations arranged in a diamond-shaped pattern (versus the
$n (n - 1) / 2$ brick-wall of the square decomposition). The decomposition contains no
diagonal phases, because a global phase and any rotation within the occupied space leave the
prepared Slater determinant unchanged.

Each Givens rotation is defined by a 4-tuple, `(c, s, i, j)`, with:

* `c`: the real-valued cosine
* `s`: the complex-valued sine
* `i`: the first index
* `j`: the second (adjacent) index

which result in a matrix of the form:

$$
\begin{pmatrix}
c & s \\
-s^\dagger & c
\end{pmatrix}
$$

* **Parameters:**
  **orbital_coeffs** – the $m \times n$ matrix of occupied-orbital coefficients.
* **Returns:**
  The sequence of Givens rotations represented as 4-tuples as explained above.

The occupied orbitals are recovered by applying the returned rotations, in order, to the
columns of the $m \times n$ reference $\begin{pmatrix} I_m & 0 \end{pmatrix}$, where
each rotation acting on indices $i$ and $j$ sends

$$
v_i \mapsto c \, v_i + s^\dagger v_j, \qquad v_j \mapsto c \, v_j - s \, v_i.
$$

The result spans the same occupied space as `orbital_coeffs` (they define the same Slater
determinant), so the squared overlap $\lvert \det(A B^\dagger) \rvert^2$ between the
reconstructed orbitals $A$ and the target $B$ is one.

```pycon
>>> import numpy as np
>>> from qiskit_fermions.linalg import givens_decomposition_slater
>>> # two occupied orbitals in a basis of three, with orthonormal rows
>>> base = np.array([[0.8, 0.6, 0.0], [-0.48, 0.64, 0.6]])
>>> orbital_coeffs = (base * np.array([[1.0], [1j]])).astype(complex)
>>> rotations = givens_decomposition_slater(orbital_coeffs)
>>> m, n = orbital_coeffs.shape
>>> reconstructed = np.eye(m, n, dtype=complex)
>>> for c, s, i, j in rotations:
...     col_i, col_j = reconstructed[:, i].copy(), reconstructed[:, j].copy()
...     reconstructed[:, i] = c * col_i + s.conjugate() * col_j
...     reconstructed[:, j] = c * col_j - s * col_i
>>> overlap = abs(np.linalg.det(reconstructed @ orbital_coeffs.conj().T)) ** 2
>>> bool(np.isclose(overlap, 1.0))
True
```

#### SEE ALSO
[`givens_decomposition()`](qiskit_fermions.linalg.givens_decomposition.html.md#qiskit_fermions.linalg.givens_decomposition) for the square (full orbital rotation) decomposition.