# qml.templates.embeddings.AmplitudeEmbedding¶

class AmplitudeEmbedding(features, wires, pad_with=None, normalize=False, pad=None, do_queue=True)[source]

Encodes $$2^n$$ features into the amplitude vector of $$n$$ qubits.

By setting pad_with to a real or complex number, features is automatically padded to dimension $$2^n$$ where $$n$$ is the number of qubits used in the embedding.

To represent a valid quantum state vector, the L2-norm of features must be one. The argument normalize can be set to True to automatically normalize the features.

If both automatic padding and normalization are used, padding is executed before normalizing.

Note

On some devices, AmplitudeEmbedding must be the first operation of a quantum circuit.

Warning

At the moment, the features argument is not differentiable when using the template, and gradients with respect to the features cannot be computed by PennyLane.

Parameters
• features (tensor_like) – input tensor of dimension (2^n,), or less if pad_with is specified

• wires (Iterable) – wires that the template acts on

• pad_with (float or complex) – if not None, the input is padded with this constant to size $$2^n$$

• normalize (bool) – whether to automatically normalize the features

• pad (float or complex) – same as pad, to be deprecated

Example

Amplitude embedding encodes a normalized $$2^n$$-dimensional feature vector into the state of $$n$$ qubits:

import pennylane as qml
from pennylane.templates import AmplitudeEmbedding

dev = qml.device('default.qubit', wires=2)

@qml.qnode(dev)
def circuit(f=None):
AmplitudeEmbedding(features=f, wires=range(2))
return qml.expval(qml.PauliZ(0))

circuit(f=[1/2, 1/2, 1/2, 1/2])


The final state of the device is - up to a global phase - equivalent to the input passed to the circuit:

>>> dev.state
[0.5+0.j 0.5+0.j 0.5+0.j 0.5+0.j]


Differentiating with respect to the features

Due to non-trivial classical processing to construct the state preparation circuit, the features argument is in general not differentiable.

Normalization

The template will raise an error if the feature input is not normalized. One can set normalize=True to automatically normalize it:

@qml.qnode(dev)
def circuit(f=None):
AmplitudeEmbedding(features=f, wires=range(2), normalize=True)
return qml.expval(qml.PauliZ(0))

circuit(f=[15, 15, 15, 15])

>>> dev.state
[0.5 + 0.j, 0.5 + 0.j, 0.5 + 0.j, 0.5 + 0.j]


If the dimension of the feature vector is smaller than the number of amplitudes, one can automatically pad it with a constant for the missing dimensions using the pad_with option:

from math import sqrt

@qml.qnode(dev)
def circuit(f=None):
return qml.expval(qml.PauliZ(0))

circuit(f=[1/sqrt(2), 1/sqrt(2)])

>>> dev.state
[0.70710678 + 0.j, 0.70710678 + 0.j, 0.0 + 0.j, 0.0 + 0.j]

 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. 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

• '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

inverse

Boolean determining if the inverse of the operation was requested.

matrix
name

Get and set the name of the operator.

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

Current parameter values.

string_for_inverse = '.inv'
wires

Wires of this operator.

Returns

wires

Return type

Wires

 Create an operation that is the adjoint of this one. decomposition(*params, wires) Returns a template decomposing the operation into other quantum operations. 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. Inverts the operation, such that the inverse will be used for the computations by the specific device. Append the operator to the Operator queue.
adjoint()[source]

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

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

multiplier, shift

Return type

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.