# qml.ops.op_math.Sum¶

class Sum(*operands, grouping_type=None, method='rlf', id=None, _grouping_indices=None, _pauli_rep=None)[source]

Bases: pennylane.ops.op_math.composite.CompositeOp

Symbolic operator representing the sum of operators.

Parameters

*summands (tuple[Operator]) – a tuple of operators which will be summed together.

Keyword Arguments
• grouping_type (str) – The type of binary relation between Pauli words used to compute the grouping. Can be 'qwc', 'commuting', or 'anticommuting'.

• method (str) – The graph colouring heuristic to use in solving minimum clique cover for grouping, which can be 'lf' (Largest First) or 'rlf' (Recursive Largest First). This keyword argument is ignored if grouping_type is None.

• id (str or None) – id for the sum operator. Default is None.

Note

Currently this operator can not be queued in a circuit as an operation, only measured terminally.

Note

This operator supports batched operands:

>>> op = qml.sum(qml.RX(np.array([1, 2, 3]), wires=0), qml.X(1))
>>> op.matrix().shape
(3, 4, 4)


But it doesn’t support batching of operators: >>> op = qml.sum(np.array([qml.RX(0.4, 0), qml.RZ(0.3, 0)]), qml.Z(0)) AttributeError: ‘numpy.ndarray’ object has no attribute ‘wires’

Note

If grouping is requested, the computed groupings are stored as a list of list of indices in Sum.grouping_indices. The indices refer to the operators and coefficients returned by Sum.terms(), not Sum.operands, as these are not guaranteed to be equivalent.

sum()

Example

>>> summed_op = Sum(qml.X(0), qml.Z(0))
>>> summed_op
X(0) + Z(0)
>>> qml.matrix(summed_op)
array([[ 1,  1],
[ 1, -1]])
>>> summed_op.terms()
([1.0, 1.0], (X(0), Z(0)))


We can combine parameterized operators, and support sums between operators acting on different wires.

>>> summed_op = Sum(qml.RZ(1.23, wires=0), qml.I(wires=1))
>>> summed_op.matrix()
array([[1.81677345-0.57695852j, 0.        +0.j        ,
0.        +0.j        , 0.        +0.j        ],
[0.        +0.j        , 1.81677345-0.57695852j,
0.        +0.j        , 0.        +0.j        ],
[0.        +0.j        , 0.        +0.j        ,
1.81677345+0.57695852j, 0.        +0.j        ],
[0.        +0.j        , 0.        +0.j        ,
0.        +0.j        , 1.81677345+0.57695852j]])


The Sum operation can also be measured inside a qnode as an observable. If the circuit is parameterized, then we can also differentiate through the sum observable.

sum_op = Sum(qml.X(0), qml.Z(1))
dev = qml.device("default.qubit", wires=2)

@qml.qnode(dev, diff_method="best")
def circuit(weights):
qml.RX(weights[0], wires=0)
qml.RY(weights[1], wires=1)
qml.CNOT(wires=[0, 1])
qml.RX(weights[2], wires=1)
return qml.expval(sum_op)

>>> weights = qnp.array([0.1, 0.2, 0.3], requires_grad=True)
array([-0.09347337, -0.18884787, -0.28818254])

 arithmetic_depth Arithmetic depth of the operator. basis batch_size Batch size of the operator if it is used with broadcasted parameters. coeffs Scalar coefficients of the operator when flattened out. data Create data property eigendecomposition Return the eigendecomposition of the matrix specified by the operator. grad_method grouping_indices Return the grouping indices attribute. has_adjoint bool(x) -> bool has_decomposition has_diagonalizing_gates bool(x) -> bool has_generator has_matrix bool(x) -> bool has_overlapping_wires Boolean expression that indicates if the factors have overlapping wires. 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 If all of the terms in the sum are hermitian, then the Sum is 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 that the operator depends on. num_wires Number of wires the operator acts on. ops Operator terms without scalar coefficients of the operator when flattened out. overlapping_ops Groups all operands of the composite operator that act on overlapping wires. 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
basis
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

coeffs

Scalar coefficients of the operator when flattened out.

This is a deprecated attribute, please use terms() instead.

data

Create data property

eigendecomposition

Return the eigendecomposition of the matrix specified by the operator.

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 operator.

Return type

dict[str, array]

grad_method = 'A'
grouping_indices

Return the grouping indices attribute.

Returns

indices needed to form groups of commuting observables

Return type

list[list[int]]

has_adjoint
has_decomposition = False
has_diagonalizing_gates
has_generator = False
has_matrix
has_overlapping_wires

Boolean expression that indicates if the factors have overlapping wires.

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

If all of the terms in the sum are hermitian, then the Sum is 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
num_wires: Union[int, pennylane.operation.WiresEnum]

Number of wires the operator acts on.

ops

Operator terms without scalar coefficients of the operator when flattened out.

This is a deprecated attribute, please use terms() instead.

overlapping_ops

Groups all operands of the composite operator that act on overlapping wires.

Returns

List of lists of operators that act on overlapping wires. All the inner lists commute with each other.

Return type

List[List[Operator]]

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. compute_decomposition(*params[, 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(*params, **hyperparams) Eigenvalues of the operator in the computational basis (static method). compute_grouping([grouping_type, method]) Compute groups of operators and coefficients corresponding to commuting observables of this Sum. compute_matrix(*params, **hyperparams) 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. Return the eigenvalues of the specified operator. 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]) How the composite operator is represented in diagrams and drawings. 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]) Updates each operator’s owner to self, this ensures that the operators are not applied to the circuit repeatedly. simplify([cutoff]) 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()[source]

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

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(*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.

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

list of diagonalizing gates

Return type

list[Operator]

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

compute_grouping(grouping_type='qwc', method='lf')[source]

Compute groups of operators and coefficients corresponding to commuting observables of this Sum.

Note

If grouping is requested, the computed groupings are stored as a list of list of indices in Sum.grouping_indices. The indices refer to operators and coefficients returned by Sum.terms(), not Sum.operands, as these are not guaranteed to be equivalent.

Parameters
• grouping_type (str) – The type of binary relation between Pauli words used to compute the grouping. Can be 'qwc', 'commuting', or 'anticommuting'.

• method (str) – The graph colouring heuristic to use in solving minimum clique cover for grouping, which can be 'lf' (Largest First) or 'rlf' (Recursive Largest First).

Example

import pennylane as qml

a = qml.X(0)
b = qml.prod(qml.X(0), qml.X(1))
c = qml.Z(0)
obs = [a, b, c]
coeffs = [1.0, 2.0, 3.0]

op = qml.dot(coeffs, obs)

>>> op.grouping_indices is None
True
>>> op.compute_grouping(grouping_type="qwc")
>>> op.grouping_indices
((2,), (0, 1))

static compute_matrix(*params, **hyperparams)

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
• *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

tensor_like

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()

Return the eigenvalues of the specified operator.

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 operator

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)[source]

How the composite operator is represented in diagrams and drawings.

Parameters
• decimals (int) – If None, no parameters are included. Else, how to round the parameters. Defaults to None.

• base_label (Iterable[str]) – Overwrite the non-parameter component of the label. Must be same length as operands attribute. Defaults to None.

• cache (dict) – Dictionary that carries information between label calls in the same drawing. Defaults to None.

Returns

label to use in drawings

Return type

str

Example (using the Sum composite operator)

>>> op = qml.S(0) + qml.X(0) + qml.Rot(1,2,3, wires=[1])
>>> op.label()
'(S+X)+Rot'
>>> op.label(decimals=2, base_label=[["my_s", "my_x"], "inc_rot"])
'(my_s+my_x)+inc_rot\n(1.00,\n2.00,\n3.00)'

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)[source]

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

• wires (operator's) –

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'>)

Updates each operator’s owner to self, this ensures that the operators are not applied to the circuit repeatedly.

simplify(cutoff=1e-12)[source]

Reduce the depth of nested operators to the minimum.

Returns

simplified operator

Return type

Operator

sparse_matrix(wire_order=None)[source]

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()[source]

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]]

Example

>>> op = 0.5 * X(0) + 0.7 * X(1) + 1.5 * Y(0) @ Y(1)
>>> op.terms()
([0.5, 0.7, 1.5],
[X(0), X(1), Y(1) @ Y(0)])


Note that this method disentangles nested structures of Sum instances like so.

>>> op = 0.5 * X(0) + (2. * (X(1) + 3. * X(2)))
>>> print(op)
(0.5*(PauliX(wires=[0]))) + (2.0*((0.5*(PauliX(wires=[1]))) + (3.0*(PauliX(wires=[2])))))
>>> print(op.terms())
([0.5, 1.0, 6.0], [PauliX(wires=[0]), PauliX(wires=[1]), PauliX(wires=[2])])