qml.PauliRot

class PauliRot(theta, pauli_word, wires, id=None)

Bases: pennylane.operation.Operation

Arbitrary Pauli word rotation.

\[RP(\theta, P) = \exp\left(-i \frac{\theta}{2} P\right)\]

Details:

• Number of wires: Any

• Number of parameters: 1

• Number of dimensions per parameter: (0,)

• Gradient recipe: \(\frac{d}{d\theta}f(RP(\theta)) = \frac{1}{2}\left[f(RP(\theta +\pi/2)) - f(RP(\theta-\pi/2))\right]\) where \(f\) is an expectation value depending on \(RP(\theta)\).

Note

If the PauliRot gate is not supported on the targeted device, PennyLane will decompose the gate using RX, Hadamard, RZ and CNOT gates.

Parameters

• theta (float) – rotation angle \(\theta\)

• pauli_word (string) – the Pauli word defining the rotation

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

• id (str or None) – String representing the operation (optional)

Example

>>> dev = qml.device('default.qubit', wires=1)
>>> @qml.qnode(dev)
... def example_circuit():
...     qml.PauliRot(0.5, 'X',  wires=0)
...     return qml.expval(qml.Z(0))
>>> print(example_circuit())
0.8775825618903724

Attributes

arithmetic_depth	Arithmetic depth of the operator.
basis	The basis of an operation, or for controlled gates, of the target operation.
batch_size	Batch size of the operator if it is used with broadcasted parameters.
control_wires	Control wires of the operator.
do_check_domain
grad_method
grad_recipe	Gradient recipe for the parameter-shift method.
has_adjoint
has_decomposition
has_diagonalizing_gates
has_generator
has_matrix
has_sparse_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.
is_hermitian	This property determines if an operator is hermitian.
name	String for the 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	Number of wires the operator acts on.
parameter_frequencies
parameters	Trainable parameters that the operator depends on.
pauli_rep	A PauliSentence representation of the Operator, or None if it doesn't have one.
wires	Wires that the operator acts on.

arithmetic_depth

Arithmetic depth of the operator.

basis

The basis of an operation, or for controlled gates, of the 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

do_check_domain = False

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_adjoint = True

has_decomposition = True

has_diagonalizing_gates = False

has_generator = True

has_matrix = True

has_sparse_matrix = False

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.

is_hermitian

This property determines if an operator is hermitian.

name

String for the name of the operator.

ndim_params = (0,)

Number of dimensions per trainable parameter that the operator depends on.

Type

tuple[int]

num_params = 1

Number of trainable parameters that the operator depends on.

Type

int

num_wires = -1

Number of wires the operator acts on.

parameter_frequencies = [(1,)]

parameters

Trainable parameters that the operator depends on.

pauli_rep

A PauliSentence representation of the Operator, or None if it doesn’t have one.

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, wires, pauli_word)	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(theta, pauli_word)	Eigenvalues of the operator in the computational basis (static method).
compute_matrix(theta, pauli_word)	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).
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.
generator()	Generator of an operator that is in single-parameter-form.
label([decimals, base_label, cache])	A customizable string representation of the operator.
map_wires(wire_map)	Returns a copy of the current operator with its wires changed according to the given wire map.
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.
simplify()	Reduce the depth of nested operators to the minimum.
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()

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.

Returns

The adjointed operation.

static compute_decomposition(theta, wires, pauli_word)

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) – rotation angle \(\theta\)

• wires (Iterable, Wires) – the wires the operation acts on

• pauli_word (string) – the Pauli word defining the rotation

Returns

decomposition into lower level operations

Return type

list[Operator]

Example:

>>> qml.PauliRot.compute_decomposition(1.2, "XY", wires=(0,1))
[Hadamard(wires=[0]),
RX(1.5707963267948966, wires=[1]),
MultiRZ(1.2, wires=[0, 1]),
Hadamard(wires=[0]),
RX(-1.5707963267948966, wires=[1])]

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 \(U^{\dagger}\).

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(theta, pauli_word)

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

If diagonalizing_gates are specified and implement a unitary \(U^{\dagger}\), 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()

Returns

eigenvalues

Return type

tensor_like

Example

>>> qml.PauliRot.compute_eigvals(torch.tensor(0.5), "X")
tensor([0.9689-0.2474j, 0.9689+0.2474j])

static compute_matrix(theta, pauli_word)

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) – rotation angle

• pauli_word (str) – string representation of Pauli word

Returns

canonical matrix

Return type

tensor_like

Example

>>> qml.PauliRot.compute_matrix(0.5, 'X')
[[9.6891e-01+4.9796e-18j 2.7357e-17-2.4740e-01j]
 [2.7357e-17-2.4740e-01j 9.6891e-01+4.9796e-18j]]

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

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 \(U^{\dagger}\).

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.

If diagonalizing_gates are specified and implement a unitary \(U^{\dagger}\), 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

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 * Y(0) + Z(0) @ X(1)

The generator may also be provided in the form of a dense or sparse Hamiltonian (using Hamiltonian and SparseHamiltonian respectively).

The default value to return is None, indicating that the operation has no defined generator.

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.PauliRot(0.1, "XYY", wires=(0,1,2))
>>> op.label()
'RXYY'
>>> op.label(decimals=2)
'RXYY\n(0.10)'
>>> op.label(base_label="PauliRot")
'PauliRot\n(0.10)'

map_wires(wire_map)

Returns a copy of the current operator with its wires changed according to the given wire map.

Parameters

wire_map (dict) – dictionary containing the old wires as keys and the new wires as values

Returns

new operator

Return type

Operator

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.QueuingManager'>)

Append the operator to the Operator queue.

simplify()

Reduce the depth of nested operators to the minimum.

Returns

simplified operator

Return type

Operator

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.

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.

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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