View source on GitHub
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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 |
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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
\[ \begin{pmatrix} \cos(\theta) & -e^{i \varphi} \sin(\theta) \\ \sin(\theta) & e^{i \varphi} \cos(\theta) \end{pmatrix}. \]
Args | |
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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\\).