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(), 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(), 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.

>>> 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