qml.templates.embeddings.QAOAEmbedding¶

QAOAEmbedding
(features, weights, wires, local_field='Y')[source]¶ 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 (array) – array of features to encode
weights (array) – array of weights
wires (Sequence[int] or int) – n qubit indices 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))
Using parameter initialization functions
The initial weight parameters can alternatively be generated by utility functions from the
pennylane.init
module, for example using the functionqaoa_embedding_normal()
:from pennylane.init import qaoa_embedding_normal weights = qaoa_embedding_normal(n_layers=2, n_wires=2, mean=0, std=0.2)
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:
o = GradientDescentOptimizer() for i in range(10): weights = o.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(pars): weights = pars[0] features = pars[1] 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]] pars = [weights, features] o = GradientDescentOptimizer() for i in range(10): pars = o.step(circuit2, pars) print("Step ", i, " weights = ", pars[0], " features = ", pars[1])
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.
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