# qml.operation.Operator¶

class Operator(*params, wires=None, do_queue=True, id=None)[source]

Bases: abc.ABC

Base class representing quantum operators.

Operators are uniquely defined by their name, the wires they act on, their (trainable) parameters, and their (non-trainable) hyperparameters. The trainable parameters can be tensors of any supported auto-differentiation framework.

An operator can define any of the following representations:

Each representation method comes with a static method prefixed by compute_, which takes the signature (*parameters, **hyperparameters) (for numerical representations that do not need to know about wire labels) or (*parameters, wires, **hyperparameters), where parameters, wires, and hyperparameters are the respective attributes of the operator class.

Parameters
• params (tuple[tensor_like]) – trainable parameters

• wires (Iterable[Any] or Any) – Wire label(s) that the operator acts on. If not given, args[-1] is interpreted as wires.

• do_queue (bool) – indicates whether the operator should be recorded when created in a tape context

• id (str) – custom label given to an operator instance, can be useful for some applications where the instance has to be identified

Example

A custom operator can be created by inheriting from Operator or one of its subclasses.

The following is an example for a custom gate that inherits from the Operation subclass. It acts by potentially flipping a qubit and rotating another qubit. The custom operator defines a decomposition, which the devices can use (since it is unlikely that a device knows a native implementation for FlipAndRotate). It also defines an adjoint operator.

import pennylane as qml

class FlipAndRotate(qml.operation.Operation):

# Define how many wires the operator acts on in total.
# In our case this may be one or two, which is why we
# use the AnyWires Enumeration to indicate a variable number.
num_wires = qml.operation.AnyWires

# This attribute tells PennyLane what differentiation method to use. Here
# we request parameter-shift (or "analytic") differentiation.

def __init__(self, angle, wire_rot, wire_flip=None, do_flip=False,
do_queue=True, id=None):

# checking the inputs --------------

if do_flip and wire_flip is None:
raise ValueError("Expected a wire to flip; got None.")

#------------------------------------

# do_flip is not trainable but influences the action of the operator,
# which is why we define it to be a hyperparameter
self._hyperparameters = {
"do_flip": do_flip
}

# we extract all wires that the operator acts on,
# relying on the Wire class arithmetic
all_wires = qml.wires.Wires(wire_rot) + qml.wires.Wires(wire_flip)

# The parent class expects all trainable parameters to be fed as positional
# arguments, and all wires acted on fed as a keyword argument.
# The id keyword argument allows users to give their instance a custom name.
# The do_queue keyword argument specifies whether or not
# the operator is queued when created in a tape context.
super().__init__(angle, wires=all_wires, do_queue=do_queue, id=id)

@property
def num_params(self):
# if it is known before creation, define the number of parameters to expect here,
# which makes sure an error is raised if the wrong number was passed. The angle
# parameter is the only trainable parameter of the operation
return 1

@property
def ndim_params(self):
# if it is known before creation, define the number of dimensions each parameter
# is expected to have. This makes sure to raise an error if a wrongly-shaped
# parameter was passed. The angle parameter is expected to be a scalar
return (0,)

@staticmethod
def compute_decomposition(angle, wires, do_flip):  # pylint: disable=arguments-differ
# Overwriting this method defines the decomposition of the new gate, as it is
# called by Operator.decomposition().
# The general signature of this function is (*parameters, wires, **hyperparameters).
op_list = []
if do_flip:
op_list.append(qml.PauliX(wires=wires[1]))
op_list.append(qml.RX(angle, wires=wires[0]))
return op_list

# the adjoint operator of this gate simply negates the angle
return FlipAndRotate(-self.parameters[0], self.wires[0], self.wires[1], do_flip=self.hyperparameters["do_flip"])


We can use the operation as follows:

from pennylane import numpy as np

dev = qml.device("default.qubit", wires=["q1", "q2", "q3"])

@qml.qnode(dev)
def circuit(angle):
FlipAndRotate(angle, wire_rot="q1", wire_flip="q1")
return qml.expval(qml.PauliZ("q1"))

>>> a = np.array(3.14)
>>> circuit(a)
-0.9999987318946099

 arithmetic_depth Arithmetic depth of the operator. batch_size Batch size of the operator if it is used with broadcasted parameters. has_matrix hash Integer hash that uniquely represents the operator. hyperparameters Dictionary of non-trainable variables that this operation depends on. id Custom string to label a specific operator instance. is_hermitian This property determines if an operator is hermitian. name String for the name of the operator. ndim_params Number of dimensions per trainable parameter of the operator. num_params Number of trainable parameters that the operator depends on. num_wires Number of wires the operator acts on. parameters Trainable parameters that the operator depends on. wires Wires that the operator acts on.
arithmetic_depth

Arithmetic depth of the operator.

batch_size

Batch size of the operator if it is used with broadcasted parameters.

The batch_size is determined based on ndim_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 is None.

Returns

Size of the parameter broadcasting dimension if present, else None.

Return type

int or None

has_matrix = False
hash

Integer hash that uniquely represents the operator.

Type

int

hyperparameters

Dictionary of non-trainable variables that this operation depends on.

Type

dict

id

Custom string to label a specific operator instance.

is_hermitian

This property determines if an operator is hermitian.

name

String for the name of the operator.

ndim_params

Number of dimensions per trainable parameter of the operator.

By default, this property returns the numbers of dimensions of the parameters used for the operator creation. If the parameter sizes for an operator subclass are fixed, this property can be overwritten to return the fixed value.

Returns

Number of dimensions for each trainable parameter.

Return type

tuple

num_params

Number of trainable parameters that the operator depends on.

By default, this property returns as many parameters as were used for the operator creation. If the number of parameters for an operator subclass is fixed, this property can be overwritten to return the fixed value.

Returns

number of parameters

Return type

int

num_wires

Number of wires the operator acts on.

parameters

Trainable parameters that the operator depends on.

wires

Wires that the operator acts on.

Returns

wires

Return type

Wires

 Create an operation that is the adjoint of this one. compute_decomposition(*params[, wires]) Representation of the operator as a product of other operators (static method). 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. Eigenvalues of the operator in the computational basis (static method). Returns a tape that has recorded the decomposition of the operator. Generator of an operator that is in single-parameter-form. 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. A list of new operators equal to this one raised to the given power. queue([context]) Append the operator to the Operator queue. Reduce the depth of nested operators to the minimum. sparse_matrix([wire_order]) Representation of the operator as a sparse matrix in the computational basis. Representation of the operator as a linear combination of other operators.
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 compute_decomposition(*params, wires=None, **hyperparameters)[source]

Representation of the operator as a product of other operators (static method).

$O = O_1 O_2 \dots O_n.$

Note

Operations making up the decomposition should be queued within the compute_decomposition method.

Parameters
• params (list) – trainable parameters of the operator, as stored in the parameters attribute

• wires (Iterable[Any], Wires) – wires that the operator acts on

• hyperparams (dict) – non-trainable hyperparameters of the operator, as stored in the hyperparameters attribute

Returns

decomposition of the operator

Return type

list[Operator]

static compute_diagonalizing_gates(*params, wires, **hyperparams)[source]

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 $$U$$.

The diagonalizing gates rotate the state into the eigenbasis of the operator.

Parameters
• params (list) – trainable parameters of the operator, as stored in the parameters attribute

• wires (Iterable[Any], Wires) – wires that the operator acts on

• hyperparams (dict) – non-trainable hyperparameters of the operator, as stored in the hyperparameters attribute

Returns

list of diagonalizing gates

Return type

list[Operator]

static compute_eigvals(*params, **hyperparams)[source]

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 attribute

• hyperparams (dict) – non-trainable hyperparameters of the operator, as stored in the hyperparameters attribute

Returns

eigenvalues

Return type

tensor_like

static compute_matrix(*params, **hyperparams)[source]

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 attribute

• hyperparams (dict) – non-trainable hyperparameters of the operator, as stored in the hyperparameters attribute

Returns

matrix representation

Return type

tensor_like

static compute_sparse_matrix(*params, **hyperparams)[source]

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.

Parameters
• params (list) – trainable parameters of the operator, as stored in the parameters attribute

• hyperparams (dict) – non-trainable 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)[source]

Representation of the operator as a linear combination of other operators (static method).

$O = \sum_i c_i O_i$
Parameters
• params (list) – trainable parameters of the operator, as stored in the parameters attribute

• hyperparams (dict) – non-trainable 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()[source]

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.

Returns

decomposition of the operator

Return type

list[Operator]

diagonalizing_gates()[source]

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 $$U$$.

The diagonalizing gates rotate the state into the eigenbasis of the operator.

A DiagGatesUndefinedError is raised if no representation by decomposition is defined.

Returns

a list of operators

Return type

list[Operator] or None

eigvals()[source]

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.

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.

Returns

eigenvalues

Return type

tensor_like

expand()[source]

Returns a tape that has recorded the decomposition of the operator.

Returns

quantum tape

Return type

QuantumTape

generator()[source]

Generator of an operator that is in single-parameter-form.

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 and SparseHamiltonian respectively).

The default value to return is None, indicating that the operation has no defined generator.

label(decimals=None, base_label=None, cache=None)[source]

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 non-parameter component of the label

• cache=None (dict) – dictionary that caries 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 matrix-valued 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.],
>>> op3 = qml.QubitUnitary(np.eye(4), wires=(0,1))
>>> op3.label(cache=cache)
'U(M1)'
>>> cache['matrices']
[tensor([[1., 0.],
tensor([[1., 0., 0., 0.],
[0., 1., 0., 0.],
[0., 0., 1., 0.],

matrix(wire_order=None)[source]

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.

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)[source]

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'>)[source]

Append the operator to the Operator queue.

simplify()pennylane.operation.Operator[source]

Reduce the depth of nested operators to the minimum.

Returns

simplified operator

Return type

Operator

sparse_matrix(wire_order=None)[source]

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.

Note

The wire_order argument is currently not implemented, and using it will raise an error.

A SparseMatrixUndefinedError is raised if the sparse matrix representation has not been defined.

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()[source]

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.

Returns

list of coefficients $$c_i$$ and list of operations $$O_i$$

Return type

tuple[list[tensor_like or float], list[Operation]]