qml.GellMann¶

class GellMann(wires, index=1, id=None)[source]

The Gell-Mann observables for qutrits

The Gell-Mann matrices are a set of 8 linearly independent $$3 \times 3$$ traceless, Hermitian matrices which naturally generalize the Pauli matrices from $$SU(2)$$ to $$SU(3)$$.

\begin{split}\displaystyle \begin{align} \lambda_{1} &= \left(\begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0\\ 0 & 0 & 0\end{array}\right) \;\;\;\;\;\;\;\;\;\; \lambda_{2} = \left(\begin{array}{ccc} 0 & -i & 0 \\ i & 0 & 0\\ 0 & 0 & 0\end{array}\right)\;\;\;\;\;\;\;\;\;\; \lambda_{3} = \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0\\ 0 & 0 & 0\end{array}\right) \\ \lambda_{4} &= \left(\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0\\ 1 & 0 & 0\end{array}\right)\;\;\;\;\;\;\;\;\;\; \lambda_{5} = \left(\begin{array}{ccc} 0 & 0 & -i \\ 0 & 0 & 0\\ i & 0 & 0\end{array}\right) \\ \lambda_{6} &= \left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 1\\ 0 & 1 & 0\end{array}\right)\;\;\;\;\;\;\;\;\;\; \lambda_{7} = \left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -i\\ 0 & i & 0\end{array}\right)\;\;\;\;\;\;\;\;\;\; \lambda_{8} = \frac{1}{\sqrt{3}}\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & -2\end{array}\right)\\ \end{align}\end{split}

Details:

• Number of wires: 1

• Number of parameters: 0

Parameters
• wires (Sequence[int] or int) – the wire(s) the observable acts on

• index (int) – The index of the Gell-Mann matrix to be used. Must be between 1 and 8 inclusive

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

Example:

>>> dev = qml.device("default.qutrit", wires=2)
>>> @qml.qnode(dev)
... def test_qnode():
...     qml.TShift(wires=0)
...     qml.TClock(wires=0)
...     qml.TShift(wires=1)
...     return qml.expval(qml.GellMann(wires=0, index=1))
>>> print(test_qnode())
0.0
>>> print(qml.draw(test_qnode)())
0: ──TShift──TClock─╭●────┤  <GellMann(1)>

 arithmetic_depth Arithmetic depth of the operator. batch_size Batch size of the operator if it is used with broadcasted parameters. 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 of the operator. num_params Number of trainable parameters 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

has_adjoint = False
has_decomposition = False
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

Number of dimensions per trainable parameter of the operator.

By default, this property returns the numbers of dimensions of the parameters used for the operator creation. If the parameter sizes for an operator subclass are fixed, this property can be overwritten to return the fixed value.

Returns

Number of dimensions for each trainable parameter.

Return type

tuple

num_params = 0

Number of trainable parameters the operator depends on

Type

int

num_wires = 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

 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(*params[, wires]) Representation of the operator as a product of other operators (static method). compute_diagonalizing_gates(wires, index) Sequence of gates that diagonalize the operator in the computational basis (static method). compute_eigvals(index) Eigenvalues of the operator in the computational basis (static method). compute_matrix(index) 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. Eigenvalues of the operator in the computational basis. Returns a tape that contains the decomposition of the operator. 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. A list of new operators equal to this one raised to the given power. queue([context]) Append the operator to the Operator queue. 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. 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

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(*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.

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(wires, index)[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.

Parameters
• index (int) – The index of the Gell-Mann matrix to be used. Must be between 1 and 8 inclusive

• wires (Iterable[Any], Wires) – wires that the operator acts on

Returns

list of diagonalizing gates

Return type

list[Operator]

Example

>>> qml.GellMann.compute_diagonalizing_gates(wires=0, index=4)
[QutritUnitary(array([[ 0.70710678-0.j,  0.        -0.j,  0.70710678-0.j],
[ 0.70710678-0.j,  0.        -0.j, -0.70710678-0.j],
[ 0.        -0.j,  1.        -0.j,  0.        -0.j]]), wires=[0])]

static compute_eigvals(index)[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.

Parameters
• index (int) – The index of the Gell-Mann matrix to be used. Must be between 1

• 8 inclusive (and) –

Returns

eigenvalues

Return type

array

Example

>>> qml.GellMann.compute_eigvals(1)
[1. -1.  0.]

static compute_matrix(index)[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.

Parameters
• index (int) – The index of the Gell-Mann matrix to be used. Must be between 1

• 8 inclusive (and) –

Returns

canonical matrix

Return type

tensor_like

Example

>>> qml.GellMann.compute_matrix(8)
array([[ 0.57735027+0.j,  0.        +0.j,  0.        +0.j],
[ 0.        +0.j,  0.57735027+0.j,  0.        +0.j],
[ 0.        +0.j,  0.        +0.j, -1.15470054+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.

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.

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.

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.

Returns

eigenvalues

Return type

tensor_like

expand()

Returns a tape that contains 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 * 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)[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 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.],
>>> op3 = qml.QubitUnitary(np.eye(4), wires=(0,1))
>>> op3.label(cache=cache)
'U(M1)'
>>> cache['matrices']
[tensor([[1., 0.],
tensor([[1., 0., 0., 0.],
[0., 1., 0., 0.],
[0., 0., 1., 0.],

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.

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.

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]]