# 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 $V^T D^*$.

[{ "type": "thumb-down", "id": "missingTheInformationINeed", "label":"Missing the information I need" },{ "type": "thumb-down", "id": "tooComplicatedTooManySteps", "label":"Too complicated / too many steps" },{ "type": "thumb-down", "id": "outOfDate", "label":"Out of date" },{ "type": "thumb-down", "id": "samplesCodeIssue", "label":"Samples / code issue" },{ "type": "thumb-down", "id": "otherDown", "label":"Other" }]
[{ "type": "thumb-up", "id": "easyToUnderstand", "label":"Easy to understand" },{ "type": "thumb-up", "id": "solvedMyProblem", "label":"Solved my problem" },{ "type": "thumb-up", "id": "otherUp", "label":"Other" }]