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 1dimensional Ising model. The featureencoding circuit associates features with the angles of
RX
rotations. The Ising ansatz consists of trainable twoqubit ZZ interactions \(e^{i \frac{\alpha}{2} \sigma_z \otimes \sigma_z}\) (in PennyLane represented by theMultiRZ
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 theRY
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 featureencoding 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 ofweights
, 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 featureencoding
RX
gates is applied.This is an example for the full embedding circuit using 2 layers, 3 features, 4 wires, and
RY
local fields:Note
QAOAEmbedding
supports gradient computations with respect to both thefeatures
and theweights
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
Usage Details
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 builtin 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 chooselocal_field='Z'
orlocal_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 1dimensional Ising model.Attributes
Get base name of the operator.
Eigenvalues of an instantiated operator.
Generator of the operation.
Gradient computation method.
Gradient recipe for the parametershift method.
String for the ID of the operator.
Boolean determining if the inverse of the operation was requested.
Matrix representation of an instantiated operator in the computational basis.
Get and set the name of the operator.
Current parameter values.
Wires of this operator.

base_name
¶ Get base name of the operator.

eigvals
¶

generator
¶ Generator of the operation.
A length2 list
[generator, scaling_factor]
, wheregenerator
is an existing PennyLane operation class or \(2\times 2\) Hermitian array that acts as the generator of the current operationscaling_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 parametershift method.'F'
: finite difference numerical differentiation.None
: the operation may not be differentiated.
Default is
'F'
, orNone
if the Operation has zero parameters.

grad_recipe
= None¶ Gradient recipe for the parametershift 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'¶
Methods
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

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.
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