# qml.ControlledPhase¶

class ControlledPhase(s, wires)[source]

Controlled phase operation.

$\text{CZ}(s) = \iint dx dy \: e^{i sxy/\hbar} \ket{x,y}\bra{x,y} = e^{i s \: \hat{x} \otimes \hat{x}/\hbar}.$

Details:

• Number of wires: 2

• Number of parameters: 1

• Gradient recipe: $$\frac{d}{ds}f(\text{CZ}(s)) = \frac{1}{2 a} \left[f(\text{CZ}(s+a)) - f(\text{CZ}(s-a))\right]$$, where $$a$$ is an arbitrary real number ($$0.1$$ by default) and $$f$$ is an expectation value depending on $$\text{CZ}(s)$$.

• Heisenberg representation:

$\begin{split}M = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & s & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & s & 0 & 0 & 1 \end{bmatrix}\end{split}$
Parameters
• s (float) – phase shift multiplier

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

 a base_name Get base name of the operator. do_check_domain eigvals Eigenvalues of an instantiated operator. generator Generator of the operation. grad_method grad_recipe inverse Boolean determining if the inverse of the operation was requested. matrix Matrix representation of an instantiated operator in the computational basis. multiplier name Get and set the name of the operator. num_params num_wires par_domain parameters Current parameter values. shift string_for_inverse supports_heisenberg supports_parameter_shift wires Wires of this operator.
a = 1
base_name

Get base name of the operator.

do_check_domain = True
eigvals

Eigenvalues of an instantiated operator.

Note that the eigenvalues are not guaranteed to be in any particular order.

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

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 = ([[5.0, 1, 0.1], [-5.0, 1, -0.1]],)
inverse

Boolean determining if the inverse of the operation was requested.

matrix

Matrix representation of an instantiated operator in the computational basis.

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

multiplier = 5.0
name

Get and set the name of the operator.

num_params = 1
num_wires = 2
par_domain = 'R'
parameters

Current parameter values.

Fixed parameters are returned as is, free parameters represented by Variable instances are replaced by their current numerical value.

Returns

parameter values

Return type

list[Any]

shift = 0.1
string_for_inverse = '.inv'
supports_heisenberg = True
supports_parameter_shift = True
wires

Wires of this operator.

Returns

wires

Return type

Wires

 check_domain(p[, flattened]) Check the validity of a parameter. decomposition(*params, wires) Returns a template decomposing the operation into other quantum operations. get_parameter_shift(idx[, shift]) Multiplier and shift for the given parameter, based on its gradient recipe. heisenberg_expand(U, wires) Expand the given local Heisenberg-picture array into a full-system one. Partial derivative of the Heisenberg picture transform matrix. heisenberg_tr(wires[, inverse]) Heisenberg picture representation of the linear transformation carried out by the gate at current parameter values. Inverts the operation, such that the inverse will be used for the computations by the specific device. Append the operator to the Operator queue.
check_domain(p, flattened=False)

Check the validity of a parameter.

Variable instances can represent any real scalars (but not arrays).

Parameters
• p (Number, array, Variable) – parameter to check

• flattened (bool) – True means p is an element of a flattened parameter sequence (affects the handling of ‘A’ parameters)

Raises
• TypeError – parameter is not an element of the expected domain

• ValueError – parameter is an element of an unknown domain

Returns

p

Return type

Number, array, Variable

static decomposition(*params, wires)

Returns a template decomposing the operation into other quantum operations.

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

heisenberg_expand(U, wires)

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)

• wires (Wires) – wires on the device the array U should be expanded to apply to

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_pd(idx)

Partial derivative of the Heisenberg picture transform matrix.

Parameters

idx (int) – index of the parameter with respect to which the partial derivative is computed.

Returns

partial derivative

Return type

array[float]

heisenberg_tr(wires, inverse=False)

Heisenberg picture representation of the linear transformation carried out by the gate at current parameter values.

Given a unitary quantum gate $$U$$, we may consider its linear transformation in the Heisenberg picture, $$U^\dagger(\cdot) U$$.

If the gate is Gaussian, this linear transformation preserves the polynomial order of any observables that are polynomials in $$\mathbf{r} = (\I, \x_0, \p_0, \x_1, \p_1, \ldots)$$. This also means it maps $$\text{span}(\mathbf{r})$$ into itself:

$U^\dagger \mathbf{r}_i U = \sum_j \tilde{U}_{ij} \mathbf{r}_j$

For Gaussian CV gates, this method returns the transformation matrix for the current parameter values of the Operation. The method is not defined for non-Gaussian (and non-CV) gates.

Parameters
• wires (Wires) – wires on the device that the observable gets applied to

• inverse (bool) – if True, return the inverse transformation instead

Raises

RuntimeError – if the specified operation is not Gaussian or is missing the _heisenberg_rep method

Returns

$$\tilde{U}$$, the Heisenberg picture representation of the linear transformation

Return type

array[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.