qml.Hermitian¶
-
class
Hermitian
(A, wires, do_queue=True, id=None)[source]¶ Bases:
pennylane.operation.Observable
An arbitrary Hermitian observable.
For a Hermitian matrix \(A\), the expectation command returns the value
\[\braket{A} = \braketT{\psi}{\cdots \otimes I\otimes A\otimes I\cdots}{\psi}\]where \(A\) acts on the requested wires.
If acting on \(N\) wires, then the matrix \(A\) must be of size \(2^N\times 2^N\).
Details:
Number of wires: Any
Number of parameters: 1
Gradient recipe: None
- Parameters
A (array) – square hermitian matrix
wires (Sequence[int] or int) – the wire(s) the operation 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
Return the eigendecomposition of the matrix specified by the Hermitian observable.
Eigenvalues of an instantiated operator.
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.
Matrix representation of an instantiated operator in the computational basis.
String for the name of the operator.
Number of trainable parameters that the operator depends on.
Trainable parameters that the operator depends on.
Measurement type that this observable is called with.
Wires that the operator acts on.
-
eigendecomposition
¶ Return the eigendecomposition of the matrix specified by the Hermitian observable.
This method uses pre-stored eigenvalues for standard observables where possible and stores the corresponding eigenvectors from the eigendecomposition.
It transforms the input operator according to the wires specified.
- Returns
dictionary containing the eigenvalues and the eigenvectors of the Hermitian observable
- Return type
dict[str, array]
-
eigvals
¶ Eigenvalues of an instantiated operator. Note that the eigenvalues are not guaranteed to be in any particular order.
Warning
The
eigvals
property is deprecated and will be removed in an upcoming release. Please useqml.eigvals
instead.Example:
>>> U = qml.RZ(0.5, wires=1) >>> U.eigvals >>> array([0.96891242-0.24740396j, 0.96891242+0.24740396j])
- Returns
eigvals representation
- Return type
array
-
grad_method
= 'F'¶
-
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.
-
matrix
¶ Matrix representation of an instantiated operator in the computational basis.
Warning
The
matrix
property is deprecated and will be removed in an upcoming release. Please useqml.matrix
instead.Example:
>>> U = qml.RY(0.5, wires=1) >>> U.matrix >>> array([[ 0.96891242+0.j, -0.24740396+0.j], [ 0.24740396+0.j, 0.96891242+0.j]])
- Returns
matrix representation
- Return type
array
-
name
¶ String for the name of the operator.
-
num_params
= 1¶ Number of trainable parameters that the operator depends on.
- Type
int
-
num_wires
= -1¶
-
parameters
¶ Trainable parameters that the operator depends on.
-
return_type
= None¶ Measurement type that this observable is called with.
- Type
None or ObservableReturnTypes
Methods
compare
(other)Compares with another
Hamiltonian
,Tensor
, orObservable
, to determine if they are equivalent.compute_decomposition
(*params[, wires])Representation of the operator as a product of other operators (static method).
compute_diagonalizing_gates
(eigenvectors, 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).
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).
Representation of the operator as a product of other operators.
Return the gate set that diagonalizes a circuit according to the specified Hermitian observable.
expand
()Returns a tape that has recorded the decomposition of the operator.
Generator of an operator that is in single-parameter-form.
Return the eigenvalues of the specified Hermitian observable.
get_matrix
([wire_order])Representation of the operator as a matrix in the computational basis.
label
([decimals, base_label, cache])A customizable string representation of the operator.
queue
([context])Append the operator to the Operator queue.
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.
-
compare
(other)¶ Compares with another
Hamiltonian
,Tensor
, orObservable
, to determine if they are equivalent.Observables/Hamiltonians are equivalent if they represent the same operator (their matrix representations are equal), and they are defined on the same wires.
Warning
The compare method does not check if the matrix representation of a
Hermitian
observable is equal to an equivalent observable expressed in terms of Pauli matrices. To do so would require the matrix form of Hamiltonians and Tensors be calculated, which would drastically increase runtime.- Returns
True if equivalent.
- Return type
(bool)
Examples
>>> ob1 = qml.PauliX(0) @ qml.Identity(1) >>> ob2 = qml.Hamiltonian([1], [qml.PauliX(0)]) >>> ob1.compare(ob2) True >>> ob1 = qml.PauliX(0) >>> ob2 = qml.Hermitian(np.array([[0, 1], [1, 0]]), 0) >>> ob1.compare(ob2) False
-
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
- 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
decomposition of the operator
- Return type
list[Operator]
-
static
compute_diagonalizing_gates
(eigenvectors, wires)[source]¶ 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
- Parameters
eigenvectors (array) – eigenvectors of the operator, as extracted from op.eigendecomposition[“eigvec”]
wires (Iterable[Any], Wires) – wires that the operator acts on
- Returns
list of diagonalizing gates
- Return type
list[Operator]
Example
>>> A = np.array([[-6, 2 + 1j], [2 - 1j, 0]]) >>> _, evecs = np.linalg.eigh(A) >>> qml.Hermitian.compute_diagonalizing_gates(evecs, wires=[0]) [QubitUnitary(tensor([[-0.94915323-0.j, 0.2815786 +0.1407893j ], [ 0.31481445-0.j, 0.84894846+0.42447423j]], requires_grad=True), wires=[0])]
-
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
- Parameters
params (list) – trainable parameters of the operator, as stored in the
parameters
attributehyperparams (dict) – non-trainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
- Returns
eigenvalues
- Return type
tensor_like
-
static
compute_matrix
(A)[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
A (tensor_like) – hermitian matrix
- Returns
canonical matrix
- Return type
tensor_like
Example
>>> A = np.array([[6+0j, 1-2j],[1+2j, -1]]) >>> qml.Hermitian.compute_matrix(A) [[ 6.+0.j 1.-2.j] [ 1.+2.j -1.+0.j]]
-
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
attributehyperparams (dict) – non-trainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
- Returns
matrix representation
- Return type
scipy.sparse.coo.coo_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
- Parameters
params (list) – trainable parameters of the operator, as stored in the
parameters
attributehyperparams (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
- Returns
decomposition of the operator
- Return type
list[Operator]
-
diagonalizing_gates
()[source]¶ Return the gate set that diagonalizes a circuit according to the specified Hermitian observable.
- Returns
list containing the gates diagonalizing the Hermitian observable
- Return type
list
-
expand
()¶ Returns a tape that has recorded the decomposition of the operator.
- Returns
quantum tape
- Return type
-
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
andSparseHamiltonian
respectively).The default value to return is
None
, indicating that the operation has no defined generator.
-
get_eigvals
()[source]¶ Return the eigenvalues of the specified Hermitian observable.
This method uses pre-stored eigenvalues for standard observables where possible and stores the corresponding eigenvectors from the eigendecomposition.
- Returns
array containing the eigenvalues of the Hermitian observable
- Return type
array
-
get_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
-
label
(decimals=None, base_label=None, cache=None)[source]¶ 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)]
-
queue
(context=<class 'pennylane.queuing.QueuingContext'>)¶ Append the operator to the Operator queue.
-
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
- Parameters
wire_order (Iterable) – global wire order, must contain all wire labels from the operator’s wires
- Returns
matrix representation
- Return type
scipy.sparse.coo.coo_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
- 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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