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.
See also
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)]
Usage Details
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 , 0.816773450.57695852j, 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 , 0.816773450.57695852j, 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))
>>> par = np.array(1.23, requires_grad=True) >>> circuit(par) tensor(0.33423773, requires_grad=True) >>> qml.grad(circuit)(par) tensor(0.9424888, requires_grad=True)
The Prod operation can also be measured as an observable. If the circuit is parameterized, then we can also differentiate through the product observable.
prod_op = Prod(qml.Z(0), qml.Hadamard(wires=1)) dev = qml.device("default.qubit", wires=2) @qml.qnode(dev) def circuit(weights): qml.RX(weights[0], wires=0) return qml.expval(prod_op)
>>> weights = np.array([0.1], requires_grad=True) >>> qml.grad(circuit)(weights) 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)])
Attributes
Arithmetic depth of the operator.
Batch size of the operator if it is used with broadcasted parameters.
Scalar coefficients of the operator when flattened out.
Create data property
Return the eigendecomposition of the matrix specified by the operator.
bool(x) > bool
bool(x) > bool
bool(x) > bool
bool(x) > bool
Boolean expression that indicates if the factors have overlapping wires.
Integer hash that uniquely represents the operator.
Dictionary of nontrainable variables that this operation depends on.
Custom string to label a specific operator instance.
Check if the product operator is hermitian.
String for the name of the operator.
Number of dimensions per trainable parameter of the operator.
Number of trainable parameters that the operator depends on.
Number of wires the operator acts on.
Access the operands of a
Prod
instanceOperator terms without scalar coefficients of the operator when flattened out.
Groups all operands of the composite operator that act on overlapping wires taking into account operator commutivity.
Trainable parameters that the operator depends on.
A
PauliSentence
representation of the Operator, orNone
if it doesn’t have one.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 onndim_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 isNone
. 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 prestored 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_adjoint
¶

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

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, orNone
if it doesn’t have one.
Methods
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 singleparameterform.
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.

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
The adjointed operation.

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
 Parameters
*params (list) – trainable parameters of the operator, as stored in the
parameters
attributewires (Iterable[Any], Wires) – wires that the operator acts on
**hyperparams (dict) – nontrainable 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
 Parameters
params (list) – trainable parameters of the operator, as stored in the
parameters
attributewires (Iterable[Any], Wires) – wires that the operator acts on
hyperparams (dict) – nontrainable 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
 Parameters
*params (list) – trainable parameters of the operator, as stored in the
parameters
attribute**hyperparams (dict) – nontrainable 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.
See also
 Parameters
*params (list) – trainable parameters of the operator, as stored in the
parameters
attribute**hyperparams (dict) – nontrainable 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
 Parameters
*params (list) – trainable parameters of the operator, as stored in the
parameters
attribute**hyperparams (dict) – nontrainable 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.See also
 Returns
a list of operators
 Return type
list[Operator] or None

eigvals
()¶ Return the eigenvalues of the specified operator.
This method uses prestored 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

generator
()¶ Generator of an operator that is in singleparameterform.
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
andSparseHamiltonian
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 toNone
.base_label (Iterable[str]) – Overwrite the nonparameter component of the label. Must be same length as
operands
attribute. Defaults toNone
.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

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.See also
 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)])