openfermion.circuits.bogoliubov_transform

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Perform a Bogoliubov transformation.

openfermion.circuits.bogoliubov_transform(
    qubits: Sequence[cirq.Qid],
    transformation_matrix: numpy.ndarray,
    initial_state: Optional[Union[int, Sequence[int]]] = None
) -> cirq.OP_TREE

Used in the notebooks

This circuit performs the transformation to a basis determined by a new set of fermionic ladder operators. It performs the unitary $U$ such that

$$U a^\dagger_p U^{-1} = b^\dagger_p$$

where the $a^\dagger_p$ are the original creation operators and the $b^\dagger_p$ are the new creation operators. The new creation operators are linear combinations of the original ladder operators with coefficients given by the matrix transformation_matrix, which will be referred to as $W$ in the following.

If $W$ is an $N \times N$ matrix, then the $b^\dagger_p$ are given by

$$b^\dagger_p = \sum_{q=1}^N W_{pq} a^\dagger_q.$$

If $W$ is an $N \times 2N$ matrix, then the $b^\dagger_p$ are given by

$$ b^\daggerp = \sum{q=1}^N W_{pq} a^\dagger_q

              + \sum_{q=N+1}^{2N} W_{pq} a_q.

$$

This algorithm assumes the Jordan-Wigner Transform.