openfermion.linalg.givens_decomposition_square

Decompose a square matrix into a sequence of Givens rotations.

openfermion.linalg.givens_decomposition_square(
    unitary_matrix, always_insert=False
)

Used in the notebooks

Used in the tutorials

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

.. math::

Q = DU

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

.. math::

U = G_k ... G_1

where :math: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

.. math::

\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 :math:`(i, j, \theta, \varphi)`, which represents a
    Givens rotation of coordinates
    :math:`i` and :math:`j` by angles :math:`\theta` and
    :math:`\varphi`.
diagonal (ndarray):
    A list of the nonzero entries of :math:`D`.