# qml.CRot¶

class CRot(phi, theta, omega, wires, id=None)[source]

Bases: pennylane.ops.op_math.controlled.ControlledOp

The controlled-Rot operator

$\begin{split}CR(\phi, \theta, \omega) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\\ 0 & 0 & e^{-i(\phi+\omega)/2}\cos(\theta/2) & -e^{i(\phi-\omega)/2}\sin(\theta/2)\\ 0 & 0 & e^{-i(\phi-\omega)/2}\sin(\theta/2) & e^{i(\phi+\omega)/2}\cos(\theta/2) \end{bmatrix}.\end{split}$

Note

The first wire provided corresponds to the control qubit.

Details:

• Number of wires: 2

• Number of parameters: 3

• Number of dimensions per parameter: (0, 0, 0)

• Gradient recipe: The controlled-Rot operator satisfies a four-term parameter-shift rule (see Appendix F, https://doi.org/10.1088/1367-2630/ac2cb3):

$\frac{d}{d\mathbf{x}_i}f(CR(\mathbf{x}_i)) = c_+ \left[f(CR(\mathbf{x}_i+a)) - f(CR(\mathbf{x}_i-a))\right] - c_- \left[f(CR(\mathbf{x}_i+b)) - f(CR(\mathbf{x}_i-b))\right]$

where $$f$$ is an expectation value depending on $$CR(\mathbf{x}_i)$$, and

• $$\mathbf{x} = (\phi, \theta, \omega)$$ and i is an index to $$\mathbf{x}$$

• $$a = \pi/2$$

• $$b = 3\pi/2$$

• $$c_{\pm} = (\sqrt{2} \pm 1)/{4\sqrt{2}}$$

Parameters
• phi (float) – rotation angle $$\phi$$

• theta (float) – rotation angle $$\theta$$

• omega (float) – rotation angle $$\omega$$

• wires (Sequence[int]) – the wire the operation acts on

• id (str or None) – String representing the operation (optional)

 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 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 this operator 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
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 = 'CRot'
ndim_params = (0, 0, 0)

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

Type

tuple[int]

num_params = 3

Number of trainable parameters that the operator depends on.

Type

int

num_wires = 2

Number of wires this operator acts on.

Type

int

parameter_frequencies = [(0.5, 1.0), (0.5, 1.0), (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.

 Create an operation that is the adjoint of this one. compute_decomposition(phi, theta, omega, 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(phi, theta, omega) 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). 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. Returns a tape that contains 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. 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. 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. 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. 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

static compute_decomposition(phi, theta, omega, wires)[source]

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

$O = O_1 O_2 \dots O_n.$

Parameters
• phi (float) – rotation angle $$\phi$$

• theta (float) – rotation angle $$\theta$$

• omega (float) – rotation angle $$\omega$$

• wires (Iterable, Wires) – the wires the operation acts on

Returns

decomposition into lower level operations

Return type

list[Operator]

Example:

>>> qml.CRot.compute_decomposition(1.234, 2.34, 3.45, wires=[0, 1])
[RZ(-1.108, wires=[1]),
CNOT(wires=[0, 1]),
RZ(-2.342, wires=[1]),
RY(-1.17, wires=[1]),
CNOT(wires=[0, 1]),
RY(1.17, wires=[1]),
RZ(3.45, wires=[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.

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)

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.

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(phi, theta, omega)[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
• phi (tensor_like or float) – first rotation angle

• theta (tensor_like or float) – second rotation angle

• omega (tensor_like or float) – third rotation angle

Returns

canonical matrix

Return type

tensor_like

Example

>>> qml.CRot.compute_matrix(torch.tensor(0.1), torch.tensor(0.2), torch.tensor(0.3))
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,  0.9752-0.1977j, -0.0993+0.0100j],
[ 0.0+0.0j,  0.0+0.0j,  0.0993+0.0100j,  0.9752+0.1977j]])

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.

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.

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.

Returns

a list of operators

Return type

list[Operator] or None

eigvals()

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.

Returns

eigenvalues

Return type

tensor_like

expand()

Returns a tape that contains 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 * Y(0) + Z(0) @ X(1)


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)

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.],
>>> 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.],

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.

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.

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]]