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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^*\\).