# qml.ops.op_math.Exp¶

class Exp(base, coeff=1, num_steps=None, do_queue=True, id=None)[source]

Bases: pennylane.ops.op_math.symbolicop.ScalarSymbolicOp, pennylane.operation.Operation

A symbolic operator representing the exponential of a operator.

Parameters
• base (Operator) – The Operator to be exponentiated

• coeff=1 (Number) – A scalar coefficient of the operator.

• num_steps (int) – The number of steps used in the decomposition of the exponential operator, also known as the Trotter number. If this value is None and the Suzuki-Trotter decomposition is needed, an error will be raised.

• do_queue (bool) – determines if the sum operator will be queued. Default is True.

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

Example

This symbolic operator can be used to make general rotation operators:

>>> x = np.array(1.23)
>>> op = Exp( qml.PauliX(0), -0.5j * x)
>>> qml.math.allclose(op.matrix(), qml.RX(x, wires=0).matrix())
True


This can even be used for more complicated generators:

>>> t = qml.PauliX(0) @ qml.PauliX(1) + qml.PauliY(0) @ qml.PauliY(1)
>>> isingxy = Exp(t, 0.25j * x)
>>> qml.math.allclose(isingxy.matrix(), qml.IsingXY(x, wires=(0,1)).matrix())
True


If the coefficient is purely imaginary and the base operator is Hermitian, then the gate can be used in a circuit, though it may not be supported by the device and may not be differentiable.

>>> @qml.qnode(qml.device('default.qubit', wires=1))
... def circuit(x):
...     Exp(qml.PauliX(0), -0.5j * x)
...     return qml.expval(qml.PauliZ(0))
>>> print(qml.draw(circuit)(1.23))
0: ──Exp─┤  <Z>


If the base operator is Hermitian and the coefficient is real, then the Exp operator can be measured as an observable:

>>> obs = Exp(qml.PauliZ(0), 3)
>>> @qml.qnode(qml.device('default.qubit', wires=1))
... def circuit():
...     return qml.expval(obs)
>>> circuit()

 arithmetic_depth Arithmetic depth of the operator. base The base operator. base_name Holdover from when in-place inversion changed then name. basis The basis of an operation, or for controlled gates, of the target operation. batch_size Batch size of the operator if it is used with broadcasted parameters. coeff The numerical coefficient of the operator in the exponent. control_wires data The trainable parameters grad_method Gradient computation method. grad_recipe Gradient recipe for the parameter-shift method. has_adjoint has_decomposition bool(x) -> bool has_diagonalizing_gates bool(x) -> bool has_matrix bool(x) -> bool 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 This property determines if an 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. parameter_frequencies Returns the frequencies for each operator parameter with respect to an expectation value of the form $$\langle \psi | U(\mathbf{p})^\dagger \hat{O} U(\mathbf{p})|\psi\rangle$$. parameters Trainable parameters that the operator depends on. wires Wires that the operator acts on.
arithmetic_depth
base

The base operator.

base_name

Holdover from when in-place inversion changed then name. To be removed.

basis = None

The basis of an operation, or for controlled gates, of the target operation. If not None, should take a value of "X", "Y", or "Z".

For example, X and CNOT have basis = "X", whereas ControlledPhaseShift and RZ have basis = "Z".

Type

str or None

batch_size
coeff

The numerical coefficient of the operator in the exponent.

control_wires = <Wires = []>
data
grad_method

• 'A': analytic differentiation using the parameter-shift method.

• 'F': finite difference numerical differentiation.

• None: the operation may not be differentiated.

Default is 'F', or None if the Operation has zero parameters.

grad_recipe = None

Gradient recipe for the parameter-shift method.

This is a tuple with one nested list per operation parameter. For parameter $$\phi_k$$, the nested list contains elements of the form $$[c_i, a_i, s_i]$$ where $$i$$ is the index of the term, resulting in a gradient recipe of

$\frac{\partial}{\partial\phi_k}f = \sum_{i} c_i f(a_i \phi_k + s_i).$

If None, the default gradient recipe containing the two terms $$[c_0, a_0, s_0]=[1/2, 1, \pi/2]$$ and $$[c_1, a_1, s_1]=[-1/2, 1, -\pi/2]$$ is assumed for every parameter.

Type

tuple(Union(list[list[float]], None)) or None

has_adjoint = False
has_decomposition
has_diagonalizing_gates
has_matrix
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
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
parameter_frequencies

Returns the frequencies for each operator parameter with respect to an expectation value of the form $$\langle \psi | U(\mathbf{p})^\dagger \hat{O} U(\mathbf{p})|\psi\rangle$$.

These frequencies encode the behaviour of the operator $$U(\mathbf{p})$$ on the value of the expectation value as the parameters are modified. For more details, please see the pennylane.fourier module.

Returns

Tuple of frequencies for each parameter. Note that only non-negative frequency values are returned.

Return type

list[tuple[int or float]]

Example

>>> op = qml.CRot(0.4, 0.1, 0.3, wires=[0, 1])
>>> op.parameter_frequencies
[(0.5, 1), (0.5, 1), (0.5, 1)]


For operators that define a generator, the parameter frequencies are directly related to the eigenvalues of the generator:

>>> op = qml.ControlledPhaseShift(0.1, wires=[0, 1])
>>> op.parameter_frequencies
[(1,)]
>>> gen = qml.generator(op, format="observable")
>>> gen_eigvals = qml.eigvals(gen)
(1.0,)


For more details on this relationship, see eigvals_to_frequencies().

parameters

Trainable parameters that the operator depends on.

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_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. Eigenvalues of the operator in the computational basis. Returns a tape that has recorded 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. The parameters required to implement a single-qubit gate as an equivalent Rot gate, up to a global phase. sparse_matrix([wire_order, format]) Representation of the operator as a sparse matrix in the computational basis. Representation of the operator as a linear combination of other operators. validate_subspace(subspace) Validate the subspace for qutrit operations.
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

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.

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]

Representation of the operator as a product of other operators. Decomposes into PauliRot if the coefficient is imaginary and the base is a Pauli Word.

$O = O_1 O_2 \dots O_n$

A DecompositionUndefinedError is raised if the coefficient is not imaginary or the base is not a Pauli Word.

Returns

decomposition of the operator

Return type

list[PauliRot]

diagonalizing_gates()[source]

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()[source]

Eigenvalues of the operator in the computational basis.

$c \mathbf{M} \mathbf{v} = c \lambda \mathbf{v} \quad \Longrightarrow \quad e^{c \mathbf{M}} \mathbf{v} = e^{c \lambda} \mathbf{v}$
>>> obs = Exp(qml.PauliX(0), 3)
>>> qml.eigvals(obs)
array([20.08553692,  0.04978707])
>>> np.exp(3 * qml.eigvals(qml.PauliX(0)))

expand()

Returns a tape that has recorded the decomposition of the operator.

Returns

quantum tape

Return type

QuantumTape

generator()[source]

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) [Y0]
+ (1.0) [Z0 X1]

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(decimals=2)
"RX\n(1.23)"
>>> op.label(base_label="my_label")
"my_label"
>>> op.label(decimals=2, base_label="my_label")
"my_label\n(1.23)"


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)[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 base 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)[source]

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()[source]

Reduce the depth of nested operators to the minimum.

Returns

simplified operator

Return type

Operator

single_qubit_rot_angles()

The parameters required to implement a single-qubit gate as an equivalent Rot gate, up to a global phase.

Returns

A list of values $$[\phi, \theta, \omega]$$ such that $$RZ(\omega) RY(\theta) RZ(\phi)$$ is equivalent to the original operation.

Return type

tuple[float, float, float]

sparse_matrix(wire_order=None, format='csr')[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()

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]]

static validate_subspace(subspace)

Validate the subspace for qutrit operations.

This method determines whether a given subspace for qutrit operations is defined correctly or not. If not, a ValueError is thrown.

Parameters

subspace (tuple[int]) – Subspace to check for correctness