qml.Identity¶

class
Identity
(wires)[source]¶ Bases:
pennylane.operation.CVObservable
,pennylane.operation.Operation
The identity observable \(\I\).
The expectation of this observable
\[E[\I] = \text{Tr}(\I \rho)\]corresponds to the trace of the quantum state, which in exact simulators should always be equal to 1.
Attributes
If inverse is requested, this is the name of the original operator to be inverted.
The target operation for controlled gates.
Control wires of the operator.
Eigenvalues of an instantiated operator.
Order in (x, p) that a CV observable is a polynomial of.
Gradient computation method.
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.
Matrix representation of an instantiated operator in the computational basis.
Name of the operator.
Number of wires that the operator acts 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.
Measurement type that this observable is called with.
The parameters required to implement a singlequbit gate as an equivalent
Rot
gate, up to a global phase.Wires that the operator acts on.

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

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

eigvals
¶ Eigenvalues of an instantiated operator. Note that the eigenvalues are not guaranteed to be in any particular order.
Warning
The
eigvals
property is deprecated and will be removed in an upcoming release. Please useqml.eigvals
instead.Example:
>>> U = qml.RZ(0.5, wires=1) >>> U.eigvals >>> array([0.968912420.24740396j, 0.96891242+0.24740396j])
 Returns
eigvals representation
 Return type
array

ev_order
= 1¶ Order in (x, p) that a CV observable is a polynomial of.
 Type
None, int

grad_method
= None¶ Gradient computation method.

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
= True¶

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.

matrix
¶ Matrix representation of an instantiated operator in the computational basis.
Warning
The
matrix
property is deprecated and will be removed in an upcoming release. Please useqml.matrix
instead.Example:
>>> U = qml.RY(0.5, wires=1) >>> U.matrix >>> array([[ 0.96891242+0.j, 0.24740396+0.j], [ 0.24740396+0.j, 0.96891242+0.j]])
 Returns
matrix representation
 Return type
array

name
¶ Name of the operator.

num_params
= 0¶

num_wires
= 1¶ Number of wires that the operator acts on.
 Type
int

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.

return_type
= None¶ Measurement type that this observable is called with.
 Type
None or ObservableReturnTypes

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]

supports_heisenberg
= True¶
Methods
adjoint
()Create an operation that is the adjoint of this one.
compare
(other)Compares with another
Hamiltonian
,Tensor
, orObservable
, to determine if they are equivalent.compute_decomposition
([wires])Representation of the operator as a product of other operators (static method).
compute_diagonalizing_gates
(wires)Sequence of gates that diagonalize the operator in the computational basis (static method).
Eigenvalues of the operator in the computational basis (static method).
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.
expand
()Returns a tape that has recorded the decomposition of the operator.
Generator of an operator that is in singleparameterform.
Eigenvalues of the operator in the computational basis (static method).
get_matrix
([wire_order])Representation of the operator as a matrix in the computational basis.
get_parameter_shift
(idx)Multiplier and shift for the given parameter, based on its gradient recipe.
heisenberg_expand
(U, wire_order)Expand the given local Heisenbergpicture array into a fullsystem one.
heisenberg_obs
(wire_order)Representation of the observable in the position/momentum operator basis.
identity_op
(*params)Alias for matrix representation of the identity operator.
inv
()Inverts the operator.
label
([decimals, base_label, cache])A customizable string representation of the operator.
queue
([context])Append the operator to the Operator queue.
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
()[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
The adjointed operation.

compare
(other)¶ Compares with another
Hamiltonian
,Tensor
, orObservable
, to determine if they are equivalent.Observables/Hamiltonians are equivalent if they represent the same operator (their matrix representations are equal), and they are defined on the same wires.
Warning
The compare method does not check if the matrix representation of a
Hermitian
observable is equal to an equivalent observable expressed in terms of Pauli matrices. To do so would require the matrix form of Hamiltonians and Tensors be calculated, which would drastically increase runtime. Returns
True if equivalent.
 Return type
(bool)
Examples
>>> ob1 = qml.PauliX(0) @ qml.Identity(1) >>> ob2 = qml.Hamiltonian([1], [qml.PauliX(0)]) >>> ob1.compare(ob2) True >>> ob1 = qml.PauliX(0) >>> ob2 = qml.Hermitian(np.array([[0, 1], [1, 0]]), 0) >>> ob1.compare(ob2) False

static
compute_decomposition
(wires=None)[source]¶ Representation of the operator as a product of other operators (static method).
\[O = O_1 O_2 \dots O_n.\]See also
 Parameters
wires (Any, Wires) – A single wire that the operator acts on.
 Returns
decomposition into lower level operations
 Return type
list[Operator]
Example:
>>> qml.Identity.compute_decomposition(wires=0) []

static
compute_diagonalizing_gates
(wires)[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.
See also
 Parameters
wires (Iterable[Any], Wires) – wires that the operator acts on
 Returns
list of diagonalizing gates
 Return type
list[Operator]
Example
>>> qml.Identity.compute_diagonalizing_gates(wires=[0]) []

static
compute_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.
See also
eigvals()
 Returns
eigenvalues
 Return type
array
Example
>>> print(qml.Identity.compute_eigvals()) [ 1 1]

static
compute_matrix
()[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.
See also
matrix()
 Returns
matrix
 Return type
ndarray
Example
>>> print(qml.Identity.compute_matrix()) [[1. 0.] [0. 1.]]

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
matrix representation
 Return type
scipy.sparse.coo.coo_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 \(U\).
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

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_eigvals
()¶ 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.See also
 Returns
eigenvalues
 Return type
tensor_like

get_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

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 \(\pi/2\).

heisenberg_expand
(U, wire_order)¶ Expand the given local Heisenbergpicture array into a fullsystem one.
 Parameters
U (array[float]) – array to expand (expected to be of the dimension
1+2*self.num_wires
)wire_order (Wires) – global wire order defining which subspace the operator acts on
 Raises
ValueError – if the size of the input matrix is invalid or num_wires is incorrect
 Returns
expanded array, dimension
1+2*num_wires
 Return type
array[float]

heisenberg_obs
(wire_order)¶ Representation of the observable in the position/momentum operator basis.
Returns the expansion \(q\) of the observable, \(Q\), in the basis \(\mathbf{r} = (\I, \x_0, \p_0, \x_1, \p_1, \ldots)\).
For firstorder observables returns a real vector such that \(Q = \sum_i q_i \mathbf{r}_i\).
For secondorder observables returns a real symmetric matrix such that \(Q = \sum_{ij} q_{ij} \mathbf{r}_i \mathbf{r}_j\).
 Parameters
wire_order (Wires) – global wire order defining which subspace the operator acts on
 Returns
\(q\)
 Return type
array[float]

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)[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 nonparameter 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 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)]

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

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.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.See also
 Parameters
wire_order (Iterable) – global wire order, must contain all wire labels from the operator’s wires
 Returns
matrix representation
 Return type
scipy.sparse.coo.coo_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]]

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