qml.IQPEmbedding¶

class
IQPEmbedding
(features, wires, n_repeats=1, pattern=None, do_queue=True, id=None)[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 nontrivial classical processing of the features. Thefeatures
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 singlequbit
RZ
rotations, applied to each qubit and encoding the \(n\) features, followed by twoqubit 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 featuresfeatures[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: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:Since diagonal gates commute, the order of the entanglers does not change the result.
 Parameters
features (tensor_like) – tensor of features to encode
wires (Any or Iterable[Any]) – 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
Usage Details
A typical usage example of the template is the following:
import pennylane as qml dev = qml.device('default.qubit', wires=3) @qml.qnode(dev) def circuit(features): qml.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): qml.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)
, whereK
is the number of entanglers to apply.pattern = [[1, 2], [0, 2], [1, 0]] @qml.qnode(dev) def circuit(features): qml.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): qml.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)
Nonconsecutive wires
In principle, the user can also pass a nonconsecutive 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]
theRZ
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.Attributes
Arithmetic depth of the operator.
If inverse is requested, this is the name of the original operator to be inverted.
The target operation for controlled gates.
Batch size of the operator if it is used with broadcasted parameters.
Control wires of the operator.
Gradient recipe for the parametershift method.
Integer hash that uniquely represents the operator.
Dictionary of nontrainable variables that this operation depends on.
Custom string to label a specific operator instance.
Boolean determining if the inverse of the operation was requested.
This property determines if an operator is hermitian.
Name of the operator.
Number of dimensions per trainable parameter of the operator.
Number of trainable parameters that the operator depends on.
Returns the frequencies for each operator parameter with respect to an expectation value of the form \(\langle \psi  U(\mathbf{p})^\dagger \hat{O} U(\mathbf{p})\psi\rangle\).
Trainable parameters that the operator depends on.
Wires that the operator acts on.

arithmetic_depth
¶ Arithmetic depth of the operator.

base_name
¶ If inverse is requested, this is the name of the original operator to be inverted.

basis
= None¶ The target operation for controlled gates. target operation. If not
None
, should take a value of"X"
,"Y"
, or"Z"
.For example,
X
andCNOT
havebasis = "X"
, whereasControlledPhaseShift
andRZ
havebasis = "Z"
. Type
str or None

batch_size
¶ Batch size of the operator if it is used with broadcasted parameters.
The
batch_size
is determined based onndim_params
and the provided parameters for the operator. If (some of) the latter have an additional dimension, and this dimension has the same size for all parameters, its size is the batch size of the operator. If no parameter has an additional dimension, the batch size isNone
. Returns
Size of the parameter broadcasting dimension if present, else
None
. Return type
int or None

control_wires
¶ Control wires of the operator.
For operations that are not controlled, this is an empty
Wires
object of length0
. Returns
The control wires of the operation.
 Return type

grad_method
= None¶

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

has_matrix
= False¶

hash
¶ Integer hash that uniquely represents the operator.
 Type
int

hyperparameters
¶ Dictionary of nontrainable variables that this operation depends on.
 Type
dict

id
¶ Custom string to label a specific operator instance.

inverse
¶ Boolean determining if the inverse of the operation was requested.

is_hermitian
¶ This property determines if an operator is hermitian.

name
¶ Name of the operator.

ndim_params
¶

num_params
¶

num_wires
= 1¶

parameter_frequencies
¶ Returns the frequencies for each operator parameter with respect to an expectation value of the form \(\langle \psi  U(\mathbf{p})^\dagger \hat{O} U(\mathbf{p})\psi\rangle\).
These frequencies encode the behaviour of the operator \(U(\mathbf{p})\) on the value of the expectation value as the parameters are modified. For more details, please see the
pennylane.fourier
module. Returns
Tuple of frequencies for each parameter. Note that only nonnegative frequency values are returned.
 Return type
list[tuple[int or float]]
Example
>>> op = qml.CRot(0.4, 0.1, 0.3, wires=[0, 1]) >>> op.parameter_frequencies [(0.5, 1), (0.5, 1), (0.5, 1)]
For operators that define a generator, the parameter frequencies are directly related to the eigenvalues of the generator:
>>> op = qml.ControlledPhaseShift(0.1, wires=[0, 1]) >>> op.parameter_frequencies [(1,)] >>> gen = qml.generator(op, format="observable") >>> gen_eigvals = qml.eigvals(gen) >>> qml.gradients.eigvals_to_frequencies(tuple(gen_eigvals)) (1.0,)
For more details on this relationship, see
eigvals_to_frequencies()
.

parameters
¶ Trainable parameters that the operator depends on.
Methods
adjoint
()Create an operation that is the adjoint of this one.
compute_decomposition
(features, wires, …)Representation of the operator as a product of other operators.
compute_diagonalizing_gates
(*params, wires, …)Sequence of gates that diagonalize the operator in the computational basis (static method).
compute_eigvals
(*params, **hyperparams)Eigenvalues of the operator in the computational basis (static method).
compute_matrix
(*params, **hyperparams)Representation of the operator as a canonical matrix in the computational basis (static method).
compute_sparse_matrix
(*params, **hyperparams)Representation of the operator as a sparse matrix in the computational basis (static method).
compute_terms
(*params, **hyperparams)Representation of the operator as a linear combination of other operators (static method).
Representation of the operator as a product of other operators.
Sequence of gates that diagonalize the operator in the computational basis.
eigvals
()Eigenvalues of the operator in the computational basis.
expand
()Returns a tape that has recorded the decomposition of the operator.
Generator of an operator that is in singleparameterform.
get_parameter_shift
(idx)Multiplier and shift for the given parameter, based on its gradient recipe.
inv
()Inverts the operator.
label
([decimals, base_label, cache])A customizable string representation of the operator.
matrix
([wire_order])Representation of the operator as a matrix in the computational basis.
pow
(z)A list of new operators equal to this one raised to the given power.
queue
([context])Append the operator to the Operator queue.
simplify
()Reduce the depth of nested operators to the minimum.
The parameters required to implement a singlequbit gate as an equivalent
Rot
gate, up to a global phase.sparse_matrix
([wire_order])Representation of the operator as a sparse matrix in the computational basis.
terms
()Representation of the operator as a linear combination of other operators.

adjoint
() → pennylane.operation.Operator¶ 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.
 Returns
The adjointed operation.

static
compute_decomposition
(features, wires, n_repeats, pattern)[source]¶ Representation of the operator as a product of other operators.
\[O = O_1 O_2 \dots O_n.\]See also
 Parameters
features (tensor_like) – tensor of features to encode
wires (Any or Iterable[Any]) – wires that the template acts on
 Returns
decomposition of the operator
 Return type
list[Operator]
Example
>>> features = torch.tensor([1., 2., 3.]) >>> pattern = [(0, 1), (0, 2), (1, 2)] >>> qml.IQPEmbedding.compute_decomposition(features, wires=[0, 1, 2], n_repeats=2, pattern=pattern) [Hadamard(wires=[0]), RZ(tensor(1.), wires=[0]), Hadamard(wires=[1]), RZ(tensor(2.), wires=[1]), Hadamard(wires=[2]), RZ(tensor(3.), wires=[2]), MultiRZ(tensor(2.), wires=[0, 1]), MultiRZ(tensor(3.), wires=[0, 2]), MultiRZ(tensor(6.), wires=[1, 2]), Hadamard(wires=[0]), RZ(tensor(1.), wires=[0]), Hadamard(wires=[1]), RZ(tensor(2.), wires=[1]), Hadamard(wires=[2]), RZ(tensor(3.), wires=[2]), MultiRZ(tensor(2.), wires=[0, 1]), MultiRZ(tensor(3.), wires=[0, 2]), MultiRZ(tensor(6.), wires=[1, 2])]

static
compute_diagonalizing_gates
(*params, wires, **hyperparams)¶ Sequence of gates that diagonalize the operator in the computational basis (static method).
Given the eigendecomposition \(O = U \Sigma U^{\dagger}\) where \(\Sigma\) is a diagonal matrix containing the eigenvalues, the sequence of diagonalizing gates implements the unitary \(U^{\dagger}\).
The diagonalizing gates rotate the state into the eigenbasis of the operator.
See also
 Parameters
params (list) – trainable parameters of the operator, as stored in the
parameters
attributewires (Iterable[Any], Wires) – wires that the operator acts on
hyperparams (dict) – nontrainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
 Returns
list of diagonalizing gates
 Return type
list[Operator]

static
compute_eigvals
(*params, **hyperparams)¶ Eigenvalues of the operator in the computational basis (static method).
If
diagonalizing_gates
are specified and implement a unitary \(U\), the operator can be reconstructed as\[O = U \Sigma U^{\dagger},\]where \(\Sigma\) is the diagonal matrix containing the eigenvalues.
Otherwise, no particular order for the eigenvalues is guaranteed.
 Parameters
params (list) – trainable parameters of the operator, as stored in the
parameters
attributehyperparams (dict) – nontrainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
 Returns
eigenvalues
 Return type
tensor_like

static
compute_matrix
(*params, **hyperparams)¶ Representation of the operator as a canonical matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires. Implicitly, this assumes that the wires of the operator correspond to the global wire order.
 Parameters
params (list) – trainable parameters of the operator, as stored in the
parameters
attributehyperparams (dict) – nontrainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
 Returns
matrix representation
 Return type
tensor_like

static
compute_sparse_matrix
(*params, **hyperparams)¶ Representation of the operator as a sparse matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires. Implicitly, this assumes that the wires of the operator correspond to the global wire order.
See also
 Parameters
params (list) – trainable parameters of the operator, as stored in the
parameters
attributehyperparams (dict) – nontrainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
 Returns
sparse matrix representation
 Return type
scipy.sparse._csr.csr_matrix

static
compute_terms
(*params, **hyperparams)¶ Representation of the operator as a linear combination of other operators (static method).
\[O = \sum_i c_i O_i\]See also
 Parameters
params (list) – trainable parameters of the operator, as stored in the
parameters
attributehyperparams (dict) – nontrainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
 Returns
list of coefficients and list of operations
 Return type
tuple[list[tensor_like or float], list[Operation]]

decomposition
()¶ Representation of the operator as a product of other operators.
\[O = O_1 O_2 \dots O_n\]A
DecompositionUndefinedError
is raised if no representation by decomposition is defined.See also
 Returns
decomposition of the operator
 Return type
list[Operator]

diagonalizing_gates
()¶ Sequence of gates that diagonalize the operator in the computational basis.
Given the eigendecomposition \(O = U \Sigma U^{\dagger}\) where \(\Sigma\) is a diagonal matrix containing the eigenvalues, the sequence of diagonalizing gates implements the unitary \(U^{\dagger}\).
The diagonalizing gates rotate the state into the eigenbasis of the operator.
A
DiagGatesUndefinedError
is raised if no representation by decomposition is defined.See also
 Returns
a list of operators
 Return type
list[Operator] or None

eigvals
()¶ Eigenvalues of the operator in the computational basis.
If
diagonalizing_gates
are specified and implement a unitary \(U\), the operator can be reconstructed as\[O = U \Sigma U^{\dagger},\]where \(\Sigma\) is the diagonal matrix containing the eigenvalues.
Otherwise, no particular order for the eigenvalues is guaranteed.
Note
When eigenvalues are not explicitly defined, they are computed automatically from the matrix representation. Currently, this computation is not differentiable.
A
EigvalsUndefinedError
is raised if the eigenvalues have not been defined and cannot be inferred from the matrix representation.See also
 Returns
eigenvalues
 Return type
tensor_like

expand
()¶ Returns a tape that has recorded the decomposition of the operator.
 Returns
quantum tape
 Return type

generator
()¶ Generator of an operator that is in singleparameterform.
For example, for operator
\[U(\phi) = e^{i\phi (0.5 Y + Z\otimes X)}\]we get the generator
>>> U.generator() (0.5) [Y0] + (1.0) [Z0 X1]
The generator may also be provided in the form of a dense or sparse Hamiltonian (using
Hermitian
andSparseHamiltonian
respectively).The default value to return is
None
, indicating that the operation has no defined generator.

get_parameter_shift
(idx)¶ Multiplier and shift for the given parameter, based on its gradient recipe.
 Parameters
idx (int) – parameter index within the operation
 Returns
list of multiplier, coefficient, shift for each term in the gradient recipe
 Return type
list[[float, float, float]]
Note that the default value for
shift
is None, which is replaced by the default shift \(\pi/2\).

inv
()¶ Inverts the operator.
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

label
(decimals=None, base_label=None, cache=None)¶ A customizable string representation of the operator.
 Parameters
decimals=None (int) – If
None
, no parameters are included. Else, specifies how to round the parameters.base_label=None (str) – overwrite the nonparameter component of the label
cache=None (dict) – dictionary that carries information between label calls in the same drawing
 Returns
label to use in drawings
 Return type
str
Example:
>>> op = qml.RX(1.23456, wires=0) >>> op.label() "RX" >>> op.label(decimals=2) "RX\n(1.23)" >>> op.label(base_label="my_label") "my_label" >>> op.label(decimals=2, base_label="my_label") "my_label\n(1.23)" >>> op.inv() >>> op.label() "RX⁻¹"
If the operation has a matrixvalued parameter and a cache dictionary is provided, unique matrices will be cached in the
'matrices'
key list. The label will contain the index of the matrix in the'matrices'
list.>>> op2 = qml.QubitUnitary(np.eye(2), wires=0) >>> cache = {'matrices': []} >>> op2.label(cache=cache) 'U(M0)' >>> cache['matrices'] [tensor([[1., 0.], [0., 1.]], requires_grad=True)] >>> op3 = qml.QubitUnitary(np.eye(4), wires=(0,1)) >>> op3.label(cache=cache) 'U(M1)' >>> cache['matrices'] [tensor([[1., 0.], [0., 1.]], requires_grad=True), tensor([[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]], requires_grad=True)]

matrix
(wire_order=None)¶ Representation of the operator as a matrix in the computational basis.
If
wire_order
is provided, the numerical representation considers the position of the operator’s wires in the global wire order. Otherwise, the wire order defaults to the operator’s wires.If the matrix depends on trainable parameters, the result will be cast in the same autodifferentiation framework as the parameters.
A
MatrixUndefinedError
is raised if the matrix representation has not been defined.See also
 Parameters
wire_order (Iterable) – global wire order, must contain all wire labels from the operator’s wires
 Returns
matrix representation
 Return type
tensor_like

pow
(z) → List[pennylane.operation.Operator]¶ A list of new operators equal to this one raised to the given power.
 Parameters
z (float) – exponent for the operator
 Returns
list[
Operator
]

queue
(context=<class 'pennylane.queuing.QueuingContext'>)¶ Append the operator to the Operator queue.

simplify
() → pennylane.operation.Operator¶ Reduce the depth of nested operators to the minimum.
 Returns
simplified operator
 Return type

single_qubit_rot_angles
()¶ The parameters required to implement a singlequbit gate as an equivalent
Rot
gate, up to a global phase. Returns
A list of values \([\phi, \theta, \omega]\) such that \(RZ(\omega) RY(\theta) RZ(\phi)\) is equivalent to the original operation.
 Return type
tuple[float, float, float]

sparse_matrix
(wire_order=None)¶ Representation of the operator as a sparse matrix in the computational basis.
If
wire_order
is provided, the numerical representation considers the position of the operator’s wires in the global wire order. Otherwise, the wire order defaults to the operator’s wires.A
SparseMatrixUndefinedError
is raised if the sparse matrix representation has not been defined.See also
 Parameters
wire_order (Iterable) – global wire order, must contain all wire labels from the operator’s wires
 Returns
sparse matrix representation
 Return type
scipy.sparse._csr.csr_matrix

terms
()¶ Representation of the operator as a linear combination of other operators.
\[O = \sum_i c_i O_i\]A
TermsUndefinedError
is raised if no representation by terms is defined.See also
 Returns
list of coefficients \(c_i\) and list of operations \(O_i\)
 Return type
tuple[list[tensor_like or float], list[Operation]]