qml.templates.embeddings.IQPEmbedding

class IQPEmbedding(features, wires, n_repeats=1, pattern=None, do_queue=True)[source]

Bases: pennylane.operation.Operation

Encodes \(n\) features into \(n\) qubits using diagonal gates of an IQP circuit.

The embedding has been proposed by Havlicek et al. (2018).

The basic IQP circuit can be repeated by specifying n_repeats. Repetitions can make the embedding “richer” through interference.

Warning

IQPEmbedding calls a circuit that involves non-trivial classical processing of the features. The features argument is therefore not differentiable when using the template, and gradients with respect to the features cannot be computed by PennyLane.

An IQP circuit is a quantum circuit of a block of Hadamards, followed by a block of gates that are diagonal in the computational basis. Here, the diagonal gates are single-qubit RZ rotations, applied to each qubit and encoding the \(n\) features, followed by two-qubit ZZ entanglers, \(e^{-i x_i x_j \sigma_z \otimes \sigma_z}\). The entangler applied to wires (wires[i], wires[j]) encodes the product of features features[i]*features[j]. The pattern in which the entanglers are applied is either the default, or a custom pattern:

  • If pattern is not specified, the default pattern will be used, in which the entangling gates connect all pairs of neighbours:


    ../../_images/iqp.png

  • Else, pattern is a list of wire pairs [[a, b], [c, d],...], applying the entangler on wires [a, b], [c, d], etc. For example, pattern = [[0, 1], [1, 2]] produces the following entangler pattern:


    ../../_images/iqp_custom.png

    Since diagonal gates commute, the order of the entanglers does not change the result.

Parameters
  • features (tensor_like) – tensor of features to encode

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

  • n_repeats (int) – number of times the basic embedding is repeated

  • pattern (list[int]) – specifies the wires and features of the entanglers

Raises

ValueError – if inputs do not have the correct format

A typical usage example of the template is the following:

import pennylane as qml
from pennylane.templates import IQPEmbedding

dev = qml.device('default.qubit', wires=3)

@qml.qnode(dev)
def circuit(features):
    IQPEmbedding(features, wires=range(3))
    return [qml.expval(qml.PauliZ(w)) for w in range(3)]

circuit([1., 2., 3.])

Repeating the embedding

The embedding can be repeated by specifying the n_repeats argument:

@qml.qnode(dev)
def circuit(features):
    IQPEmbedding(features, wires=range(3), n_repeats=4)
    return [qml.expval(qml.PauliZ(w)) for w in range(3)]

circuit([1., 2., 3.])

Every repetition uses exactly the same quantum circuit.

Using a custom entangler pattern

A custom entangler pattern can be used by specifying the pattern argument. A pattern has to be a nested list of dimension (K, 2), where K is the number of entanglers to apply.

pattern = [[1, 2], [0, 2], [1, 0]]

@qml.qnode(dev)
def circuit(features):
    IQPEmbedding(features, wires=range(3), pattern=pattern)
    return [qml.expval(qml.PauliZ(w)) for w in range(3)]

circuit([1., 2., 3.])

Since diagonal gates commute, the order of the wire pairs has no effect on the result.

from pennylane import numpy as np

pattern1 = [[1, 2], [0, 2], [1, 0]]
pattern2 = [[1, 0], [0, 2], [1, 2]]  # a reshuffling of pattern1

@qml.qnode(dev)
def circuit(features, pattern):
    IQPEmbedding(features, wires=range(3), pattern=pattern, n_repeats=3)
    return [qml.expval(qml.PauliZ(w)) for w in range(3)]

res1 = circuit([1., 2., 3.], pattern=pattern1)
res2 = circuit([1., 2., 3.], pattern=pattern2)

assert np.allclose(res1, res2)

Non-consecutive wires

In principle, the user can also pass a non-consecutive wire list to the template. For single qubit gates, the i’th feature is applied to the i’th wire index (which may not be the i’th wire). For the entanglers, the product of i’th and j’th features is applied to the wire indices at the i’th and j’th position in wires.

For example, for wires=[2, 0, 1] the RZ block applies the first feature to wire 2, the second feature to wire 0, and the third feature to wire 1.

Likewise, using the default pattern, the entangler block applies the product of the first and second feature to the wire pair [2, 0], the product of the second and third feature to [2, 1], and so forth.

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

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

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

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.

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

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.