qml.ops.op_math.Pow

class Pow(base=None, z=1, id=None)

Bases: pennylane.ops.op_math.symbolicop.ScalarSymbolicOp

Symbolic operator denoting an operator raised to a power.

Parameters

• base (Operator) – the operator to be raised to a power

• z=1 (float) – the exponent

Example

>>> sqrt_x = Pow(qml.X(0), 0.5)
>>> sqrt_x.decomposition()
[SX(wires=[0])]
>>> qml.matrix(sqrt_x)
array([[0.5+0.5j, 0.5-0.5j],
            [0.5-0.5j, 0.5+0.5j]])
>>> qml.matrix(qml.SX(0))
array([[0.5+0.5j, 0.5-0.5j],
   [0.5-0.5j, 0.5+0.5j]])
>>> qml.matrix(Pow(qml.T(0), 1.234))
array([[1.        +0.j        , 0.        +0.j        ],
   [0.        +0.j        , 0.56597465+0.82442265j]])

Attributes

arithmetic_depth	Arithmetic depth of the operator.
base	The base operator.
basis
batch_size	Batch size of the operator if it is used with broadcasted parameters.
data	The trainable parameters
has_adjoint	bool(x) -> bool
has_decomposition	bool(x) -> bool
has_diagonalizing_gates	bool(x) -> bool
has_generator	bool(x) -> bool
has_matrix	bool(x) -> bool
has_sparse_matrix	bool(x) -> bool
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 of the operator.
num_params	Number of trainable parameters that the operator depends on.
num_wires	Number of wires the operator acts on.
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.
z	The exponent.

arithmetic_depth

base

The base operator.

basis

batch_size

data

The trainable parameters

has_adjoint

has_decomposition

has_diagonalizing_gates

has_generator

has_matrix

has_sparse_matrix

hash

hyperparameters

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

Type

dict

id

Custom string to label a specific operator instance.

is_hermitian

name

String for the name of the operator.

ndim_params

num_params

num_wires

Number of wires the operator acts on.

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

z

The exponent.

Methods

adjoint()	Create an operation that is the adjoint of this one.
compute_decomposition(*params[, 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(*params, **hyperparams)	Representation of the operator as a canonical matrix in the computational basis (static method).
compute_sparse_matrix(*params[, base, z])	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.
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.

Warning

The adjoint of a fractional power of an operator is not well-defined due to branch cuts in the power function. Therefore, an AdjointUndefinedError is raised when the power z is not an integer.

The integer power check is a type check, so that floats like 2.0 are not considered to be integers.

Returns

The adjointed operation.

Raises

AdjointUndefinedError – If the exponent z is not of type int.

static compute_decomposition(*params, wires=None, **hyperparameters)

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

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

Note

Operations making up the decomposition should be queued within the compute_decomposition method.

See also

decomposition().

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

decomposition of the operator

Return type

list[Operator]

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(*params, **hyperparams)

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

Operator.eigvals() and qml.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(*params, **hyperparams)

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

Operator.matrix() and qml.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

matrix representation

Return type

tensor_like

static compute_sparse_matrix(*params, base=None, z=0)

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 of an operator to a power is the same as the diagonalizing gates as the original operator. As we can see,

\[O^2 = U \Sigma U^{\dagger} U \Sigma U^{\dagger} = U \Sigma^2 U^{\dagger}\]

This formula can be extended to inversion and any rational number.

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.

The generator of a power operator is z times the generator of the base matrix.

\[U(\phi)^z = e^{i\phi (z G)}\]

See also 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 carries 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(base_label="my_label")
"my_label"
>>> op = qml.RX(1.23456, wires=0, id="test_data")
>>> op.label()
"RX("test_data")"
>>> op.label(decimals=2)
"RX\n(1.23,"test_data")"
>>> op.label(base_label="my_label")
"my_label("test_data")"
>>> op.label(decimals=2, base_label="my_label")
"my_label\n(1.23,"test_data")"

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)]

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 base matrix representation has not been defined.

See also

compute_matrix()

Parameters

• wire_order (Iterable) – global wire order, must contain all wire labels from the

• wires (operator's) –

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

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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