openfermion.linalg.givens_decomposition_square

Decompose a square matrix into a sequence of Givens rotations.

Used in the notebooks

Used in the tutorials

The input is a square $n \times n$ matrix $Q$. $Q$ can be decomposed as follows:

$$ Q = DU $$

where $U$ is unitary and $D$ is diagonal. Furthermore, we can decompose $U$ as

$$ U = G_k ... G_1 $$

where $G_1, \ldots, G_k$ are complex Givens rotations. A Givens rotation is a rotation within the two-dimensional subspace spanned by two coordinate axes. Within the two relevant coordinate axes, a Givens rotation has the form

$$ \begin{pmatrix} \cos(\theta) & -e^{i \varphi} \sin(\theta) \\ \sin(\theta) & e^{i \varphi} \cos(\theta) \end{pmatrix}. $$

unitary_matrix A numpy array with orthonormal rows, representing the matrix Q.

Returns

decomposition (list[tuple]):
    A list of tuples of objects describing Givens
    rotations. The list looks like [(G_1, ), (G_2, G_3), ... ].
    The Givens rotations within a tuple can be implemented in parallel.
    The description of a Givens rotation is itself a tuple of the
    form $(i, j, \theta, \varphi)$, which represents a
    Givens rotation of coordinates
    $i$ and $j$ by angles $\theta$ and
    $\varphi$.
diagonal (ndarray):
    A list of the nonzero entries of $D$.