# qml.U3¶

class U3(theta, phi, lambda, wires)[source]

Arbitrary single qubit unitary.

$\begin{split}U_3(\theta, \phi, \lambda) = \begin{bmatrix} \cos(\theta/2) & -\exp(i \lambda)\sin(\theta/2) \\ \exp(i \phi)\sin(\theta/2) & \exp(i (\phi + \lambda))\cos(\theta/2) \end{bmatrix}\end{split}$

The $$U_3$$ gate is related to the single-qubit rotation $$R$$ (Rot) and the $$R_\phi$$ (PhaseShift) gates via the following relation:

$U_3(\theta, \phi, \lambda) = R_\phi(\phi+\lambda) R(\lambda,\theta,-\lambda)$

Note

If the U3 gate is not supported on the targeted device, PennyLane will attempt to decompose the gate into PhaseShift and Rot gates.

Details:

• Number of wires: 1

• Number of parameters: 3

• Gradient recipe: $$\frac{d}{d\phi}f(U_3(\theta, \phi, \lambda)) = \frac{1}{2}\left[f(U_3(\theta+\pi/2, \phi, \lambda)) - f(U_3(\theta-\pi/2, \phi, \lambda))\right]$$ where $$f$$ is an expectation value depending on $$U_3(\theta, \phi, \lambda)$$. This gradient recipe applies for each angle argument $$\{\theta, \phi, \lambda\}$$.

Parameters
• theta (float) – polar angle $$\theta$$

• phi (float) – azimuthal angle $$\phi$$

• lambda (float) – quantum phase $$\lambda$$

• wires (Sequence[int] or int) – the subsystem the gate acts on

 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. 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.

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 = None
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

name

Get and set the name of the operator.

num_params = 3
num_wires = 1
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]

string_for_inverse = '.inv'
wires

Wires of this operator.

Returns

wires

Return type

Wires

 check_domain(p[, flattened]) Check the validity of a parameter. decomposition(theta, phi, lam, 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. 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(theta, phi, lam, wires)[source]

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

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.