qml.CRZ¶
- class CRZ(phi, wires, id=None)[source]¶
Bases:
pennylane.ops.op_math.controlled.ControlledOp
The controlled-RZ operator
\[\begin{split}\begin{align} CR_z(\phi) &= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\\ 0 & 0 & e^{-i\phi/2} & 0\\ 0 & 0 & 0 & e^{i\phi/2} \end{bmatrix}. \end{align}\end{split}\]Note
The subscripts of the operations in the formula refer to the wires they act on, e.g. 1 corresponds to the first element in
wires
that is the control qubit.Details:
Number of wires: 2
Number of parameters: 1
Number of dimensions per parameter: (0,)
Gradient recipe: The controlled-RZ operator satisfies a four-term parameter-shift rule (see Appendix F, https://doi.org/10.1088/1367-2630/ac2cb3):
\[\frac{d}{d\phi}f(CR_z(\phi)) = c_+ \left[f(CR_z(\phi+a)) - f(CR_z(\phi-a))\right] - c_- \left[f(CR_z(\phi+b)) - f(CR_z(\phi-b))\right]\]where \(f\) is an expectation value depending on \(CR_z(\phi)\), and
\(a = \pi/2\)
\(b = 3\pi/2\)
\(c_{\pm} = (\sqrt{2} \pm 1)/{4\sqrt{2}}\)
- Parameters
phi (float) – rotation angle \(\phi\)
wires (Sequence[int]) – the wire the operation acts on
id (str or None) – String representing the operation (optional)
Attributes
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 parameter-shift 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 non-trainable variables that this operation depends on.
Custom string to label a specific operator instance.
This property determines if an operator is hermitian.
Number of dimensions per trainable parameter that the operator depends on.
Number of trainable parameters that the operator depends on.
Number of wires that the operation acts on.
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.
- 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 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¶
- has_decomposition¶
- has_diagonalizing_gates¶
- has_generator¶
- has_matrix¶
- has_sparse_matrix = True¶
- 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 = 'CRZ'¶
- 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 = 2¶
Number of wires that the operation acts on.
- Type
int
- parameter_frequencies = [(0.5, 1.0)]¶
- 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
(phi, 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
(theta, **_)Eigenvalues of the operator in the computational basis (static method).
compute_matrix
(theta)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.
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.
The parameters required to implement a single-qubit 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(phi, wires)[source]¶
Representation of the operator as a product of other operators (static method). :
\[O = O_1 O_2 \dots O_n.\]See also
- Parameters
phi (float) – rotation angle \(\phi\)
wires (Iterable, Wires) – wires that the operator acts on
- Returns
decomposition into lower level operations
- Return type
list[Operator]
Example:
>>> qml.CRZ.compute_decomposition(1.2, wires=(0,1)) [PhaseShift(0.6, wires=[1]), CNOT(wires=[0, 1]), PhaseShift(-0.6, wires=[1]), CNOT(wires=[0, 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
- 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) – 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, **_)[source]¶
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
theta (tensor_like or float) – rotation angle
- Returns
eigenvalues
- Return type
tensor_like
Example
>>> qml.CRZ.compute_eigvals(torch.tensor(0.5)) tensor([1.0000+0.0000j, 1.0000+0.0000j, 0.9689-0.2474j, 0.9689+0.2474j])
- static compute_matrix(theta)[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
theta (tensor_like or float) – rotation angle
- Returns
canonical matrix
- Return type
tensor_like
Example
>>> qml.CRZ.compute_matrix(torch.tensor(0.5)) tensor([[1.0+0.0j, 0.0+0.0j, 0.0+0.0j, 0.0+0.0j], [0.0+0.0j, 1.0+0.0j, 0.0+0.0j, 0.0+0.0j], [0.0+0.0j, 0.0+0.0j, 0.9689-0.2474j, 0.0+0.0j], [0.0+0.0j, 0.0+0.0j, 0.0+0.0j, 0.9689+0.2474j]])
- 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) – 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
- 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()[source]¶
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
- 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
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 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
- 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
- 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 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, 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]]