Source code for pennylane.templates.layers.particle_conserving_u1

# 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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r"""
Contains the hardware-efficient ParticleConservingU1 template.
"""
# pylint: disable-msg=too-many-branches,too-many-arguments,protected-access
import numpy as np
import pennylane as qml
from pennylane.operation import Operation, AnyWires


def decompose_ua(phi, wires=None):
    r"""Appends the circuit decomposing the controlled application of the unitary
    :math:`U_A(\phi)`

    .. math::

        U_A(\phi) = \left(\begin{array}{cc} 0 & e^{-i\phi} \\ e^{-i\phi} & 0 \\ \end{array}\right)

    in terms of the quantum operations supported by PennyLane.

    :math:`U_A(\phi)` is used in `arXiv:1805.04340 <https://arxiv.org/abs/1805.04340>`_,
    to define two-qubit exchange gates required to build particle-conserving
    VQE ansatze for quantum chemistry simulations. See :func:`~.ParticleConservingU1`.

    :math:`U_A(\phi)` is expressed in terms of ``PhaseShift``, ``Rot`` and ``PauliZ`` operations
    :math:`U_A(\phi) = R_\phi(-2\phi) R(-\phi, \pi, \phi) \sigma_z`.

    Args:
        phi (float): angle :math:`\phi` defining the unitary :math:`U_A(\phi)`
        wires (Iterable): the wires ``n`` and ``m`` the circuit acts on

    Returns:
          list[.Operator]: sequence of operators defined by this function
    """
    op_list = []
    n, m = wires

    op_list.append(qml.CZ(wires=wires))
    op_list.append(qml.CRot(-phi, np.pi, phi, wires=wires))

    # decomposition of C-PhaseShift(2*phi) gate
    op_list.append(qml.PhaseShift(-phi, wires=m))
    op_list.append(qml.CNOT(wires=wires))
    op_list.append(qml.PhaseShift(phi, wires=m))
    op_list.append(qml.CNOT(wires=wires))
    op_list.append(qml.PhaseShift(-phi, wires=n))

    return op_list


def u1_ex_gate(phi, theta, wires=None):
    r"""Appends the two-qubit exchange gate :math:`U_{1,\mathrm{ex}}` proposed
    in `arXiv:1805.04340 <https://arxiv.org/abs/1805.04340>`_ to build
    a hardware-efficient particle-conserving VQE ansatz for quantum chemistry
    simulations.

    Args:
        phi (float): angle entering the unitary :math:`U_A(\phi)`
        theta (float): angle entering the rotation :math:`R(0, 2\theta, 0)`
        wires (list[Iterable]): the two wires ``n`` and ``m`` the circuit acts on

    Returns:
        list[.Operator]: sequence of operators defined by this function
    """
    op_list = []

    # C-UA(phi)
    op_list.extend(decompose_ua(phi, wires=wires))

    op_list.append(qml.CZ(wires=wires[::-1]))
    op_list.append(qml.CRot(0, 2 * theta, 0, wires=wires[::-1]))

    # C-UA(-phi)
    op_list.extend(decompose_ua(-phi, wires=wires))

    return op_list


[docs]class ParticleConservingU1(Operation): r"""Implements the heuristic VQE ansatz for quantum chemistry simulations using the particle-conserving gate :math:`U_{1,\mathrm{ex}}` proposed by Barkoutsos *et al.* in `arXiv:1805.04340 <https://arxiv.org/abs/1805.04340>`_. This template prepares :math:`N`-qubit trial states by applying :math:`D` layers of the entangler block :math:`U_\mathrm{ent}(\vec{\phi}, \vec{\theta})` to the Hartree-Fock state .. math:: \vert \Psi(\vec{\phi}, \vec{\theta}) \rangle = \hat{U}^{(D)}_\mathrm{ent}(\vec{\phi}_D, \vec{\theta}_D) \dots \hat{U}^{(2)}_\mathrm{ent}(\vec{\phi}_2, \vec{\theta}_2) \hat{U}^{(1)}_\mathrm{ent}(\vec{\phi}_1, \vec{\theta}_1) \vert \mathrm{HF}\rangle. The circuit implementing the entangler blocks is shown in the figure below: | .. figure:: ../../_static/templates/layers/particle_conserving_u1.png :align: center :width: 50% :target: javascript:void(0); | The repeated units across several qubits are shown in dotted boxes. Each layer contains :math:`N-1` particle-conserving two-parameter exchange gates :math:`U_{1,\mathrm{ex}}(\phi, \theta)` that act on pairs of nearest neighbors qubits. The unitary matrix representing :math:`U_{1,\mathrm{ex}}(\phi, \theta)` is given by (see `arXiv:1805.04340 <https://arxiv.org/abs/1805.04340>`_), .. math:: U_{1, \mathrm{ex}}(\phi, \theta) = \left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & \mathrm{cos}(\theta) & e^{i\phi} \mathrm{sin}(\theta) & 0 \\ 0 & e^{-i\phi} \mathrm{sin}(\theta) & -\mathrm{cos}(\theta) & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right). The figure below shows the circuit decomposing :math:`U_{1, \mathrm{ex}}` in elementary gates. The Pauli matrix :math:`\sigma_z` and single-qubit rotation :math:`R(0, 2 \theta, 0)` apply the Pauli Z operator and an arbitrary rotation on the qubit ``n`` with qubit ``m`` bein the control qubit, | .. figure:: ../../_static/templates/layers/u1_decomposition.png :align: center :width: 80% :target: javascript:void(0); | :math:`U_A(\phi)` is the unitary matrix .. math:: U_A(\phi) = \left(\begin{array}{cc} 0 & e^{-i\phi} \\ e^{-i\phi} & 0 \\ \end{array}\right), which is applied controlled on the state of qubit ``m`` and can be further decomposed in terms of the `quantum operations <https://pennylane.readthedocs.io/en/stable/introduction/operations.html>`_ supported by Pennylane, | .. figure:: ../../_static/templates/layers/ua_decomposition.png :align: center :width: 70% :target: javascript:void(0); | where, | .. figure:: ../../_static/templates/layers/phaseshift_decomposition.png :align: center :width: 65% :target: javascript:void(0); | The quantum circuits above decomposing the unitaries :math:`U_{1,\mathrm{ex}}(\phi, \theta)` and :math:`U_A(\phi)` are implemented by the ``u1_ex_gate`` and ``decompose_ua`` functions, respectively. :math:`R_\phi` refers to the ``PhaseShift`` gate in the circuit diagram. Args: weights (tensor_like): Array of weights of shape ``(D, M, 2)``. ``D`` is the number of entangler block layers and :math:`M=N-1` is the number of exchange gates :math:`U_{1,\mathrm{ex}}` per layer. wires (Iterable): wires that the template acts on. init_state (tensor_like): iterable or shape ``(len(wires),)`` tensor representing the Hartree-Fock state used to initialize the wires. If ``None``, a tuple of zeros is selected as initial state. .. details:: :title: Usage Details #. The number of wires :math:`N` has to be equal to the number of spin orbitals included in the active space. #. The number of trainable parameters scales linearly with the number of layers as :math:`2D(N-1)`. An example of how to use this template is shown below: .. code-block:: python import pennylane as qml import numpy as np from functools import partial # Build the electronic Hamiltonian symbols, coordinates = (['H', 'H'], np.array([0., 0., -0.66140414, 0., 0., 0.66140414])) h, qubits = qml.qchem.molecular_hamiltonian(symbols, coordinates) # Define the Hartree-Fock state electrons = 2 ref_state = qml.qchem.hf_state(electrons, qubits) # Define the device dev = qml.device('default.qubit', wires=qubits) # Define the ansatz ansatz = partial(qml.ParticleConservingU1, init_state=ref_state, wires=dev.wires) # Define the cost function @qml.qnode(dev) def cost_fn(params): ansatz(params) return qml.expval(h) # Compute the expectation value of 'h' layers = 2 shape = qml.ParticleConservingU1.shape(layers, qubits) params = np.random.random(shape) print(cost_fn(params)) **Parameter shape** The shape of the trainable weights tensor can be computed by the static method :meth:`~.ParticleConservingU1.shape` and used when creating randomly initialised weight tensors: .. code-block:: python shape = qml.ParticleConservingU1.shape(n_layers=2, n_wires=2) params = np.random.random(size=shape) """ num_wires = AnyWires grad_method = None def __init__(self, weights, wires, init_state=None, id=None): if len(wires) < 2: raise ValueError( f"Expected the number of qubits to be greater than one; " f"got wires {wires}" ) shape = qml.math.shape(weights) if len(shape) != 3: raise ValueError(f"Weights tensor must be 3-dimensional; got shape {shape}") if shape[1] != len(wires) - 1: raise ValueError( f"Weights tensor must have second dimension of length {len(wires) - 1}; got {shape[1]}" ) if shape[2] != 2: raise ValueError( f"Weights tensor must have third dimension of length 2; got {shape[2]}" ) init_state = tuple(0 for _ in wires) if init_state is None else init_state self._hyperparameters = {"init_state": tuple(init_state)} super().__init__(weights, wires=wires, id=id) @property def num_params(self): return 1
[docs] @staticmethod def compute_decomposition(weights, wires, init_state): # pylint: disable=arguments-differ r"""Representation of the ParticleConservingU1operator as a product of other operators. .. math:: O = O_1 O_2 \dots O_n. .. seealso:: :meth:`~.ParticleConservingU1.decomposition`. Args: weights (tensor_like): Array of weights of shape ``(D, M, 2)``. ``D`` is the number of entangler block layers and :math:`M=N-1` is the number of exchange gates :math:`U_{1,\mathrm{ex}}` per layer. wires (Any or Iterable[Any]): wires that the operator acts on init_state (tensor_like): iterable or shape ``(len(wires),)`` tensor representing the Hartree-Fock state used to initialize the wires Returns: list[.Operator]: decomposition of the operator **Example** >>> weights = torch.tensor([[[0.3, 1.]]]) >>> qml.ParticleConservingU1.compute_decomposition(weights, wires=["a", "b"], init_state=[0, 1]) [BasisEmbedding(wires=['a', 'b']), CZ(wires=['a', 'b']), CRot(tensor(-0.3000), 3.141592653589793, tensor(0.3000), wires=['a', 'b']), PhaseShift(tensor(-0.3000), wires=['b']), CNOT(wires=['a', 'b']), PhaseShift(tensor(0.3000), wires=['b']), CNOT(wires=['a', 'b']), PhaseShift(tensor(-0.3000), wires=['a']), CZ(wires=['b', 'a']), CRot(0, tensor(2.), 0, wires=['b', 'a']), CZ(wires=['a', 'b']), CRot(tensor(0.3000), 3.141592653589793, tensor(-0.3000), wires=['a', 'b']), PhaseShift(tensor(0.3000), wires=['b']), CNOT(wires=['a', 'b']), PhaseShift(tensor(-0.3000), wires=['b']), CNOT(wires=['a', 'b']), PhaseShift(tensor(0.3000), wires=['a'])] """ nm_wires = [wires[l : l + 2] for l in range(0, len(wires) - 1, 2)] nm_wires += [wires[l : l + 2] for l in range(1, len(wires) - 1, 2)] n_layers = qml.math.shape(weights)[0] op_list = [qml.BasisEmbedding(init_state, wires=wires)] for l in range(n_layers): for i, wires_ in enumerate(nm_wires): op_list.extend(u1_ex_gate(weights[l, i, 0], weights[l, i, 1], wires=wires_)) return op_list
[docs] @staticmethod def shape(n_layers, n_wires): r"""Returns the shape of the weight tensor required for this template. Args: n_layers (int): number of layers n_wires (int): number of qubits Returns: tuple[int]: shape """ if n_wires < 2: raise ValueError( f"The number of qubits must be greater than one; got 'n_wires' = {n_wires}" ) return n_layers, n_wires - 1, 2