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Decompose a matrix into a sequence of Givens rotations and particle-hole transformations on the last fermionic mode.
openfermion.linalg.fermionic_gaussian_decomposition(
unitary_rows
)
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.
Args | |
|---|---|
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^*\\).