qml.ops.op_math.LinearCombination

class LinearCombination(coeffs, observables, simplify=False, grouping_type=None, method='lf', _grouping_indices=None, _pauli_rep=None, id=None)

Bases: pennylane.ops.op_math.sum.Sum

Operator representing a linear combination of operators.

The LinearCombination is represented as a linear combination of other operators, e.g., \(\sum_{k=0}^{N-1} c_k O_k\), where the \(c_k\) are trainable parameters.

Parameters

• coeffs (tensor_like) – coefficients of the LinearCombination expression

• observables (Iterable[Observable]) – observables in the LinearCombination expression, of same length as coeffs

• simplify (bool) – Specifies whether the LinearCombination is simplified upon initialization (like-terms are combined). The default value is False. Note that coeffs cannot be differentiated when using the 'torch' interface and simplify=True. Use of this argument is deprecated.

• grouping_type (str) – If not None, compute and store information on how to group commuting observables upon initialization. This information may be accessed when a QNode containing this LinearCombination is executed on devices. The string refers to the type of binary relation between Pauli words. Can be 'qwc' (qubit-wise commuting), 'commuting', or 'anticommuting'.

• method (str) – The graph colouring heuristic to use in solving minimum clique cover for grouping, which can be 'lf' (Largest First), 'rlf' (Recursive Largest First), 'dsatur' (Degree of Saturation), or 'gis' (IndependentSet). Defaults to 'lf'. Ignored if grouping_type=None.

• id (str) – name to be assigned to this LinearCombination instance

See also

rustworkx.ColoringStrategy for more information on the ('lf', 'dsatur', 'gis') strategies.

Warning

The simplify argument is deprecated and will be removed in a future release. Instead, you can call qml.simplify on the constructed operator.

Example:

A LinearCombination can be created by simply passing the list of coefficients as well as the list of observables:

>>> coeffs = [0.2, -0.543]
>>> obs = [qml.X(0) @ qml.Z(1), qml.Z(0) @ qml.Hadamard(2)]
>>> H = qml.ops.LinearCombination(coeffs, obs)
>>> print(H)
0.2 * (X(0) @ Z(1)) + -0.543 * (Z(0) @ Hadamard(wires=[2]))

The coefficients can be a trainable tensor, for example:

>>> coeffs = tf.Variable([0.2, -0.543], dtype=tf.double)
>>> obs = [qml.X(0) @ qml.Z(1), qml.Z(0) @ qml.Hadamard(2)]
>>> H = qml.ops.LinearCombination(coeffs, obs)
>>> print(H)
0.2 * (X(0) @ Z(1)) + -0.543 * (Z(0) @ Hadamard(wires=[2]))

A LinearCombination can store information on which commuting observables should be measured together in a circuit:

>>> obs = [qml.X(0), qml.X(1), qml.Z(0)]
>>> coeffs = np.array([1., 2., 3.])
>>> H = qml.ops.LinearCombination(coeffs, obs, grouping_type='qwc')
>>> H.grouping_indices
((0, 1), (2,))

This attribute can be used to compute groups of coefficients and observables:

>>> grouped_coeffs = [coeffs[list(indices)] for indices in H.grouping_indices]
>>> grouped_obs = [[H.ops[i] for i in indices] for indices in H.grouping_indices]
>>> grouped_coeffs
[array([1., 2.]), array([3.])]
>>> grouped_obs
[[X(0), X(1)], [Z(0)]]

Devices that evaluate a LinearCombination expectation by splitting it into its local observables can use this information to reduce the number of circuits evaluated.

Note that one can compute the grouping_indices for an already initialized LinearCombination by using the compute_grouping method.

Attributes

arithmetic_depth	Arithmetic depth of the operator.
basis
batch_size
coeffs	Return the coefficients defining the LinearCombination.
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
num_params	Number of trainable parameters that the operator depends on.
num_wires	Number of wires the operator acts on.
ops	Return the operators defining the LinearCombination.
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	The sorted union of wires from all operators.

arithmetic_depth

basis

batch_size = None

coeffs

Return the coefficients defining the LinearCombination.

Returns

coefficients in the LinearCombination expression

Return type

Iterable[float])

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

ndim_params = None

num_params

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

Number of wires the operator acts on.

ops

Return the operators defining the LinearCombination.

Returns

observables in the LinearCombination expression

Return type

Iterable[Observable])

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

The sorted union of wires from all operators.

Returns

Combined wires present in all terms, sorted.

Return type

(Wires)

Methods

adjoint()	Create an operation that is the adjoint of this one.
compare(other)	Determines mathematical equivalence between operators
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 LinearCombination.
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).
decomposition()	Representation of the operator as a product of other operators.
diagonalizing_gates()	Sequence of gates that diagonalize the operator in the computational basis.
eigvals()	Return the eigenvalues of the specified operator.
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])	How the composite operator is represented in diagrams and drawings.
map_wires(wire_map)	Returns a copy of the current LinearCombination 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])	Queues a qml.ops.LinearCombination instance
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.
terms()	Retrieve the coefficients and operators of the LinearCombination.

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)

Determines mathematical equivalence between operators

LinearCombination and other operators are equivalent if they mathematically represent the same operator (their matrix representations are equal), acting on the same wires.

Warning

This method does not compute explicit matrices but uses the underlyding operators and coefficients for comparisons. When both operators consist purely of Pauli operators, and therefore have a valid op.pauli_rep, the comparison is cheap. When that is not the case (e.g. one of the operators contains a Hadamard gate), it can be more expensive as it involves mathematical simplification of both operators.

Returns

True if equivalent.

Return type

(bool)

Examples

>>> H = qml.ops.LinearCombination(
...     [0.5, 0.5],
...     [qml.PauliZ(0) @ qml.PauliY(1), qml.PauliY(1) @ qml.PauliZ(0) @ qml.Identity("a")]
... )
>>> obs = qml.PauliZ(0) @ qml.PauliY(1)
>>> print(H.compare(obs))
True

>>> H1 = qml.ops.LinearCombination([1, 1], [qml.PauliX(0), qml.PauliZ(1)])
>>> H2 = qml.ops.LinearCombination([1, 1], [qml.PauliZ(0), qml.PauliX(1)])
>>> H1.compare(H2)
False

>>> ob1 = qml.ops.LinearCombination([1], [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(*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.

See also

diagonalizing_gates().

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.

See also

Operator.eigvals() and qml.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

compute_grouping(grouping_type='qwc', method='lf')

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

Note

If grouping is requested, the computed groupings are stored as a list of list of indices in LinearCombination.grouping_indices.

Parameters

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

• 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). Defaults to 'lf'.

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.ops.LinearCombination(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.

See also

Operator.matrix() and qml.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

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.

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()

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.

See also

compute_diagonalizing_gates().

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)

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 LinearCombination 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 LinearCombination

Return type

LinearCombination

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

• 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'>)

Queues a qml.ops.LinearCombination instance

simplify(cutoff=1e-12)

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()

Retrieve the coefficients and operators of the LinearCombination.

Returns

list of coefficients \(c_i\) and list of operations \(O_i\)

Return type

tuple[list[tensor_like or float], list[Operation]]

Example

>>> coeffs = [1., 2., 3.]
>>> ops = [X(0), X(0) @ X(1), X(1) @ X(2)]
>>> op = qml.ops.LinearCombination(coeffs, ops)
>>> op.terms()
([1.0, 2.0, 3.0], [X(0), X(0) @ X(1), X(1) @ X(2)])

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