qml.CRZ¶

class CRZ(phi, wires, id=None)[source]¶

Bases: pennylane.ops.op_math.controlled.ControlledOp

The controlled-RZ operator

\[\begin{split}\begin{align} CR_z(\phi) &= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\\ 0 & 0 & e^{-i\phi/2} & 0\\ 0 & 0 & 0 & e^{i\phi/2} \end{bmatrix}. \end{align}\end{split}\]

Note

The subscripts of the operations in the formula refer to the wires they act on, e.g. 1 corresponds to the first element in wires that is the control qubit.

Details:

Number of wires: 2
Number of parameters: 1
Number of dimensions per parameter: (0,)
Gradient recipe: The controlled-RZ operator satisfies a four-term parameter-shift rule (see Appendix F, https://doi.org/10.1088/1367-2630/ac2cb3):

\[\frac{d}{d\phi}f(CR_z(\phi)) = c_+ \left[f(CR_z(\phi+a)) - f(CR_z(\phi-a))\right] - c_- \left[f(CR_z(\phi+b)) - f(CR_z(\phi-b))\right]\]

where \(f\) is an expectation value depending on \(CR_z(\phi)\), and
- \(a = \pi/2\)
- \(b = 3\pi/2\)
- \(c_{\pm} = (\sqrt{2} \pm 1)/{4\sqrt{2}}\)

Parameters

phi (float) – rotation angle \(\phi\)
wires (Sequence[int]) – the wire the operation acts on
id (str or None) – String representing the operation (optional)

Attributes

`arithmetic_depth`	Arithmetic depth of the operator.
`base`	The base operator.
`basis`	The basis of an operation, or for controlled gates, of the target operation.
`batch_size`	Batch size of the operator if it is used with broadcasted parameters.
`control_values`	Iterable[Bool].
`control_wires`	The control wires.
`data`	The trainable parameters
`grad_method`	Gradient computation method.
`grad_recipe`	Gradient recipe for the parameter-shift method.
`has_adjoint`	bool(x) -> bool
`has_decomposition`	bool(x) -> bool
`has_diagonalizing_gates`	bool(x) -> bool
`has_generator`	bool(x) -> bool
`has_matrix`	bool(x) -> bool
`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 hermitian.
`name`
`ndim_params`	Number of dimensions per trainable parameter that the operator depends on.
`num_params`	Number of trainable parameters that the operator depends on.
`num_wires`	Number of wires that the operation acts on.
`parameter_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.
`target_wires`	The wires of the target operator.
`wires`	Wires that the operator acts on.
`work_wires`	Additional wires that can be used in the decomposition.

arithmetic_depth¶

base¶: The base operator.

basis¶

batch_size¶

control_values¶: Iterable[Bool]. For each control wire, denotes whether to control on True or False.

control_wires¶: The control wires.

data¶: The trainable parameters

grad_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_adjoint¶

has_decomposition¶

has_diagonalizing_gates¶

has_generator¶

has_matrix¶

has_sparse_matrix = True¶

hash¶

hyperparameters¶

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

Type: dict

id¶: Custom string to label a specific operator instance.

is_hermitian¶

name = 'CRZ'¶

ndim_params = (0,)¶

Number of dimensions per trainable parameter that the operator depends on.

Type: tuple[int]

num_params = 1¶

Number of trainable parameters that the operator depends on.

Type: int

num_wires = 2¶

Number of wires that the operation acts on.

Type: int

parameter_frequencies = [(0.5, 1.0)]¶

parameters¶: Trainable parameters that the operator depends on.

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

target_wires¶: The wires of the target operator.

wires¶

work_wires¶: Additional wires that can be used in the decomposition. Not modified by the operation.

Methods

`adjoint`()	Create an operation that is the adjoint of this one.
`compute_decomposition`(phi, 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`(theta, **_)	Eigenvalues of the operator in the computational basis (static method).
`compute_matrix`(theta)	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).
`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)	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.
`single_qubit_rot_angles`()	The parameters required to implement a single-qubit gate as an equivalent `Rot` gate, up to a global phase.
`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()¶

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(phi, wires)[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

phi (float) – rotation angle \(\phi\)
wires (Iterable, Wires) – wires that the operator acts on

Returns

decomposition into lower level operations

Return type

list[Operator]

Example:

>>> qml.CRZ.compute_decomposition(1.2, wires=(0,1))
[PhaseShift(0.6, wires=[1]),
CNOT(wires=[0, 1]),
PhaseShift(-0.6, wires=[1]),
CNOT(wires=[0, 1])]

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

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(theta, **_)[source]¶

Eigenvalues of the operator in the computational basis (static method).

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.

See also

eigvals()

Parameters: theta (tensor_like or float) – rotation angle
Returns: eigenvalues
Return type: tensor_like

Example

>>> qml.CRZ.compute_eigvals(torch.tensor(0.5))
tensor([1.0000+0.0000j, 1.0000+0.0000j, 0.9689-0.2474j, 0.9689+0.2474j])

static compute_matrix(theta)[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()

Parameters: theta (tensor_like or float) – rotation angle
Returns: canonical matrix
Return type: tensor_like

Example

>>> qml.CRZ.compute_matrix(torch.tensor(0.5))
tensor([[1.0+0.0j, 0.0+0.0j,       0.0+0.0j,       0.0+0.0j],
        [0.0+0.0j, 1.0+0.0j,       0.0+0.0j,       0.0+0.0j],
        [0.0+0.0j, 0.0+0.0j, 0.9689-0.2474j,       0.0+0.0j],
        [0.0+0.0j, 0.0+0.0j,       0.0+0.0j, 0.9689+0.2474j]])

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

sparse matrix representation

Return type

scipy.sparse._csr.csr_matrix

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 \(U^{\dagger}\).

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

eigvals()[source]¶

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()

Returns: eigenvalues
Return type: tensor_like

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

The generator may also be provided in the form of a dense or sparse Hamiltonian (using Hamiltonian 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)¶

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("test_data")"
>>> op.label(decimals=2)
"RX\n(1.23,"test_data")"
>>> op.label(base_label="my_label")
"my_label("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(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)]

map_wires(wire_map)¶

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

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

pow(z)¶

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.QueuingManager'>)¶: Append the operator to the Operator queue.

simplify()¶

Reduce the depth of nested operators to the minimum.

Returns: simplified operator
Return type: Operator

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]

sparse_matrix(wire_order=None, format='csr')¶

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
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.

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

code/api/pennylane.CRZ

Download Python script

Download Notebook

View on GitHub

qml.CRZ¶

Attributes

Methods

Contents

Downloads

qml.CRZ¶

Attributes

Methods

Contents

Downloads

Related