# qml.ops.op_math.Prod¶

class Prod(*operands, id=None, _pauli_rep=None)[source]

Bases: pennylane.ops.op_math.composite.CompositeOp

Symbolic operator representing the product of operators.

Parameters

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

Keyword Arguments

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

prod()

Example

>>> prod_op = Prod(qml.X(0), qml.PauliZ(1))
>>> prod_op
X(0) @ Z(1)
>>> qml.matrix(prod_op, wire_order=prod_op.wires)
array([[ 0,  0,  1,  0],
[ 0,  0,  0, -1],
[ 1,  0,  0,  0],
[ 0, -1,  0,  0]])
>>> prod_op.terms()
([1.0], [Z(1) @ X(0)])

Note

When a Prod operator is applied in a circuit, its factors are applied in the reverse order. (i.e Prod(op1, op2) corresponds to $$\hat{op}_{1}\cdot\hat{op}_{2}$$ which indicates first applying $$\hat{op}_{2}$$ then $$\hat{op}_{1}$$ in the circuit). We can see this in the decomposition of the operator.

>>> op = Prod(qml.X(0), qml.Z(1))
>>> op.decomposition()
[Z(1), X(0)]

The Prod operator represents both matrix composition and tensor products between operators.

>>> prod_op = Prod(qml.RZ(1.23, wires=0), qml.X(0), qml.Z(1))
>>> prod_op.matrix()
array([[ 0.        +0.j        ,  0.        +0.j        ,
0.81677345-0.57695852j,  0.        +0.j        ],
[ 0.        +0.j        ,  0.        +0.j        ,
0.        +0.j        , -0.81677345+0.57695852j],
[ 0.81677345+0.57695852j,  0.        +0.j        ,
0.        +0.j        ,  0.        +0.j        ],
[ 0.        +0.j        , -0.81677345-0.57695852j,
0.        +0.j        ,  0.        +0.j        ]])

The Prod operation can be used inside a qnode as an operation which, if parameterized, can be differentiated.

dev = qml.device("default.qubit", wires=3)

@qml.qnode(dev)
def circuit(theta):
qml.prod(qml.Z(0), qml.RX(theta, 1))
return qml.expval(qml.Z(1))
>>> circuit(par)

The Prod operation can also be measured as an observable. If the circuit is parameterized, then we can also differentiate through the product observable.

dev = qml.device("default.qubit", wires=2)

@qml.qnode(dev)
def circuit(weights):
qml.RX(weights[0], wires=0)
return qml.expval(prod_op)
array([-0.07059289])

Note that the terms() method always simplifies and flattens the operands.

>>> op = qml.ops.Prod(qml.X(0), qml.sum(qml.Y(0), qml.Z(1)))
>>> op.terms()
([1j, 1.0], [Z(0), Z(1) @ X(0)])
 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. has_adjoint bool(x) -> bool has_decomposition bool(x) -> bool 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 Check if the product operator 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. obs Access the operands of a Prod instance 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 taking into account operator commutivity. 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.

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]

has_decomposition
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

Check if the product operator is hermitian.

Note, this check is not exhaustive. There can be hermitian operators for which this check yields false, which ARE hermitian. So a false result only implies a more explicit check must be performed.

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

Number of wires the operator acts on.

obs

Access the operands of a Prod instance

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 taking into account operator commutivity.

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

 adjoint() 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_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 of the product operator is given by each factor applied in succession. 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 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. pow(z) 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() Transforms any nested Prod instance into the form $$\sum c_i O_i$$ where $$c_i$$ is a scalar coefficient and $$O_i$$ is a single PL operator or pure product of single PL operators. 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.

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

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.

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()[source]

Decomposition of the product operator is given by each factor applied in succession.

Note that the decomposition is the list of factors returned in reversed order. This is to support the intuition that when we write $$\hat{O} = \hat{A} \cdot \hat{B}$$ it is implied that $$\hat{B}$$ is applied to the state before $$\hat{A}$$ in the quantum circuit.

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.

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 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.

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()[source]

Transforms any nested Prod instance into the form $$\sum c_i O_i$$ where $$c_i$$ is a scalar coefficient and $$O_i$$ is a single PL operator or pure product of single PL operators.

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 = X(0) @ (0.5 * X(1) + X(2))
>>> op.terms()
([0.5, 1.0],
[X(1) @ X(0),
X(2) @ X(0)])