qml.qchem.basis_rotation

basis_rotation(one_electron, two_electron, tol_factor=1e-05, **factorization_kwargs)

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.

Keyword Arguments

• tol_eigval (float) – Threshold error value for discarding the negligible factor eigenvalues. This can be used only when compressed=False.

• cholesky (bool) – Use Cholesky decomposition for the two_electron instead of eigendecomposition. Default is False.

• compressed (bool) – Use compressed double factorization for decomposing the two_electron.

• regularization (string | None) – Type of regularization ("L1" or "L2") to be used for optimizing the factors. Default is to not include any regularization term.

• **compression_kwargs – Look at the keyword arguments (compression_kwargs) in the factorize() method for all the available options with compressed=True.

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([-2.59579282,  0.84064649,  0.84064649,  0.45724992,  0.45724992]),
 array([ 5.60006390e-03, -9.73801723e-05, -9.73801723e-05,  2.84747318e-03,
         9.57150297e-05, -2.79878310e-03,  9.57150297e-05, -2.79878310e-03,
        -2.79878310e-03, -2.79878310e-03,  2.75092558e-03]),
 array([ 0.09060523,  0.04530262, -0.04530262, -0.04530262, -0.04530262,
        -0.04530262,  0.04530262]),
 array([ 1.6874169 , -0.68077716, -0.68077716,  0.17166195, -0.66913628,
         0.16872663, -0.66913628,  0.16872663,  0.16872663,  0.16872663,
         0.16584151])]

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.

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