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

`eigendecomposition`	Return the eigendecomposition of the matrix specified by the Hermitian observable.
`eigvals`	Eigenvalues of an instantiated operator.
`grad_method`
`has_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.
`matrix`	Matrix representation of an instantiated operator in the computational basis.
`name`	String for the name of the operator.
`num_params`	Number of trainable parameters that the operator depends on.
`num_wires`
`parameters`	Trainable parameters that the operator depends on.
`return_type`	Measurement type that this observable is called with.
`wires`	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 use qml.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 use qml.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

wires¶

Wires that the operator acts on.

Returns: wires
Return type: Wires

Methods

`compare`(other)	Compares with another `Hamiltonian`, `Tensor`, or `Observable`, 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).
`compute_matrix`(A)	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).
`decomposition`()	Representation of the operator as a product of other operators.
`diagonalizing_gates`()	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`()	Generator of an operator that is in single-parameter-form.
`get_eigvals`()	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, or Observable, 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

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(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

diagonalizing_gates().

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

get_eigvals() and 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(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

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

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

terms()

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

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

compute_decomposition().

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: QuantumTape

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 and SparseHamiltonian 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

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

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

compute_sparse_matrix()

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

compute_terms()

Returns: list of coefficients \(c_i\) and list of operations \(O_i\)
Return type: tuple[list[tensor_like or float], list[Operation]]

qml.Hermitian¶

Attributes

Methods

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