qml.U3¶

class U3(theta, phi, delta, wires, do_queue=True, id=None)[source]¶

Bases: pennylane.operation.Operation

Arbitrary single qubit unitary.

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

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

\[U_3(\theta, \phi, \delta) = R_\phi(\phi+\delta) R(\delta,\theta,-\delta)\]

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
Number of dimensions per parameter: (0, 0, 0)
Gradient recipe: \(\frac{d}{d\phi}f(U_3(\theta, \phi, \delta)) = \frac{1}{2}\left[f(U_3(\theta+\pi/2, \phi, \delta)) - f(U_3(\theta-\pi/2, \phi, \delta))\right]\) where \(f\) is an expectation value depending on \(U_3(\theta, \phi, \delta)\). This gradient recipe applies for each angle argument \(\{\theta, \phi, \delta\}\).

Parameters

theta (float) – polar angle \(\theta\)
phi (float) – azimuthal angle \(\phi\)
delta (float) – quantum phase \(\delta\)
wires (Sequence[int] or int) – the subsystem the gate acts on
do_queue (bool) – Indicates whether the operator should be immediately pushed into the Operator queue (optional)
id (str or None) – String representing the operation (optional)

Attributes

`base_name`	If inverse is requested, this is the name of the original operator to be inverted.
`basis`	The target operation for controlled gates.
`batch_size`	Batch size of the operator if it is used with broadcasted parameters.
`control_wires`	Control wires of the operator.
`grad_method`
`grad_recipe`	Gradient recipe for the parameter-shift method.
`has_matrix`
`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.
`inverse`	Boolean determining if the inverse of the operation was requested.
`is_hermitian`	This property determines if an operator is hermitian.
`name`	Name of the operator.
`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`
`parameter_frequencies`
`parameters`	Trainable parameters that the operator depends on.
`wires`	Wires that the operator acts on.

base_name¶: If inverse is requested, this is the name of the original operator to be inverted.

basis = None¶

The target operation for controlled gates. target operation. If not None, should take a value of "X", "Y", or "Z".

For example, X and CNOT have basis = "X", whereas ControlledPhaseShift and RZ have basis = "Z".

Type: str or None

batch_size¶

Batch size of the operator if it is used with broadcasted parameters.

The batch_size is determined based on ndim_params and the provided parameters for the operator. If (some of) the latter have an additional dimension, and this dimension has the same size for all parameters, its size is the batch size of the operator. If no parameter has an additional dimension, the batch size is None.

Returns: Size of the parameter broadcasting dimension if present, else None.
Return type: int or None

control_wires¶

Control wires of the operator.

For operations that are not controlled, this is an empty Wires object of length 0.

Returns: The control wires of the operation.
Return type: Wires

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

has_matrix = True¶

hash¶

Integer hash that uniquely represents the operator.

Type: int

hyperparameters¶

Dictionary of non-trainable variables that this operation depends on.

Type: dict

id¶: Custom string to label a specific operator instance.

inverse¶: Boolean determining if the inverse of the operation was requested.

is_hermitian¶: This property determines if an operator is hermitian.

name¶: Name of the operator.

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 = 1¶

parameter_frequencies = [(1,), (1,), (1,)]¶

parameters¶: Trainable parameters that the operator depends on.

wires¶

Wires that the operator acts on.

Returns: wires
Return type: Wires

Methods

`adjoint`()	Create an operation that is the adjoint of this one.
`compute_decomposition`(theta, phi, delta, 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`(theta, phi, delta)	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).
`compute_terms`(params, *hyperparams)	Representation of the operator as a linear combination of other operators (static method).
`decomposition`()	Representation of the operator as a product of other operators.
`diagonalizing_gates`()	Sequence of gates that diagonalize the operator in the computational basis.
`eigvals`()	Eigenvalues of the operator in the computational basis (static method).
`expand`()	Returns a tape that has recorded the decomposition of the operator.
`generator`()	Generator of an operator that is in single-parameter-form.
`get_parameter_shift`(idx)	Multiplier and shift for the given parameter, based on its gradient recipe.
`inv`()	Inverts the operator.
`label`([decimals, base_label, cache])	A customizable string representation of the operator.
`matrix`([wire_order])	Representation of the operator as a matrix in the computational basis.
`pow`(z)	A list of new operators equal to this one raised to the given power.
`queue`([context])	Append the operator to the Operator queue.
`single_qubit_rot_angles`()	The parameters required to implement a single-qubit gate as an equivalent `Rot` gate, up to a global phase.
`sparse_matrix`([wire_order])	Representation of the operator as a sparse matrix in the computational basis.
`terms`()	Representation of the operator as a linear combination of other operators.

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 compute_decomposition(theta, phi, delta, wires)[source]¶

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

\[O = O_1 O_2 \dots O_n.\]

See also

decomposition().

Parameters

theta (float) – polar angle \(\theta\)
phi (float) – azimuthal angle \(\phi\)
delta (float) – quantum phase \(\delta\)
wires (Iterable, Wires) – the subsystem the gate acts on

Returns

decomposition into lower level operations

Return type

list[Operator]

Example:

>>> qml.U3.compute_decomposition(1.23, 2.34, 3.45, wires=0)
[Rot(3.45, 1.23, -3.45, wires=[0]),
PhaseShift(3.45, wires=[0]),
PhaseShift(2.34, wires=[0])]

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 \(U\).

The diagonalizing gates rotate the state into the eigenbasis of the operator.

See also

diagonalizing_gates().

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 \(U\), 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.

See also

eigvals() and eigvals()

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(theta, phi, delta)[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.

See also

matrix()

Parameters

theta (tensor_like or float) – polar angle
phi (tensor_like or float) – azimuthal angle
delta (tensor_like or float) – quantum phase

Returns

canonical matrix

Return type

tensor_like

Example

>>> qml.U3.compute_matrix(torch.tensor(0.1), torch.tensor(0.2), torch.tensor(0.3))
tensor([[ 0.9988+0.0000j, -0.0477-0.0148j],
        [ 0.0490+0.0099j,  0.8765+0.4788j]])

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.

See also

sparse_matrix()

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

static compute_terms(*params, **hyperparams)¶

Representation of the operator as a linear combination of other operators (static method).

\[O = \sum_i c_i O_i\]

See also

terms()

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

list of coefficients and list of operations

Return type

tuple[list[tensor_like or float], list[Operation]]

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.

See also

compute_decomposition().

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 \(U\).

The diagonalizing gates rotate the state into the eigenbasis of the operator.

A DiagGatesUndefinedError is raised if no representation by decomposition is defined.

See also

compute_diagonalizing_gates().

Returns: a list of operators
Return type: list[Operator] or None

eigvals()¶

Eigenvalues of the operator in the computational basis (static method).

If diagonalizing_gates are specified and implement a unitary \(U\), 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.

See also

compute_eigvals()

Returns: eigenvalues
Return type: tensor_like

expand()¶

Returns a tape that has recorded 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) [Y0]
+ (1.0) [Z0 X1]

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.

get_parameter_shift(idx)¶

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

Parameters: idx (int) – parameter index within the operation
Returns: list of multiplier, coefficient, shift for each term in the gradient recipe
Return type: list[[float, float, float]]

Note that the default value for shift is None, which is replaced by the default shift \(\pi/2\).

inv()¶

Inverts the operator.

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

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 caries 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(decimals=2)
"RX\n(1.23)"
>>> op.label(base_label="my_label")
"my_label"
>>> op.label(decimals=2, base_label="my_label")
"my_label\n(1.23)"
>>> op.inv()
>>> op.label()
"RX⁻¹"

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.],
 [0., 1.]], requires_grad=True)]
>>> op3 = qml.QubitUnitary(np.eye(4), wires=(0,1))
>>> op3.label(cache=cache)
'U(M1)'
>>> cache['matrices']
[tensor([[1., 0.],
        [0., 1.]], requires_grad=True),
tensor([[1., 0., 0., 0.],
        [0., 1., 0., 0.],
        [0., 0., 1., 0.],
        [0., 0., 0., 1.]], requires_grad=True)]

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.

See also

compute_matrix()

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.QueuingContext'>)¶: Append the operator to the Operator queue.

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)¶

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.

Note

The wire_order argument is currently not implemented, and using it will raise an error.

A SparseMatrixUndefinedError is raised if the sparse matrix representation has not been defined.

See also

compute_sparse_matrix()

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.

See also

compute_terms()

Returns: list of coefficients \(c_i\) and list of operations \(O_i\)
Return type: tuple[list[tensor_like or float], list[Operation]]

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