Z90

Z90#

Bases: Operation

The single-qubit rotation of 90 degrees around the Z axis, equivalent to \(RZ(\pi/2)\).

Details:

Number of wires: 1
Number of parameters: 0

Parameters:: wires (Sequence[int] or int) – the wire the operation acts on

Attributes

`arithmetic_depth`	Arithmetic depth of the operator.
`basis`
`batch_size`
`control_wires`	Control wires of the operator.
`grad_method`	Gradient computation method.
`grad_recipe`	Gradient recipe for the parameter-shift method.
`has_adjoint`
`has_decomposition`
`has_diagonalizing_gates`
`has_generator`
`has_matrix`
`has_qfunc_decomposition`
`has_sparse_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 likely 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.
`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.
`pauli_rep`	A `PauliSentence` representation of the Operator, or `None` if it doesn't have one.
`resource_keys`
`resource_params`	A dictionary containing the minimal information needed to compute a resource estimate of the operator's decomposition.
`wires`	Wires that the operator acts on.

arithmetic_depth#: Arithmetic depth of the operator.

basis = 'Z'#

batch_size = 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

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

has_adjoint = True#

has_decomposition = True#

has_diagonalizing_gates = False#

has_generator = False#

has_matrix = True#

has_qfunc_decomposition = False#

has_sparse_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 likely hermitian.

Note

It is recommended to use the is_hermitian() function. Although this function may be expensive to calculate, the op.is_hermitian property can lead to technically incorrect results.

If this property returns True, the operator is guaranteed to be hermitian, but if it returns False, the operator may still be hermitian.

As an example, consider the following edge case:

>>> op = (qml.X(0) @ qml.Y(0) - qml.X(0) @ qml.Z(0)) * 1j
>>> op.is_hermitian
False

On the contrary, the is_hermitian() function will give the correct answer:

>>> qml.is_hermitian(op)
True

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 = 0#

Number of trainable parameters that the operator depends on.

Type:: int

num_wires = 1#: Number of wires the operator acts on.

parameter_frequencies#

Returns the frequencies for each operator parameter with respect to an expectation value of the form \(\langle \psi | U(\mathbf{p})^\dagger \hat{O} U(\mathbf{p})|\psi\rangle\).

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), (0.5, 1.0), (0.5, 1.0)]

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))
(np.float64(1.0),)

For more details on this relationship, see eigvals_to_frequencies().

parameters#: Trainable parameters that the operator depends on.

pauli_rep#: A PauliSentence representation of the Operator, or None if it doesn’t have one.

resource_keys = {}#

resource_params#

A dictionary containing the minimal information needed to compute a resource estimate of the operator’s decomposition.

The keys of this dictionary should match the resource_keys attribute of the operator class. Two instances of the same operator type should have identical resource_params iff their decompositions exhibit the same counts for each gate type, even if the individual gate parameters differ.

Examples

The MultiRZ has non-empty resource_keys:

>>> qml.MultiRZ.resource_keys
{'num_wires'}

The resource_params of an instance of MultiRZ will contain the number of wires:

>>> op = qml.MultiRZ(0.5, wires=[0, 1])
>>> op.resource_params
{'num_wires': 2}

Note that another MultiRZ may have different parameters but the same resource_params:

>>> op2 = qml.MultiRZ(0.7, wires=[1, 2])
>>> op2.resource_params
{'num_wires': 2}

wires#

Wires that the operator acts on.

Returns:: wires
Return type:: Wires

Methods

`adjoint`()	Return the adjoint of Z90, which is ZM90.
`compute_decomposition`(wires)	Decompose Z90 into primitive PennyLane operations.
`compute_diagonalizing_gates`(*params, wires, ...)	Sequence of gates that diagonalize the operator in the computational basis (static method).
`compute_eigvals`()	Compute the eigenvalues of Z90.
`compute_matrix`()	Compute the canonical matrix representation of Z90.
`compute_qfunc_decomposition`(*args, ...)	Experimental method to compute the dynamic decomposition of the operator with program capture enabled.
`compute_sparse_matrix`(*params[, format])	Representation of the operator as a sparse matrix in the computational basis (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.
`eigvals`()	Eigenvalues of the operator in the computational basis.
`generator`()	Generator of an operator that is in single-parameter-form.
`label`([decimals, base_label, cache])	A customizable string representation of the operator.
`map_wires`(wire_map)	Returns a copy of the current operator with its wires changed according to the given wire map.
`matrix`([wire_order])	Representation of the operator as a matrix in the computational basis.
`pow`(z)	Raise Z90 to an integer power.
`queue`(context)	Append the operator to the Operator queue.
`simplify`()	Reduce the depth of nested operators to the minimum.
`single_qubit_rot_angles`()	Euler rotation angles (ZYZ convention) that reproduce Z90.
`sparse_matrix`([wire_order, format])	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()#

Return the adjoint of Z90, which is ZM90.

Returns:: ZM90 acting on the same wire
Return type:: Operation

static compute_decomposition(wires)#

Decompose Z90 into primitive PennyLane operations.

Parameters:: wires (Sequence[int] or int) – the wire the operation acts on
Returns:: [RZ(pi/2, wires)]
Return type:: list[Operation]

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

diagonalizing_gates().

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()#

Compute the eigenvalues of Z90.

Returns:: eigenvalues of \(RZ(\pi/2)\).
Return type:: numpy.ndarray

static compute_matrix()#

Compute the canonical matrix representation of Z90.

Returns:: 2x2 unitary matrix for \(RZ(\pi/2)\).
Return type:: numpy.ndarray

static compute_qfunc_decomposition(*args, **hyperparameters) → None#

Experimental method to compute the dynamic decomposition of the operator with program capture enabled.

When the program capture feature is enabled with qml.capture.enable(), the decomposition of the operator is computed with this method if it is defined. Otherwise, the compute_decomposition() method is used.

The exception to this rule is when the operator is returned from the compute_decomposition() method of another operator, in which case the decomposition is performed with compute_decomposition() (even if this method is defined), and not with this method.

When compute_qfunc_decomposition is defined for an operator, the control flow operations within the method (specifying the decomposition of the operator) are recorded in the JAX representation.

Note

This method is experimental and subject to change.

See also

compute_decomposition().

Parameters:

*args (list) – positional arguments passed to the operator, including trainable parameters and wires
**hyperparameters (dict) – non-trainable hyperparameters of the operator, as stored in the hyperparameters attribute

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
format (str) – format of the returned scipy sparse matrix, for example ‘csr’
**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

decomposition() → list[Operator]#

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() → list[Operator]#

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

compute_diagonalizing_gates().

Returns:: a list of operators
Return type:: list[.Operator] or None

Eigenvalues of the operator in the computational basis.

If diagonalizing_gates are specified and implement a unitary \(U^{\dagger}\), 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() and qml.eigvals()

Returns:: eigenvalues
Return type:: tensor_like

generator() → Operator#

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 * Y(0) + Z(0) @ X(1)

The generator may also be provided in the form of a dense or sparse Hamiltonian (using LinearCombination and SparseHamiltonian respectively).

label(decimals: int | None = None, base_label: str | None = None, cache: dict | None = None) → str#

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 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(base_label="my_label")
'my_label'
>>> op = qml.RX(1.23456, wires=0, id="test_data")
>>> op.label()
'RX\n("test_data")'
>>> op.label(decimals=2)
'RX\n(1.23,"test_data")'
>>> op.label(base_label="my_label")
'my_label\n("test_data")'
>>> op.label(decimals=2, base_label="my_label")
'my_label\n(1.23,"test_data")'

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\n(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\n(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)]

map_wires(wire_map: dict[Hashable, Hashable]) → Operator#

Returns a copy of the current operator with its wires changed according to the given wire map.

Parameters:: wire_map (dict) – dictionary containing the old wires as keys and the new wires as values
Returns:: new operator
Return type:: .Operator

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

pow(z)#

Raise Z90 to an integer power.

Parameters:: z (int) – the exponent
Returns:: [RZ(z * pi/2, wires)]
Return type:: list[Operation]

queue(context: QueuingManager = <class 'pennylane.queuing.QueuingManager'>)#: Append the operator to the Operator queue.

simplify() → Operator#

Reduce the depth of nested operators to the minimum.

Returns:: simplified operator
Return type:: .Operator

single_qubit_rot_angles()#

Euler rotation angles (ZYZ convention) that reproduce Z90.

Returns:: [phi, theta, omega] such that RZ(phi) @ RY(theta) @ RZ(omega) equals \(RZ(\pi/2)\).
Return type:: list[float]

sparse_matrix(wire_order: Wires | Iterable[Hashable] | Hashable | None = None, format='csr') → spmatrix#

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

compute_sparse_matrix()

Parameters:

wire_order (Iterable) – global wire order, must contain all wire labels from the operator’s wires
format (str) – format of the returned scipy sparse matrix, for example ‘csr’

Returns:

sparse matrix representation

Return type:

scipy.sparse._csr.csr_matrix

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]]

Z90

Contents

Z90#

Attributes

Methods