qml.templates.embeddings.QAOAEmbedding

class QAOAEmbedding(features, weights, wires, local_field='Y', do_queue=True)[source]

Bases: pennylane.operation.Operation

Encodes \(N\) features into \(n>N\) 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 RX rotations. The Ising ansatz consists of trainable two-qubit ZZ interactions \(e^{-i \frac{\alpha}{2} \sigma_z \otimes \sigma_z}\) (in PennyLane represented by the MultiRZ gate), and trainable local fields \(e^{-i \frac{\beta}{2} \sigma_{\mu}}\), where \(\sigma_{\mu}\) can be chosen to be \(\sigma_{x}\), \(\sigma_{y}\) or \(\sigma_{z}\) (default choice is \(\sigma_{y}\) or the RY gate), and \(\alpha, \beta\) 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 \(\alpha, \beta\) parameters for each layer. The number of layers \(L\) is derived from the first dimension of weights, which has the following shape:

  • \((L, 1)\), if the embedding acts on a single wire,

  • \((L, 3)\), if the embedding acts on two wires,

  • \((L, 2n)\) else.

After the \(L\) th layer, another set of feature-encoding RX gates is applied.

This is an example for the full embedding circuit using 2 layers, 3 features, 4 wires, and RY local fields:


../../_images/qaoa_layers.png

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.

Parameters
  • 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

The QAOA embedding encodes an \(n\)-dimensional feature vector into at most \(n\) qubits. The embedding applies layers of a circuit, and each layer is defined by a set of weight parameters.

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 shape() and used when creating randomly initialised weight tensors:

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:

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:

@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.

@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.

base_name

Get base name of the operator.

eigvals

Eigenvalues of an instantiated operator.

generator

Generator of the operation.

grad_method

Gradient computation method.

grad_recipe

Gradient recipe for the parameter-shift method.

id

String for the ID of the operator.

inverse

Boolean determining if the inverse of the operation was requested.

matrix

Matrix representation of an instantiated operator in the computational basis.

name

Get and set the name of the operator.

num_params

num_wires

par_domain

parameters

Current parameter values.

string_for_inverse

wires

Wires of this operator.

base_name

Get base name of the operator.

eigvals
generator

Generator of the operation.

A length-2 list [generator, scaling_factor], where

  • generator is an existing PennyLane operation class or \(2\times 2\) Hermitian array that acts as the generator of the current operation

  • scaling_factor represents a scaling factor applied to the generator operation

For example, if \(U(\theta)=e^{i0.7\theta \sigma_x}\), then \(\sigma_x\), with scaling factor \(s\), is the generator of operator \(U(\theta)\):

generator = [PauliX, 0.7]

Default is [None, 1], indicating the operation has no generator.

grad_method

Gradient computation method.

  • 'A': analytic differentiation using the parameter-shift method.

  • 'F': finite difference numerical differentiation.

  • None: the operation may not be differentiated.

Default is 'F', or None if the Operation has zero parameters.

grad_recipe = None

Gradient recipe for the parameter-shift method.

This is a tuple with one nested list per operation parameter. For parameter \(\phi_k\), the nested list contains elements of the form \([c_i, a_i, s_i]\) where \(i\) is the index of the term, resulting in a gradient recipe of

\[\frac{\partial}{\partial\phi_k}f = \sum_{i} c_i f(a_i \phi_k + s_i).\]

If None, the default gradient recipe containing the two terms \([c_0, a_0, s_0]=[1/2, 1, \pi/2]\) and \([c_1, a_1, s_1]=[-1/2, 1, -\pi/2]\) is assumed for every parameter.

Type

tuple(Union(list[list[float]], None)) or None

id

String for the ID of the operator.

inverse

Boolean determining if the inverse of the operation was requested.

matrix
name

Get and set the name of the operator.

num_params = 2
num_wires = -1
par_domain = 'A'
parameters

Current parameter values.

string_for_inverse = '.inv'
wires

Wires of this operator.

Returns

wires

Return type

Wires

adjoint([do_queue])

Create an operation that is the adjoint of this one.

decomposition(*params, wires)

Returns a template decomposing the operation into other quantum operations.

expand()

Returns a tape containing the decomposed operations, rather than a list.

get_parameter_shift(idx[, shift])

Multiplier and shift for the given parameter, based on its gradient recipe.

inv()

Inverts the operation, such that the inverse will be used for the computations by the specific device.

queue()

Append the operator to the Operator queue.

shape(n_layers, n_wires)

Returns the shape of the weight tensor required for this template.

adjoint(do_queue=False)

Create an operation that is the adjoint of this one.

Adjointed operations are the conjugated and transposed version of the original operation. Adjointed ops are equivalent to the inverted operation for unitary gates.

Parameters

do_queue – Whether to add the adjointed gate to the context queue.

Returns

The adjointed operation.

static decomposition(*params, wires)

Returns a template decomposing the operation into other quantum operations.

expand()[source]

Returns a tape containing the decomposed operations, rather than a list.

Returns

Returns a quantum tape that contains the operations decomposition, or if not implemented, simply the operation itself.

Return type

JacobianTape

get_parameter_shift(idx, shift=1.5707963267948966)

Multiplier and shift for the given parameter, based on its gradient recipe.

Parameters

idx (int) – parameter index

Returns

list of multiplier, coefficient, shift for each term in the gradient recipe

Return type

list[[float, float, float]]

inv()

Inverts the operation, such that the inverse will be used for the computations by the specific device.

This method concatenates a string to the name of the operation, to indicate that the inverse will be used for computations.

Any subsequent call of this method will toggle between the original operation and the inverse of the operation.

Returns

operation to be inverted

Return type

Operator

queue()

Append the operator to the Operator queue.

static shape(n_layers, n_wires)[source]

Returns the shape of the weight tensor required for this template.

Parameters
  • n_layers (int) – number of layers

  • n_wires (int) – number of qubits

Returns

shape

Return type

tuple[int]