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