qml.MultiControlledX¶

class
MultiControlledX
(control_wires, wires, control_values)[source]¶ Bases:
pennylane.ops.op_math.controlled.ControlledOp
Apply a Pauli X gate controlled on an arbitrary computational basis state.
Details:
Number of wires: Any (the operation can act on any number of wires)
Number of parameters: 0
Gradient recipe: None
 Parameters
control_wires (Union[Wires, Sequence[int], or int]) – Deprecated way to indicate the control wires. Now users should use “wires” to indicate both the control wires and the target wire.
wires (Union[Wires, Sequence[int], or int]) – control wire(s) followed by a single target wire where the operation acts on
control_values (Union[bool, list[bool], int, list[int]]) – The value(s) the control wire(s) should take. Integers other than 0 or 1 will be treated as
int(bool(x))
.work_wires (Union[Wires, Sequence[int], or int]) – optional work wires used to decompose the operation into a series of Toffoli gates
Note
If
MultiControlledX
is not supported on the targeted device, PennyLane will decompose the operation intoToffoli
and/orCNOT
gates. When controlling on three or more wires, the Toffolibased decompositions described in Lemmas 7.2 and 7.3 of Barenco et al. will be used. These methods require at least one work wire.The number of work wires provided determines the decomposition method used and the resulting number of Toffoli gates required. When
MultiControlledX
is controlling on \(n\) wires:If at least \(n  2\) work wires are provided, the decomposition in Lemma 7.2 will be applied using the first \(n  2\) work wires.
If fewer than \(n  2\) work wires are provided, a combination of Lemmas 7.3 and 7.2 will be applied using only the first work wire.
These methods present a tradeoff between qubit number and depth. The method in point 1 requires fewer Toffoli gates but a greater number of qubits.
Note that the state of the work wires before and after the decomposition takes place is unchanged.
Attributes
Wires modified by the operator.
Arithmetic depth of the operator.
The base operator.
The basis of an operation, or for controlled gates, of the target operation.
Batch size of the operator if it is used with broadcasted parameters.
Iterable[Bool].
The control wires.
The trainable parameters
Gradient computation method.
Gradient recipe for the parametershift method.
bool(x) > bool
bool(x) > bool
bool(x) > bool
bool(x) > bool
bool(x) > bool
Integer hash that uniquely represents the operator.
Dictionary of nontrainable variables that this operation depends on.
Custom string to label a specific operator instance.
This property determines if an operator is hermitian.
Whether or not the operator is selfinverse.
Number of dimensions per trainable parameter that the operator depends on.
Number of trainable parameters that the operator depends on.
Number of wires the operation acts on.
Returns the frequencies for each operator parameter with respect to an expectation value of the form \(\langle \psi  U(\mathbf{p})^\dagger \hat{O} U(\mathbf{p})\psi\rangle\).
Trainable parameters that the operator depends on.
A
PauliSentence
representation of the Operator, orNone
if it doesn’t have one.The wires of the target operator.
Wires that the operator acts on.
Additional wires that can be used in the decomposition.

active_wires
¶ Wires modified by the operator. This is the control wires followed by the target wires.

arithmetic_depth
¶

base
¶ The base operator.

basis
¶

batch_size
¶

control_values
¶ Iterable[Bool]. For each control wire, denotes whether to control on
True
orFalse
.

control_wires
¶ The control wires.

data
¶ The trainable parameters

grad_method
¶

grad_recipe
= None¶ Gradient recipe for the parametershift 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
¶

has_decomposition
¶

has_diagonalizing_gates
¶

has_generator
¶

has_matrix
¶

hash
¶

hyperparameters
¶ Dictionary of nontrainable variables that this operation depends on.
 Type
dict

id
¶ Custom string to label a specific operator instance.

is_hermitian
¶

is_self_inverse
= True¶ Whether or not the operator is selfinverse.
 Type
bool

name
= 'MultiControlledX'¶

ndim_params
= ()¶ Number of dimensions per trainable parameter that the operator depends on.
 Type
tuple[int]

num_params
= 0¶ Number of trainable parameters that the operator depends on.
 Type
int

num_wires
= 1¶ Number of wires the operation acts on.
 Type
int

parameter_frequencies
¶

parameters
¶ Trainable parameters that the operator depends on.

pauli_rep
¶ A
PauliSentence
representation of the Operator, orNone
if it doesn’t have one.

target_wires
¶ The wires of the target operator.

wires
¶

work_wires
¶ Additional wires that can be used in the decomposition. Not modified by the operation.
Methods
adjoint
()Create an operation that is the adjoint of this one.
compute_decomposition
([wires, work_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
(control_wires[, control_values])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).
Representation of the operator as a product of other operators.
Sequence of gates that diagonalize the operator in the computational basis.
eigvals
()Eigenvalues of the operator in the computational basis.
expand
()Returns a tape that contains the decomposition of the operator.
Generator of an operator that is in singleparameterform.
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.
The parameters required to implement a singlequbit gate as an equivalent
Rot
gate, up to a global phase.sparse_matrix
([wire_order, format])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
(wires=None, work_wires=None, control_values=None, **kwargs)[source]¶ Representation of the operator as a product of other operators (static method).
\[O = O_1 O_2 \dots O_n.\]See also
 Parameters
wires (Iterable[Any] or Wires) – wires that the operation acts on
work_wires (Wires) – optional work wires used to decompose the operation into a series of Toffoli gates.
control_values (Union[bool, list[bool], int, list[int]]) – The value(s) the control wire(s) should take. Integers other than 0 or 1 will be treated as
int(bool(x))
.
 Returns
decomposition into lower level operations
 Return type
list[Operator]
Example:
>>> print(qml.MultiControlledX.compute_decomposition( ... wires=[0,1,2,3], control_values=[1,1,1], work_wires=qml.wires.Wires("aux"))) [Toffoli(wires=[2, 'aux', 3]), Toffoli(wires=[0, 1, 'aux']), Toffoli(wires=[2, 'aux', 3]), Toffoli(wires=[0, 1, 'aux'])]

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
 Parameters
params (list) – trainable parameters of the operator, as stored in the
parameters
attributewires (Iterable[Any], Wires) – wires that the operator acts on
hyperparams (dict) – nontrainable 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
 Parameters
*params (list) – trainable parameters of the operator, as stored in the
parameters
attribute**hyperparams (dict) – nontrainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
 Returns
eigenvalues
 Return type
tensor_like

static
compute_matrix
(control_wires, control_values=None, **kwargs)[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
 Parameters
control_wires (Any or Iterable[Any]) – wires to place controls on
control_values (Union[bool, list[bool], int, list[int]]) – The value(s) the control wire(s) should take. Integers other than 0 or 1 will be treated as
int(bool(x))
.
 Returns
matrix representation
 Return type
tensor_like
Example
>>> print(qml.MultiControlledX.compute_matrix([0], [1])) [[1. 0. 0. 0.] [0. 1. 0. 0.] [0. 0. 0. 1.] [0. 0. 1. 0.]] >>> print(qml.MultiControlledX.compute_matrix([1], [0])) [[0. 1. 0. 0.] [1. 0. 0. 0.] [0. 0. 1. 0.] [0. 0. 0. 1.]]

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
 Parameters
*params (list) – trainable parameters of the operator, as stored in the
parameters
attribute**hyperparams (dict) – nontrainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
 Returns
sparse matrix representation
 Return type
scipy.sparse._csr.csr_matrix

decomposition
()[source]¶ 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
 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
 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
 Returns
eigenvalues
 Return type
tensor_like

expand
()¶ Returns a tape that contains the decomposition of the operator.
 Returns
quantum tape
 Return type

generator
()¶ Generator of an operator that is in singleparameterform.
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
Hermitian
andSparseHamiltonian
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 nonparameter 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 matrixvalued 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

matrix
(wire_order=None)[source]¶ 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
 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

single_qubit_rot_angles
()¶ The parameters required to implement a singlequbit 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, format='csr')¶ 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
 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]]