Source code for pennylane.transforms.decompositions.single_qubit_unitary

# Copyright 2018-2021 Xanadu Quantum Technologies Inc.

# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
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#     http://www.apache.org/licenses/LICENSE-2.0

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"""Contains transforms and helpers functions for decomposing arbitrary unitary
operations into elementary gates.
"""

import pennylane as qml
from pennylane import math


def _convert_to_su2(U):
    r"""Convert a 2x2 unitary matrix to :math:`SU(2)`.

    Args:
        U (array[complex]): A matrix, presumed to be :math:`2 \times 2` and unitary.

    Returns:
        array[complex]: A :math:`2 \times 2` matrix in :math:`SU(2)` that is
        equivalent to U up to a global phase.
    """
    # Compute the determinant
    det = U[0, 0] * U[1, 1] - U[0, 1] * U[1, 0]

    exp_angle = -1j * math.cast_like(math.angle(det), 1j) / 2
    return U * math.exp(exp_angle)


[docs]def zyz_decomposition(U, wire): r"""Recover the decomposition of a single-qubit matrix :math:`U` in terms of elementary operations. Diagonal operations can be converted to a single :class:`.RZ` gate, while non-diagonal operations will be converted to a :class:`.Rot` gate that implements the original operation up to a global phase in the form :math:`RZ(\omega) RY(\theta) RZ(\phi)`. .. warning:: When used with ``jax.jit``, all unitaries will be converted to :class:`.Rot` gates, including those that are diagonal. Args: U (tensor): A 2 x 2 unitary matrix. wire (Union[Wires, Sequence[int] or int]): The wire on which to apply the operation. Returns: list[qml.Operation]: A ``Rot`` gate on the specified wire that implements ``U`` up to a global phase, or an equivalent ``RZ`` gate if ``U`` is diagonal. **Example** Suppose we would like to apply the following unitary operation: .. code-block:: python3 U = np.array([ [-0.28829348-0.78829734j, 0.30364367+0.45085995j], [ 0.53396245-0.10177564j, 0.76279558-0.35024096j] ]) For PennyLane devices that cannot natively implement ``QubitUnitary``, we can instead recover a ``Rot`` gate that implements the same operation, up to a global phase: >>> decomp = zyz_decomposition(U, 0) >>> decomp [Rot(-0.24209529417800013, 1.14938178234275, 1.7330581433950871, wires=[0])] """ U = _convert_to_su2(U) # If the value of U is not abstract, we can include a conditional statement # that will check if the off-diagonal elements are 0; if so, just use one RZ if not math.is_abstract(U): if math.allclose(U[0, 1], 0.0): return [qml.RZ(2 * math.angle(U[1, 1]), wires=wire)] # Derive theta from the off-diagonal element. Clip to ensure valid arcsin input element = math.clip(math.abs(U[0, 1]), 0, 1) theta = 2 * math.arcsin(element) # Compute phi and omega from the angles of the top row; use atan2 to keep # the angle within -np.pi and np.pi, and add very small values to avoid the # undefined case of 0/0. We add a smaller value to the imaginary part than # the real part because it is imag / real in the definition of atan2. angle_U00 = math.arctan2(math.imag(U[0, 0]) + 1e-128, math.real(U[0, 0]) + 1e-64) angle_U10 = math.arctan2(math.imag(U[1, 0]) + 1e-128, math.real(U[1, 0]) + 1e-64) phi = -angle_U10 - angle_U00 omega = angle_U10 - angle_U00 return [qml.Rot(phi, theta, omega, wires=wire)]