Source code for pennylane.templates.embeddings.qaoa

# 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 QAOAEmbedding template.
"""
# pylint: disable-msg=too-many-branches,too-many-arguments,protected-access, consider-using-enumerate
import pennylane as qml
from pennylane.operation import Operation, AnyWires


def qaoa_feature_encoding_hamiltonian(features, wires):
    """Implements the encoding Hamiltonian of the QAOA embedding.

    Args:
        features (tensor_like): tensor of features to encode
        wires (Iterable): wires that the template acts on
    """
    n_features = qml.math.shape(features)[0]

    for i in range(n_features):
        qml.RX(features[i], wires=wires[i])
    for i in range(n_features, len(wires)):
        qml.Hadamard(wires=wires[i])


def qaoa_ising_hamiltonian(weights, wires, local_fields):
    """Implements the Ising-like Hamiltonian of the QAOA embedding.

    Args:
        weights (tensor_like): tensor of weights for one layer
        wires (Iterable): qubit indices that the template acts on
        local_fields (str): gate implementing the local field
    """

    if len(wires) == 1:
        local_fields(weights[0], wires=wires)

    elif len(wires) == 2:
        # deviation for 2 wires: we do not connect last to first qubit
        # with the entangling gates
        qml.MultiRZ(weights[0], wires=wires.subset([0, 1]))
        local_fields(weights[1], wires=wires[0:1])
        local_fields(weights[2], wires=wires[1:2])

    else:
        for i in range(len(wires)):
            qml.MultiRZ(weights[i], wires=wires.subset([i, i + 1], periodic_boundary=True))
        for i in range(len(wires)):
            local_fields(weights[len(wires) + i], wires=wires[i])


[docs]class QAOAEmbedding(Operation): r""" Encodes :math:`N` features into :math:`n>N` qubits, using a layered, trainable quantum circuit that is inspired by the QAOA ansatz. A single layer applies two circuits or "Hamiltonians": The first encodes the features, and the second is a variational ansatz inspired by a 1-dimensional Ising model. The feature-encoding circuit associates features with the angles of :class:`RX` rotations. The Ising ansatz consists of trainable two-qubit ZZ interactions :math:`e^{-i \frac{\alpha}{2} \sigma_z \otimes \sigma_z}` (in PennyLane represented by the :class:`~.MultiRZ` gate), and trainable local fields :math:`e^{-i \frac{\beta}{2} \sigma_{\mu}}`, where :math:`\sigma_{\mu}` can be chosen to be :math:`\sigma_{x}`, :math:`\sigma_{y}` or :math:`\sigma_{z}` (default choice is :math:`\sigma_{y}` or the ``RY`` gate), and :math:`\alpha, \beta` are adjustable gate parameters. The number of features has to be smaller or equal to the number of qubits. If there are fewer features than qubits, the feature-encoding rotation is replaced by a Hadamard gate. The argument ``weights`` contains an array of the :math:`\alpha, \beta` parameters for each layer. The number of layers :math:`L` is derived from the first dimension of ``weights``, which has the following shape: * :math:`(L, 1)`, if the embedding acts on a single wire, * :math:`(L, 3)`, if the embedding acts on two wires, * :math:`(L, 2n)` else. After the :math:`L` th layer, another set of feature-encoding :class:`RX` gates is applied. This is an example for the full embedding circuit using 2 layers, 3 features, 4 wires, and ``RY`` local fields: | .. figure:: ../../_static/qaoa_layers.png :align: center :width: 60% :target: javascript:void(0); | .. note:: ``QAOAEmbedding`` supports gradient computations with respect to both the ``features`` and the ``weights`` arguments. Note that trainable parameters need to be passed to the quantum node as positional arguments. Args: features (tensor_like): tensor of features to encode weights (tensor_like): tensor of weights wires (Iterable): wires that the template acts on local_field (str): type of local field used, one of ``'X'``, ``'Y'``, or ``'Z'`` Raises: ValueError: if inputs do not have the correct format .. UsageDetails:: The QAOA embedding encodes an :math:`n`-dimensional feature vector into at most :math:`n` qubits. The embedding applies layers of a circuit, and each layer is defined by a set of weight parameters. .. code-block:: python import pennylane as qml from pennylane.templates import QAOAEmbedding dev = qml.device('default.qubit', wires=2) @qml.qnode(dev) def circuit(weights, f=None): QAOAEmbedding(features=f, weights=weights, wires=range(2)) return qml.expval(qml.PauliZ(0)) features = [1., 2.] layer1 = [0.1, -0.3, 1.5] layer2 = [3.1, 0.2, -2.8] weights = [layer1, layer2] print(circuit(weights, f=features)) **Parameter shape** The shape of the weights argument can be computed by the static method :meth:`~.QAOAEmbedding.shape` and used when creating randomly initialised weight tensors: .. code-block:: python shape = QAOAEmbedding.shape(n_layers=2, n_wires=2) weights = np.random.random(shape) **Training the embedding** The embedding is typically trained with respect to a given cost. For example, one can train it to minimize the PauliZ expectation of the first qubit: .. code-block:: python opt = qml.GradientDescentOptimizer() for i in range(10): weights = opt.step(lambda w : circuit(w, f=features), weights) print("Step ", i, " weights = ", weights) **Training the features** In principle, also the features are trainable, which means that gradients with respect to feature values can be computed. To train both weights and features, they need to be passed to the qnode as positional arguments. If the built-in optimizer is used, they have to be merged to one input: .. code-block:: python @qml.qnode(dev) def circuit2(weights, features): QAOAEmbedding(features=features, weights=weights, wires=range(2)) return qml.expval(qml.PauliZ(0)) features = [1., 2.] weights = [[0.1, -0.3, 1.5], [3.1, 0.2, -2.8]] opt = qml.GradientDescentOptimizer() for i in range(10): weights, features = opt.step(circuit2, weights, features) print("Step ", i, "\n weights = ", weights, "\n features = ", features,"\n") **Local Fields** While by default, ``RY`` gates are used as local fields, one may also choose ``local_field='Z'`` or ``local_field='X'`` as hyperparameters of the embedding. .. code-block:: python @qml.qnode(dev) def circuit(weights, f=None): QAOAEmbedding(features=f, weights=weights, wires=range(2), local_field='Z') return qml.expval(qml.PauliZ(0)) Choosing ``'Z'`` fields implements a QAOAEmbedding where the second Hamiltonian is a 1-dimensional Ising model. """ num_params = 2 num_wires = AnyWires par_domain = "A" def __init__(self, features, weights, wires, local_field="Y", do_queue=True): if local_field == "Z": self.local_field = qml.RZ elif local_field == "X": self.local_field = qml.RX elif local_field == "Y": self.local_field = qml.RY else: raise ValueError(f"did not recognize local field {local_field}") self._preprocess(features, weights, wires) super().__init__(features, weights, wires=wires, do_queue=do_queue)
[docs] def expand(self): features = self.parameters[0] weights = self.parameters[1] # first dimension of the weights tensor determines # the number of layers repeat = qml.math.shape(weights)[0] with qml.tape.QuantumTape() as tape: for l in range(repeat): # apply alternating Hamiltonians qaoa_feature_encoding_hamiltonian(features, self.wires) qaoa_ising_hamiltonian(weights[l], self.wires, self.local_field) # repeat the feature encoding once more at the end qaoa_feature_encoding_hamiltonian(features, self.wires) return tape
@staticmethod def _preprocess(features, weights, wires): """Validate and pre-process inputs as follows: * Check that the features tensor is one-dimensional. * Check that the first dimension of the features tensor has length :math:`n` or less, where :math:`n` is the number of qubits. * Check that the shape of the weights tensor is correct for the number of qubits. Args: features (tensor_like): feature tensor weights (tensor_like): weight tensor """ shape = qml.math.shape(features) if len(shape) != 1: raise ValueError(f"Features must be a one-dimensional tensor; got shape {shape}.") n_features = shape[0] if n_features > len(wires): raise ValueError( f"Features must be of length {len(wires)} or less; got length {n_features}." ) shape = qml.math.shape(weights) repeat = shape[0] if len(wires) == 1: if shape != (repeat, 1): raise ValueError(f"Weights tensor must be of shape {(repeat, 1)}; got {shape}") elif len(wires) == 2: if shape != (repeat, 3): raise ValueError(f"Weights tensor must be of shape {(repeat, 3)}; got {shape}") else: if shape != (repeat, 2 * len(wires)): raise ValueError( f"Weights tensor must be of shape {(repeat, 2*len(wires))}; got {shape}" )
[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 == 1: return n_layers, 1 if n_wires == 2: return n_layers, 3 return n_layers, 2 * n_wires