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

`base_name`	If inverse is requested, this is the name of the original operator to be inverted.
`basis`	The target operation for controlled gates.
`control_wires`	Control wires of the operator.
`eigvals`	Eigenvalues of an instantiated operator.
`ev_order`	Order in (x, p) that a CV observable is a polynomial of.
`grad_method`	Gradient computation method.
`grad_recipe`	Gradient recipe for the parameter-shift method.
`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.
`inverse`	Boolean determining if the inverse of the operation was requested.
`matrix`	Matrix representation of an instantiated operator in the computational basis.
`name`	Name of the operator.
`num_params`
`num_wires`	Number of wires that the operator acts on.
`parameter_frequencies`	Returns the frequencies for each operator parameter with respect to an expectation value of the form \(\langle \psi \| U(\mathbf{p})^\dagger \hat{O} U(\mathbf{p})\|\psi\rangle\).
`parameters`	Trainable parameters that the operator depends on.
`return_type`	Measurement type that this observable is called with.
`single_qubit_rot_angles`	The parameters required to implement a single-qubit gate as an equivalent `Rot` gate, up to a global phase.
`supports_heisenberg`
`wires`	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 and CNOT have basis = "X", whereas ControlledPhaseShift and RZ have basis = "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 length 0.

Returns: The control wires of the operation.
Return type: Wires

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 use qml.eigvals instead.

Example:

>>> U = qml.RZ(0.5, wires=1)
>>> U.eigvals
>>> array([0.96891242-0.24740396j, 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 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

has_matrix = True¶

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.

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 use qml.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 non-negative 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 single-qubit 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¶

wires¶

Wires that the operator acts on.

Returns: wires
Return type: Wires

Methods

`adjoint`()	Create an operation that is the adjoint of this one.
`compare`(other)	Compares with another `Hamiltonian`, `Tensor`, or `Observable`, 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).
`compute_eigvals`()	Eigenvalues of the operator in the computational basis (static method).
`compute_matrix`()	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).
`decomposition`()	Representation of the operator as a product of other operators.
`diagonalizing_gates`()	Sequence of gates that diagonalize the operator in the computational basis.
`expand`()	Returns a tape that has recorded the decomposition of the operator.
`generator`()	Generator of an operator that is in single-parameter-form.
`get_eigvals`()	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 Heisenberg-picture array into a full-system 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, or Observable, 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

decomposition().

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

diagonalizing_gates().

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

sparse_matrix()

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

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

terms()

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()¶

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

compute_decomposition().

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

compute_diagonalizing_gates().

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: QuantumTape

generator()¶

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.

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

compute_eigvals()

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

compute_matrix()

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 Heisenberg-picture array into a full-system 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 first-order observables returns a real vector such that \(Q = \sum_i q_i \mathbf{r}_i\).
For second-order 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]

static identity_op(*params)[source]¶: Alias for matrix representation of the identity operator.

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 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.],
 [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

compute_sparse_matrix()

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

compute_terms()

Returns: list of coefficients \(c_i\) and list of operations \(O_i\)
Return type: tuple[list[tensor_like or float], list[Operation]]

qml.Identity¶

Attributes

Methods

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