Source code for pennylane.qchem.factorization

# Copyright 2018-2022 Xanadu Quantum Technologies Inc.

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This module contains the functions needed for two-electron tensor factorization.
import pennylane as qml
from pennylane import numpy as np

[docs]def factorize(two_electron, tol_factor=1.0e-5, tol_eigval=1.0e-5): r"""Return the double-factorized form of a two-electron integral tensor in spatial basis. The two-electron tensor :math:`V`, in `chemist notation <>`_, is first factorized in terms of symmetric matrices :math:`L^{(r)}` such that :math:`V_{ijkl} = \sum_r^R L_{ij}^{(r)} L_{kl}^{(r) T}`. The rank :math:`R` is determined by a threshold error. Then, each matrix :math:`L^{(r)}` is diagonalized and its eigenvalues (and corresponding eigenvectors) are truncated at a threshold error. Args: 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 tol_eigval (float): threshold error value for discarding the negligible factor eigenvalues Returns: tuple(array[array[float]], list[array[float]], list[array[float]]): tuple containing symmetric matrices (factors) approximating the two-electron integral tensor, truncated eigenvalues of the generated factors, and truncated eigenvectors of the generated factors **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)() >>> two = np.swapaxes(two, 1, 3) # convert to chemist notation >>> factors, eigvals, eigvecs = factorize(two, 1e-5, 1e-5) >>> print(factors) [[[ 1.06723440e-01 9.73575768e-15] [ 8.36288956e-15 -1.04898533e-01]] [[-2.20945401e-13 -4.25688222e-01] [-4.25688222e-01 -2.98228790e-13]] [[-8.14472856e-01 5.01669019e-13] [ 5.01689072e-13 -8.28642140e-01]]] .. details:: :title: Theory The second quantized electronic Hamiltonian is constructed in terms of fermionic creation, :math:`a^{\dagger}` , and annihilation, :math:`a`, operators as [`arXiv:1902.02134 <>`_] .. math:: H = \sum_{\alpha \in \{\uparrow, \downarrow \} } \sum_{pq} h_{pq} a_{p,\alpha}^{\dagger} a_{q, \alpha} + \frac{1}{2} \sum_{\alpha, \beta \in \{\uparrow, \downarrow \} } \sum_{pqrs} h_{pqrs} a_{p, \alpha}^{\dagger} a_{q, \beta}^{\dagger} a_{r, \beta} a_{s, \alpha}, where :math:`h_{pq}` and :math:`h_{pqrs}` are the one- and two-electron integrals computed as .. math:: h_{pq} = \int \phi_p(r)^* \left ( -\frac{\nabla_r^2}{2} - \sum_i \frac{Z_i}{|r-R_i|} \right) \phi_q(r) dr, and .. math:: h_{pqrs} = \int \frac{\phi_p(r_1)^* \phi_q(r_2)^* \phi_r(r_2) \phi_s(r_1)}{|r_1 - r_2|} dr_1 dr_2. The two-electron integrals can be rearranged in the so-called chemist notation which gives .. math:: V_{pqrs} = \int \frac{\phi_p(r_1)^* \phi_q(r_1)^* \phi_r(r_2) \phi_s(r_2)}{|r_1 - r_2|} dr_1 dr_2, and the molecular Hamiltonian can be rewritten as .. math:: 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}, with .. math:: T_{pq} = h_{pq} - \frac{1}{2} \sum_s h_{pssq}. This notation allows a low-rank factorization of the two-electron integral. The objective of the factorization is to find a set of symmetric matrices, :math:`L^{(r)}`, such that .. math:: V_{ijkl} = \sum_r^R L_{ij}^{(r)} L_{kl}^{(r) T}, with the rank :math:`R \leq n^2` where :math:`n` is the number of molecular orbitals. The matrices :math:`L^{(r)}` are diagonalized and for each matrix the eigenvalues that are smaller than a given threshold (and their corresponding eigenvectors) are discarded. The factorization algorithm has the following steps [`arXiv:1902.02134 <>`_]: - Reshape the :math:`n \times n \times n \times n` two-electron tensor to a :math:`n^2 \times n^2` matrix where :math:`n` is the number of orbitals. - Diagonalize the resulting matrix and keep the :math:`r` eigenvectors that have corresponding eigenvalues larger than a threshold. - Multiply the eigenvectors by the square root of the eigenvalues to obtain matrices :math:`L^{(r)}`. - Reshape the selected eigenvectors to :math:`n \times n` matrices. - Diagonalize the :math:`n \times n` matrices and for each matrix keep the eigenvalues (and their corresponding eigenvectors) that are larger than a threshold. """ shape = two_electron.shape if len(shape) != 4 or len(set(shape)) != 1: raise ValueError("The two-electron repulsion tensor must have a (N x N x N x N) shape.") n = shape[0] two = two_electron.reshape(n * n, n * n) eigvals_r, eigvecs_r = np.linalg.eigh(two) eigvals_r = np.array([val for val in eigvals_r if abs(val) > tol_factor]) eigvecs_r = eigvecs_r[:, -len(eigvals_r) :] if eigvals_r.size == 0: raise ValueError( "All factors are discarded. Consider decreasing the first threshold error." ) vectors = eigvecs_r @ np.diag(np.sqrt(eigvals_r)) r = len(eigvals_r) factors = np.array([vectors.reshape(n, n, r)[:, :, k] for k in range(r)]) eigvals, eigvecs = np.linalg.eigh(factors) eigvals_m = [] eigvecs_m = [] for n, eigval in enumerate(eigvals): idx = [i for i, v in enumerate(eigval) if abs(v) > tol_eigval] eigvals_m.append(eigval[idx]) eigvecs_m.append(eigvecs[n][idx]) if np.sum([len(v) for v in eigvecs_m]) == 0: raise ValueError( "All eigenvectors are discarded. Consider decreasing the second threshold error." ) return factors, eigvals_m, eigvecs_m
[docs]def basis_rotation(one_electron, two_electron, tol_factor=1.0e-5): r"""Return the grouped coefficients and observables of a molecular Hamiltonian and the basis rotation unitaries obtained with the basis rotation grouping method. Args: 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(list[array[float]], list[list[Observable]], list[array[float]]): tuple containing grouped coefficients, grouped observables and basis rotation transformation matrices **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])] .. details:: :title: Theory A second-quantized molecular Hamiltonian can be constructed in the `chemist notation <>`_ format following Eq. (1) of [`PRX Quantum 2, 030305, 2021 <>`_] as .. math:: 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 :math:`V_{pqrs}` denotes a two-electron integral in the chemist notation and :math:`T_{pq}` is obtained from the one- and two-electron integrals, :math:`h_{pq}` and :math:`h_{pqrs}`, as .. math:: T_{pq} = h_{pq} - \frac{1}{2} \sum_s h_{pssq}. The tensor :math:`V` can be converted to a matrix which is indexed by the indices :math:`pq` and :math:`rs` and eigendecomposed up to a rank :math:`R` to give .. math:: V_{pqrs} = \sum_r^R L_{pq}^{(r)} L_{rs}^{(r) T}, where :math:`L` denotes the matrix of eigenvectors of the matrix :math:`V`. The molecular Hamiltonian can then be rewritten following Eq. (7) of [`Phys. Rev. Research 3, 033055, 2021 <>`_] as .. math:: 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 :math:`T` and :math:`L^{(r)}` matrix is diagonal. The Hamiltonian can then be written following Eq. (2) of [`npj Quantum Information, 7, 23 (2021) <>`_] as .. math:: 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 :math:`d` are obtained by diagonalizing the :math:`T` and :math:`L^{(r)}` matrices. The number operators :math:`n_p = a_p^{\dagger} a_p` can be converted to qubit operators using .. math:: n_p = \frac{1-Z_p}{2}, where :math:`Z_p` is the Pauli :math:`Z` operator applied to qubit :math:`p`. This gives the qubit Hamiltonian .. math:: 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 :math:`O = \sum_i c_i P_i` is a linear combination of Pauli words :math:`P_i` that are a tensor product of Pauli :math:`Z` and Identity operators. This allows all the Pauli words in each of the :math:`O` terms to be measured simultaneously. This function returns the coefficients and the Pauli words grouped for each of the :math:`O` terms as well as the basis rotation transformation matrices that are constructed from the eigenvectors of the :math:`T` and :math:`L^{(r)}` matrices. Each column of the transformation matrix is an eigenvector of the corresponding :math:`T` or :math:`L^{(r)}` matrix. """ num_orbitals = one_electron.shape[0] * 2 one_body_tensor, chemist_two_body_tensor = _chemist_transform(one_electron, two_electron) chemist_one_body_tensor = np.kron(one_body_tensor, np.eye(2)) # account for spin t_eigvals, t_eigvecs = np.linalg.eigh(chemist_one_body_tensor) factors, _, _ = factorize(chemist_two_body_tensor, tol_factor=tol_factor) factors = [np.kron(factor, np.eye(2)) for factor in factors] # account for spin v_coeffs, v_unitaries = np.linalg.eigh(factors) indices = [np.argsort(v_coeff)[::-1] for v_coeff in v_coeffs] v_coeffs = [v_coeff[indices[idx]] for idx, v_coeff in enumerate(v_coeffs)] v_unitaries = [v_unitary[:, indices[idx]] for idx, v_unitary in enumerate(v_unitaries)] ops_t = 0.0 for p in range(num_orbitals): ops_t += 0.5 * t_eigvals[p] * (qml.Identity(p) - qml.PauliZ(p)) ops_l = [] for idx in range(len(factors)): ops_l_ = 0.0 for p in range(num_orbitals): for q in range(num_orbitals): ops_l_ += ( v_coeffs[idx][p] * v_coeffs[idx][q] * 0.25 * ( qml.Identity(p) - qml.PauliZ(p) - qml.PauliZ(q) + qml.pauli.pauli_mult_with_phase(qml.PauliZ(p), qml.PauliZ(q))[0] ) ) ops_l.append(ops_l_) ops = [ops_t] + ops_l c_group = [op.coeffs for op in ops] o_group = [op.ops for op in ops] u_transform = list([t_eigvecs] + list(v_unitaries)) # Inverse of diagonalizing unitaries return c_group, o_group, u_transform
def _chemist_transform(one_body_tensor=None, two_body_tensor=None, spatial_basis=True): r"""Transforms one- and two-body terms in physicists' notation to `chemists' notation <>`_\ . This converts the input two-body tensor :math:`h_{pqrs}` that constructs :math:`\sum_{pqrs} h_{pqrs} a^\dagger_p a^\dagger_q a_r a_s` to a transformed two-body tensor :math:`V_{pqrs}` that follows the chemists' convention to construct :math:`\sum_{pqrs} V_{pqrs} a^\dagger_p a_q a^\dagger_r a_s` in the spatial basis. During the tranformation, some extra one-body terms come out. These are returned as a one-body tensor :math:`T_{pq}` in the chemists' notation either as is or after summation with the input one-body tensor :math:`h_{pq}`, if provided. Args: one_body_tensor (array[float]): a one-electron integral tensor giving the :math:`h_{pq}`. two_body_tensor (array[float]): a two-electron integral tensor giving the :math:`h_{pqrs}`. spatial_basis (bool): True if the integral tensor are passed in spatial-orbital basis. False if they are in spin basis. Returns: tuple(array[float], array[float]) or tuple(array[float],): transformed one-body tensor :math:`T_{pq}` and two-body tensor :math:`V_{pqrs}` for the provided terms. **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)() >>> qml.qchem.factorization._chemist_transform(two_body_tensor=two, spatial_basis=True) (tensor([[-0.427983, -0. ], [-0. , -0.439431]], requires_grad=True), tensor([[[[0.337378, 0. ], [0. , 0.331856]], [[0. , 0.090605], [0.090605 , 0. ]]], [[[0. , 0.090605], [0.090605 , 0. ]], [[0.331856, 0. ], [0. , 0.348826]]]], requires_grad=True)) .. details:: :title: Theory The two-electron integral in physicists' notation is defined as: .. math:: \langle pq \vert rs \rangle = h_{pqrs} = \int \frac{\chi^*_{p}(x_1) \chi^*_{q}(x_2) \chi_{r}(x_1) \chi_{s}(x_2)}{|r_1 - r_2|} dx_1 dx_2, while in chemists' notation it is written as: .. math:: [pq \vert rs] = V_{pqrs} = \int \frac{\chi^*_{p}(x_1) \chi_{q}(x_1) \chi^*_{r}(x_2) \chi_{s}(x_2)}{|r_1 - r_2|} dx_1 dx_2. In the spin basis, this index reordering :math:`pqrs \rightarrow psrq` leads to formation of one-body terms :math:`h_{prrs}` that come out during the coversion: .. math:: h_{prrs} = \int \frac{\chi^*_{p}(x_1) \chi^*_{r}(x_2) \chi_{r}(x_1) \chi_{s}(x_2)}{|x_1 - x_2|} dx_1 dx_2, where both :math:`\chi_{r}(x_1)` and :math:`\chi_{r}(x_2)` will have same spin functions, i.e., :math:`\chi_{r}(x_i) = \phi(r_i)\alpha(\omega)` or :math:`\chi_{r}(x_i) = \phi(r_i)\beta(\omega)`\ . These are added to the one-electron integral tensor :math:`h_{pq}` to compute :math:`T_{pq}`\ . """ chemist_two_body_coeffs, chemist_one_body_coeffs = None, None if one_body_tensor is not None: chemist_one_body_coeffs = one_body_tensor.copy() if two_body_tensor is not None: chemist_two_body_coeffs = np.swapaxes(two_body_tensor, 1, 3) one_body_coeffs = -np.einsum("prrs", chemist_two_body_coeffs) if chemist_one_body_coeffs is None: chemist_one_body_coeffs = np.zeros_like(one_body_coeffs) if spatial_basis: chemist_two_body_coeffs = 0.5 * chemist_two_body_coeffs one_body_coeffs = 0.5 * one_body_coeffs chemist_one_body_coeffs += one_body_coeffs return (x for x in [chemist_one_body_coeffs, chemist_two_body_coeffs] if x is not None)