# openfermion.linalg.fermionic_gaussian_decomposition

Decompose a matrix into a sequence of Givens rotations and

particle-hole transformations on the last fermionic mode.

The input is an $$N \times 2N$$ matrix $$W$$ with orthonormal rows. Furthermore, $$W$$ must have the block form

$W = ( W_1 \hspace{4pt} W_2 )$

where $$W_1$$ and $$W_2$$ satisfy

$W_1 W_1^\dagger + W_2 W_2^\dagger &= I$

W_1  W_2^T + W_2  W_1^T &= 0.


Then $$W$$ can be decomposed as

$V W U^\dagger = ( 0 \hspace{6pt} D )$

where $$V$$ and $$U$$ are unitary matrices and $$D$$ is a diagonal unitary matrix. Furthermore, $$U$$ can be decomposed as follows:

$U = B G_{k} \cdots B G_3 G_2 B G_1 B,$

where each $$G_i$$ is a Givens rotation, and $$B$$ represents swapping the $$N$$-th column with the $$2N$$-th column, which corresponds to a particle-hole transformation on the last fermionic mode. This particle-hole transformation maps $$a^\dagger_N$$ to $$a_N$$ and vice versa, while leaving the other fermionic ladder operators invariant.

The decomposition of $$U$$ is returned as a list of tuples of objects describing rotations and particle-hole transformations. The list looks something like [('pht', ), (G_1, ), ('pht', G_2), ... ]. The objects within a tuple are either the string 'pht', which indicates a particle-hole transformation on the last fermionic mode, or a tuple of the form $$(i, j, \theta, \varphi)$$, which indicates a Givens rotation of rows $$i$$ and $$j$$ by angles $$\theta$$ and $$\varphi$$.

The matrix $$V^T D^*$$ can also be decomposed as a sequence of Givens rotations. This decomposition is needed for a circuit that prepares an excited state.

unitary_rows(ndarray): A matrix with orthonormal rows and additional structure described above.

## Returns

decomposition (list[tuple]):
The decomposition of \$$U\$$.
left_decomposition (list[tuple]):
The decomposition of \$$V^T D^*\$$.
diagonal (ndarray):
A list of the nonzero entries of \$$D\$$.
left_diagonal (ndarray):
A list of the nonzero entries left from the decomposition
of \$$V^T D^*\$$.

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