# qml.CRot¶

class CRot(phi, theta, omega, wires)[source]

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

• Gradient recipe: The controlled-Rot operator satisfies a four-term parameter-shift rule (see Appendix F, https://arxiv.org/abs/2104.05695):

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

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

 base_name Get base name of the operator. eigvals Eigenvalues of an instantiated operator. generator Generator of the operation. grad_method grad_recipe Gradient recipe for the parameter-shift method. inverse Boolean determining if the inverse of the operation was requested. matrix Matrix representation of an instantiated operator in the computational basis. name Get and set the name of the operator. num_params num_wires par_domain parameters Current parameter values. string_for_inverse wires Wires of this operator.
base_name

Get base name of the operator.

eigvals
generator

Generator of the operation.

A length-2 list [generator, scaling_factor], where

• generator is an existing PennyLane operation class or $$2\times 2$$ Hermitian array that acts as the generator of the current operation

• scaling_factor represents a scaling factor applied to the generator operation

For example, if $$U(\theta)=e^{i0.7\theta \sigma_x}$$, then $$\sigma_x$$, with scaling factor $$s$$, is the generator of operator $$U(\theta)$$:

generator = [PauliX, 0.7]


Default is [None, 1], indicating the operation has no generator.

grad_method = 'A'
grad_recipe = ([[0.4267766952966368, 1, 1.5707963267948966], [-0.4267766952966368, 1, -1.5707963267948966], [-0.07322330470336313, 1, 4.71238898038469], [0.07322330470336313, 1, -4.71238898038469]], [[0.4267766952966368, 1, 1.5707963267948966], [-0.4267766952966368, 1, -1.5707963267948966], [-0.07322330470336313, 1, 4.71238898038469], [0.07322330470336313, 1, -4.71238898038469]], [[0.4267766952966368, 1, 1.5707963267948966], [-0.4267766952966368, 1, -1.5707963267948966], [-0.07322330470336313, 1, 4.71238898038469], [0.07322330470336313, 1, -4.71238898038469]])

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

inverse

Boolean determining if the inverse of the operation was requested.

matrix
name

Get and set the name of the operator.

num_params = 3
num_wires = 2
par_domain = 'R'
parameters

Current parameter values.

string_for_inverse = '.inv'
wires

Wires of this operator.

Returns

wires

Return type

Wires

 Create an operation that is the adjoint of this one. decomposition(phi, theta, omega, wires) Returns a template decomposing the operation into other quantum operations. Returns a tape containing the decomposed operations, rather than a list. get_parameter_shift(idx[, shift]) Multiplier and shift for the given parameter, based on its gradient recipe. Inverts the operation, such that the inverse will be used for the computations by the specific device. Append the operator to the Operator queue.
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.

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

Returns a template decomposing the operation into other quantum operations.

expand()

Returns a tape containing the decomposed operations, rather than a list.

Returns

Returns a quantum tape that contains the operations decomposition, or if not implemented, simply the operation itself.

Return type

JacobianTape

get_parameter_shift(idx, shift=1.5707963267948966)

Multiplier and shift for the given parameter, based on its gradient recipe.

Parameters

idx (int) – parameter index

Returns

multiplier, shift

Return type

float, float

inv()

Inverts the operation, such that the inverse will be used for the computations by the specific device.

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

queue()

Append the operator to the Operator queue.