qml.U3

class U3(theta, phi, lambda, wires)

Bases: pennylane.operation.Operation

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

Attributes

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.
id	String for the ID of the operator.
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 = 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

id

String for the ID of the operator.

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 = 1

par_domain = 'R'

parameters

Current parameter values.

string_for_inverse = '.inv'

wires

Wires of this operator.

Returns

wires

Return type

Wires

Methods

adjoint()	Create an operation that is the adjoint of this one.
decomposition(theta, phi, lam, wires)	Returns a template decomposing the operation into other quantum operations.
expand()	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.
inv()	Inverts the operation, such that the inverse will be used for the computations by the specific device.
queue()	Append the operator to the Operator queue.

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.

Parameters

do_queue – Whether to add the adjointed gate to the context queue.

Returns

The adjointed operation.

static decomposition(theta, phi, lam, wires)

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

list of multiplier, coefficient, shift for each term in the gradient recipe

Return type

list[[float, 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.