qml.qchem.optimal_sector¶
- optimal_sector(qubit_op, generators, active_electrons)[source]¶
Get the optimal sector which contains the ground state.
To obtain the optimal sector, we need to choose the right eigenvalues for the symmetry generators \(\bm{\tau}\). We can do so by using the following relation between the Pauli-Z qubit operator and the occupation number under a Jordan-Wigner transform.
\[\sigma_{i}^{z} = I - 2a_{i}^{\dagger}a_{i}\]According to this relation, the occupied and unoccupied fermionic modes correspond to the -1 and +1 eigenvalues of the Pauli-Z operator, respectively. Since all of the generators \(\bm{\tau}\) consist only of \(I\) and Pauli-Z operators, the correct eigenvalue for each \(\tau\) operator can be simply obtained by applying it on the reference Hartree-Fock (HF) state, and looking at the overlap between the wires on which the Pauli-Z operators act and the wires that correspond to occupied orbitals in the HF state.
- Parameters
- Returns
eigenvalues corresponding to the optimal sector which contains the ground state
- Return type
list[int]
Example
>>> symbols = ["H", "H"] >>> geometry = np.array([[0.0, 0.0, -0.69440367], [0.0, 0.0, 0.69440367]]) >>> H, qubits = qml.qchem.molecular_hamiltonian(symbols, geometry) >>> generators = qml.qchem.symmetry_generators(H) >>> qml.qchem.optimal_sector(H, generators, 2) [1, -1, -1]