qml.qchem.basis_rotation¶
-
basis_rotation(one_electron, two_electron, tol_factor=1e-05)[source]¶ Return the grouped coefficients and observables of a molecular Hamiltonian and the basis rotation unitaries obtained with the basis rotation grouping method.
- Parameters
one_electron (array[float]) – one-electron integral matrix in the molecular orbital basis
two_electron (array[array[float]]) – two-electron integral tensor in the molecular orbital basis arranged in chemist notation
tol_factor (float) – threshold error value for discarding the negligible factors
- Returns
tuple containing grouped coefficients, grouped observables and basis rotation transformation matrices
- Return type
tuple(list[array[float]], list[list[Observable]], list[array[float]])
Example
>>> symbols = ['H', 'H'] >>> geometry = np.array([[0.0, 0.0, 0.0], ... [1.398397361, 0.0, 0.0]], requires_grad=False) >>> mol = qml.qchem.Molecule(symbols, geometry) >>> core, one, two = qml.qchem.electron_integrals(mol)() >>> coeffs, ops, unitaries = basis_rotation(one, two, tol_factor=1.0e-5) >>> print(coeffs) [array([ 0.84064649, -2.59579282, 0.84064649, 0.45724992, 0.45724992]), array([ 9.57150297e-05, 5.60006390e-03, 9.57150297e-05, 2.75092558e-03, -9.73801723e-05, -2.79878310e-03, -9.73801723e-05, -2.79878310e-03, -2.79878310e-03, -2.79878310e-03, 2.84747318e-03]), array([ 0.04530262, -0.04530262, -0.04530262, -0.04530262, -0.04530262, 0.09060523, 0.04530262]), array([-0.66913628, 1.6874169 , -0.66913628, 0.16584151, -0.68077716, 0.16872663, -0.68077716, 0.16872663, 0.16872663, 0.16872663, 0.17166195])]
Theory
A second-quantized molecular Hamiltonian can be constructed in the chemist notation format following Eq. (1) of [PRX Quantum 2, 030305, 2021] as
\[H = \sum_{\alpha \in \{\uparrow, \downarrow \} } \sum_{pq} T_{pq} a_{p,\alpha}^{\dagger} a_{q, \alpha} + \frac{1}{2} \sum_{\alpha, \beta \in \{\uparrow, \downarrow \} } \sum_{pqrs} V_{pqrs} a_{p, \alpha}^{\dagger} a_{q, \alpha} a_{r, \beta}^{\dagger} a_{s, \beta},\]where \(V_{pqrs}\) denotes a two-electron integral in the chemist notation and \(T_{pq}\) is obtained from the one- and two-electron integrals, \(h_{pq}\) and \(h_{pqrs}\), as
\[T_{pq} = h_{pq} - \frac{1}{2} \sum_s h_{pssq}.\]The tensor \(V\) can be converted to a matrix which is indexed by the indices \(pq\) and \(rs\) and eigendecomposed up to a rank \(R\) to give
\[V_{pqrs} = \sum_r^R L_{pq}^{(r)} L_{rs}^{(r) T},\]where \(L\) denotes the matrix of eigenvectors of the matrix \(V\). The molecular Hamiltonian can then be rewritten following Eq. (7) of [Phys. Rev. Research 3, 033055, 2021] as
\[H = \sum_{\alpha \in \{\uparrow, \downarrow \} } \sum_{pq} T_{pq} a_{p,\alpha}^{\dagger} a_{q, \alpha} + \frac{1}{2} \sum_r^R \left ( \sum_{\alpha \in \{\uparrow, \downarrow \} } \sum_{pq} L_{pq}^{(r)} a_{p, \alpha}^{\dagger} a_{q, \alpha} \right )^2.\]The orbital basis can be rotated such that each \(T\) and \(L^{(r)}\) matrix is diagonal. The Hamiltonian can then be written following Eq. (2) of [npj Quantum Information, 7, 23 (2021)] as
\[H = U_0 \left ( \sum_p d_p n_p \right ) U_0^{\dagger} + \sum_r^R U_r \left ( \sum_{pq} d_{pq}^{(r)} n_p n_q \right ) U_r^{\dagger},\]where the coefficients \(d\) are obtained by diagonalizing the \(T\) and \(L^{(r)}\) matrices. The number operators \(n_p = a_p^{\dagger} a_p\) can be converted to qubit operators using
\[n_p = \frac{1-Z_p}{2},\]where \(Z_p\) is the Pauli \(Z\) operator applied to qubit \(p\). This gives the qubit Hamiltonian
\[H = U_0 \left ( \sum_p O_p^{(0)} \right ) U_0^{\dagger} + \sum_r^R U_r \left ( \sum_{q} O_q^{(r)} \right ) U_r^{\dagger},\]where \(O = \sum_i c_i P_i\) is a linear combination of Pauli words \(P_i\) that are a tensor product of Pauli \(Z\) and Identity operators. This allows all the Pauli words in each of the \(O\) terms to be measured simultaneously. This function returns the coefficients and the Pauli words grouped for each of the \(O\) terms as well as the basis rotation transformation matrices that are constructed from the eigenvectors of the \(T\) and \(L^{(r)}\) matrices. Each column of the transformation matrix is an eigenvector of the corresponding \(T\) or \(L^{(r)}\) matrix.