openfermion.linalg.givens_decomposition

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

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

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

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 \(D\).