qml.THermitian¶

class THermitian(A, wires, id=None)[source]¶

Bases: pennylane.ops.qubit.observables.Hermitian

An arbitrary Hermitian observable for qutrits.

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 \(3^N\times 3^N\).

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
id (str or None) – String representing the operation (optional)

Note

Hermitian cannot be used with qutrit devices due to its use of QubitUnitary in compute_diagonalizing_gates().

Attributes

`arithmetic_depth`	Arithmetic depth of the operator.
`batch_size`	Batch size of the operator if it is used with broadcasted parameters.
`eigendecomposition`	Return the eigendecomposition of the matrix specified by the Hermitian observable.
`grad_method`
`has_adjoint`
`has_decomposition`
`has_diagonalizing_gates`
`has_generator`
`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.
`is_hermitian`	All observables must be hermitian
`name`	String for the name of the operator.
`ndim_params`	Number of dimensions per trainable parameter that the operator depends on.
`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.

arithmetic_depth¶: Arithmetic depth of the operator.

batch_size¶

Batch size of the operator if it is used with broadcasted parameters.

The batch_size is determined based on ndim_params and the provided parameters for the operator. If (some of) the latter have an additional dimension, and this dimension has the same size for all parameters, its size is the batch size of the operator. If no parameter has an additional dimension, the batch size is None.

Returns: Size of the parameter broadcasting dimension if present, else None.
Return type: int or None

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]

grad_method = 'F'¶

has_adjoint = False¶

has_decomposition = True¶

has_diagonalizing_gates = True¶

has_generator = False¶

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.

is_hermitian¶: All observables must be hermitian

name¶: String for the name of the operator.

ndim_params = (2,)¶

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: Union[int, pennylane.operation.WiresEnum] = -1¶: 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.

Returns: wires
Return type: Wires

Methods

`adjoint`()	Create an operation that is the adjoint of this one.
`compare`(other)	Compares with another `Hamiltonian`, `Tensor`, or `Observable`, to determine if they are equivalent.
`compute_decomposition`(A, wires)	Decomposes a hermitian matrix as a sum of Pauli operators.
`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).
`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.
`eigvals`()	Return the eigenvalues of the specified Hermitian observable.
`expand`()	Returns a tape that contains the decomposition of the operator.
`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.

Returns: The adjointed operation.

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.X(0) @ qml.Identity(1)
>>> ob2 = qml.Hamiltonian([1], [qml.X(0)])
>>> ob1.compare(ob2)
True
>>> ob1 = qml.X(0)
>>> ob2 = qml.Hermitian(np.array([[0, 1], [1, 0]]), 0)
>>> ob1.compare(ob2)
False

static compute_decomposition(A, wires)¶

Decomposes a hermitian matrix as a sum of Pauli operators.

Parameters

A (array or Sequence) – hermitian matrix
wires (Iterable[Any], Wires) – wires that the operator acts on

Returns

decomposition of the hermitian matrix

Return type

list[Operator]

Examples

>>> op = qml.X(0) + qml.Y(1) + 2 * qml.X(0) @ qml.Z(3)
>>> op_matrix = qml.matrix(op)
>>> qml.Hermitian.compute_decomposition(op_matrix, wires=['a', 'b', 'aux'])
[(
      1.0 * (I('a') @ Y('b') @ I('aux'))
    + 1.0 * (X('a') @ I('b') @ I('aux'))
    + 2.0 * (X('a') @ I('b') @ Z('aux'))
)]
>>> op = np.array([[1, 1], [1, -1]]) / np.sqrt(2)
>>> qml.Hermitian.compute_decomposition(op, wires=0)
[(
      0.7071067811865475 * X(0)
    + 0.7071067811865475 * Z(0)
)]

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 \(U^{\dagger}\).

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, 0], [2 - 1j, 0, 0], [0, 0, 1]])
>>> _, evecs = np.linalg.eigh(A)
>>> qml.THermitian.compute_diagonalizing_gates(evecs, wires=[0])
[QutritUnitary(tensor([[-0.94915323-0.j    0.1407893 +0.2815786j  -0.        -0.j  ]
                       [ 0.31481445-0.j    0.42447423+0.84894846j  0.        -0.j  ]
                       [ 0.        -0.j    0.        -0.j          1.        -0.j  ]], 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 \(U^{\dagger}\), 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.

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, 0],[1+2j, -1, 0], [0, 0, 1]])
>>> qml.THermitian.compute_matrix(A)
[[ 6.+0.j  1.-2.j  0.+0.j]
 [ 1.+2.j -1.+0.j  0.+0.j]
 [ 0.+0.j  0.+0.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

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()¶

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

eigvals()¶

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

expand()¶

Returns a tape that contains the decomposition of the operator.

Warning

This function is deprecated and will be removed in version 0.39. The same behaviour can be achieved simply through ‘qml.tape.QuantumScript(self.decomposition())’.

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 * Y(0) + Z(0) @ X(1)

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.

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 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

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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