openfermion.linalg.givens_decomposition

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Decompose a matrix into a sequence of Givens rotations.

openfermion.linalg.givens_decomposition(
    unitary_rows, always_insert=False
)

The input is an $m \times n$ matrix $Q$ with $m \leq n$. The rows of $Q$ are orthonormal. $Q$ can be decomposed as follows:

$$V Q U^\dagger = D$$

where $V$ and $U$ are unitary matrices, and $D$ is an $m \times n$ matrix with the first $m$ columns forming a diagonal matrix and the rest of the columns being zero. 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

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

Returns

givens_rotations (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\\).
left_unitary (ndarray):
    An \\(m \times m\\) numpy array representing the matrix
    \\(V\\).
diagonal (ndarray):
    A list of the nonzero entries of \\(D\\).