Source code for pennylane.qchem.givens_decomposition

# Copyright 2018-2023 Xanadu Quantum Technologies Inc.

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This module contains the functions needed for performing basis transformations defined by a set of fermionic ladder operators.

import pennylane as qml
from pennylane import numpy as np

def _givens_matrix(a, b, left=True, tol=1e-8):
    r"""Build a :math:`2 \times 2` Givens rotation matrix :math:`G`.

    When the matrix :math:`G` is applied to a vector :math:`[a,\ b]^T` the following would happen:

    .. math::

            G \times \begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} 0 \\ r \end{bmatrix} \quad \quad \quad \begin{bmatrix} a \\ b \end{bmatrix} \times G = \begin{bmatrix} r \\ 0 \end{bmatrix},

    where :math:`r` is a complex number.

        a (float or complex): first element of the vector for which the Givens matrix is being computed
        b (float or complex): second element of the vector for which the Givens matrix is being computed
        left (bool): determines if the Givens matrix is being applied from the left side or right side.
        tol (float): determines tolerance limits for :math:`|a|` and :math:`|b|` under which they are considered as zero.

        np.ndarray (or tensor): Givens rotation matrix

    abs_a, abs_b = np.abs(a), np.abs(b)
    if abs_a < tol:
        cosine, sine, phase = 1.0, 0.0, 1.0
    elif abs_b < tol:
        cosine, sine, phase = 0.0, 1.0, 1.0
        hypot = np.hypot(abs_a, abs_b)
        cosine = abs_b / hypot
        sine = abs_a / hypot
        phase = 1.0 * b / abs_b * a.conjugate() / abs_a

    if left:
        return np.array([[phase * cosine, -sine], [phase * sine, cosine]])

    return np.array([[phase * sine, cosine], [-phase * cosine, sine]])

[docs]def givens_decomposition(unitary): r"""Decompose a unitary into a sequence of Givens rotation gates with phase shifts and a diagonal phase matrix. This decomposition is based on the construction scheme given in `Optica, 3, 1460 (2016) <>`_\ , which allows one to write any unitary matrix :math:`U` as: .. math:: U = D \left(\prod_{(m, n) \in G} T_{m, n}(\theta, \phi)\right), where :math:`D` is a diagonal phase matrix, :math:`T(\theta, \phi)` is the Givens rotation gates with phase shifts and :math:`G` defines the specific ordered sequence of the Givens rotation gates acting on wires :math:`(m, n)`. The unitary for the :math:`T(\theta, \phi)` gates appearing in the decomposition is of the following form: .. math:: T(\theta, \phi) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & e^{i \phi} \cos(\theta) & -\sin(\theta) & 0 \\ 0 & e^{i \phi} \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}, where :math:`\theta \in [0, \pi/2]` is the angle of rotation in the :math:`\{|01\rangle, |10\rangle \}` subspace and :math:`\phi \in [0, 2 \pi]` represents the phase shift at the first wire. **Example** .. code-block:: python unitary = np.array([[ 0.73678+0.27511j, -0.5095 +0.10704j, -0.06847+0.32515j], [-0.21271+0.34938j, -0.38853+0.36497j, 0.61467-0.41317j], [ 0.41356-0.20765j, -0.00651-0.66689j, 0.32839-0.48293j]]) phase_mat, ordered_rotations = givens_decomposition(unitary) >>> phase_mat tensor([-0.20604358+0.9785369j , -0.82993272+0.55786114j, 0.56230612-0.82692833j], requires_grad=True) >>> ordered_rotations [(tensor([[-0.65087861-0.63937521j, -0.40933651-0.j ], [-0.29201359-0.28685265j, 0.91238348-0.j ]], requires_grad=True), (0, 1)), (tensor([[ 0.47970366-0.33308926j, -0.8117487 -0.j ], [ 0.66677093-0.46298215j, 0.5840069 -0.j ]], requires_grad=True), (1, 2)), (tensor([[ 0.36147547+0.73779454j, -0.57008306-0.j ], [ 0.2508207 +0.51194108j, 0.82158706-0.j ]], requires_grad=True), (0, 1))] Args: unitary (tensor): unitary matrix on which decomposition will be performed Returns: (np.ndarray, list[(np.ndarray, tuple)]): diagonal elements of the phase matrix :math:`D` and Givens rotation matrix :math:`T` with their indices. Raises: ValueError: if the provided matrix is not square. .. details:: :title: Theory and Pseudocode **Givens Rotation** Applying the Givens rotation :math:`T(\theta, \phi)` performs the following transformation of the basis states: .. math:: &|00\rangle \mapsto |00\rangle\\ &|01\rangle \mapsto e^{i \phi} \cos(\theta) |01\rangle - \sin(\theta) |10\rangle\\ &|10\rangle \mapsto e^{i \phi} \sin(\theta) |01\rangle + \cos(\theta) |10\rangle\\ &|11\rangle \mapsto |11\rangle. **Pseudocode** The algorithm that implements the decomposition is the following: .. code-block:: python def givens_decomposition(U): for i in range(1, N): if i % 2: for j in range(0, i): # Find T^-1(i-j, i-j+1) matrix that nulls element (N-j, i-j) of U # Update U = U @ T^-1(i-j, i-j+1) else: for j in range(1, i): # Find T(N+j-i-1, N+j-i) matrix that nulls element (N+j-i, j) of U # Update U = T(N+j-i-1, N+j-i) @ U """ unitary, (M, N) = qml.math.toarray(unitary).copy(), unitary.shape if M != N: raise ValueError(f"The unitary matrix should be of shape NxN, got {unitary.shape}") left_givens, right_givens = [], [] for i in range(1, N): if i % 2: for j in range(0, i): indices = [i - j - 1, i - j] grot_mat = _givens_matrix(*unitary[N - j - 1, indices].T, left=True) unitary[:, indices] = unitary[:, indices] @ grot_mat.T right_givens.append((grot_mat.conj(), indices)) else: for j in range(1, i + 1): indices = [N + j - i - 2, N + j - i - 1] grot_mat = _givens_matrix(*unitary[indices, j - 1], left=False) unitary[indices] = grot_mat @ unitary[indices] left_givens.append((grot_mat, indices)) nleft_givens = [] for grot_mat, (i, j) in reversed(left_givens): sphase_mat = np.diag(np.diag(unitary)[[i, j]]) decomp_mat = grot_mat.conj().T @ sphase_mat givens_mat = _givens_matrix(*decomp_mat[1, :].T) nphase_mat = decomp_mat @ givens_mat.T # check for T_{m,n}^{-1} x D = D x T. if not np.allclose(nphase_mat @ givens_mat.conj(), decomp_mat): # pragma: no cover raise ValueError("Failed to shift phase transposition.") unitary[i, i], unitary[j, j] = np.diag(nphase_mat) nleft_givens.append((givens_mat.conj(), (i, j))) phases, ordered_rotations = np.diag(unitary), [] for grot_mat, (i, j) in list(reversed(nleft_givens)) + list(reversed(right_givens)): if not np.all(np.isreal(grot_mat[0, 1]) and np.isreal(grot_mat[1, 1])): # pragma: no cover raise ValueError(f"Incorrect Givens Rotation encountered, {grot_mat}") ordered_rotations.append((grot_mat, (i, j))) return phases, ordered_rotations