openfermion.ops.general_basis_change

Change the basis of a general interaction tensor.

openfermion.ops.general_basis_change(
    general_tensor: numpy.ndarray,
    rotation_matrix: numpy.ndarray,
    key: tuple[int, ...],
    transpose: (bool | None) = None
) -> numpy.ndarray

Used in the notebooks

Used in the tutorials

M'^{p_1p_2...p_n} = R^{p1}{a_1} R^{p2}{a_2} ... R^{pn}{a_n} M^{a_1a_2...a_n}

where R is the rotation matrix, M is the general tensor, M' is the transformed general tensor, and a_k and p_k are indices. The formula uses the Einstein notation (implicit sum over repeated indices).

In case R is complex, the k-th R in the above formula need to be conjugated if key has a 1 in the k-th place (meaning that the corresponding operator is a creation operator).

general_tensor A square numpy array or matrix containing information about a general interaction tensor.
rotation_matrix A square numpy array or matrix having dimensions of n_qubits by n_qubits. Assumed to be unitary.
key A tuple indicating the type of general_tensor. Assumed to be non-empty. For example, a tensor storing coefficients of \(a^\dagger_p a_q\) would have a key of (1, 0) whereas a tensor storing coefficients of \(a^\dagger_p a_q a_r a^\dagger_s\) would have a key of (1, 0, 0, 1).
transpose If True, transposes the rotation matrix before applying it. If False, the rotation matrix is not transposed. If None (default), behaves as True to maintain backwards compatibility, but raises a FutureWarning.

transformed_general_tensor general_tensor in the rotated basis.