# qml.CommutingEvolution¶

class CommutingEvolution(hamiltonian, time, frequencies=None, shifts=None, do_queue=True, id=None)[source]

Applies the time-evolution operator for a Hamiltonian expressed as a linear combination of mutually commuting Pauli words.

A commuting Hamiltonian is of the form

$H \ = \ \displaystyle\sum_{j} c_j P_j,$

where $$P_j$$ are mutually commutative Pauli words and $$c_j$$ are real coefficients. The time-evolution under a commuting Hamiltonian is given by a unitary of the form

$U(t) \ = \ e^{-i H t} \ = \exp(-i t \displaystyle\sum_j c_j P_j) = \displaystyle\prod_j \exp(-i t c_j P_j).$

If the Hamiltonian has a small number of unique eigenvalues, partial derivatives of observable expectation values, i.e.

$\langle 0 | W(t)^\dagger O W(t) | 0 \rangle,$

where $$W(t) = V U(t) Y$$ for some $$V$$ and $$Y$$, taken with respect to $$t$$ may be efficiently computed through generalized parameter shift rules. When initialized, this template will automatically compute the parameter-shift rule if given the Hamiltonian’s eigenvalue frequencies, i.e., the unique positive differences between eigenvalues.

Warning

This template uses the ApproxTimeEvolution operation with n=1 in order to implement the time evolution, as a single-step Trotterization is exact for a commuting Hamiltonian.

• If the input Hamiltonian contains Pauli words which do not commute, the compilation of the time evolution operator to a sequence of gates will not equate to the exact propagation under the given Hamiltonian.

• Furthermore, if the specified frequencies do not correspond to the true eigenvalue frequency spectrum of the commuting Hamiltonian, computed gradients will be incorrect in general.

Parameters
Keyword Arguments
• frequencies (tuple[int or float]) – The unique positive differences between eigenvalues in the spectrum of the Hamiltonian. If the frequencies are not given, the cost function partial derivative will be computed using the standard two-term shift rule applied to the constituent Pauli words in the Hamiltonian individually.

• shifts (tuple[int or float]) – The parameter shifts to use in obtaining the generalized parameter shift rules. If unspecified, equidistant shifts are used.

The template is used inside a qnode:

import pennylane as qml

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

coeffs = [1, -1]
obs = [qml.PauliX(0) @ qml.PauliY(1), qml.PauliY(0) @ qml.PauliX(1)]
hamiltonian = qml.Hamiltonian(coeffs, obs)
frequencies = (2, 4)

@qml.qnode(dev)
def circuit(time):
qml.PauliX(0)
qml.CommutingEvolution(hamiltonian, time, frequencies)
return qml.expval(qml.PauliZ(0))

>>> circuit(1)
0.6536436208636115

 base_name If inverse is requested, this is the name of the original operator to be inverted. basis The target operation for controlled gates. batch_size Batch size of the operator if it is used with broadcasted parameters. control_wires Control wires of the operator. grad_method grad_recipe Gradient recipe for the parameter-shift method. has_matrix 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. inverse Boolean determining if the inverse of the operation was requested. is_hermitian This property determines if an operator is hermitian. name 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 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.
base_name

If inverse is requested, this is the name of the original operator to be inverted.

basis = None

The target operation for controlled gates. 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

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

control_wires

Control wires of the operator.

For operations that are not controlled, this is an empty Wires object of length 0.

Returns

The control wires of the operation.

Return type

Wires

grad_method = None
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_matrix = False
hash

Integer hash that uniquely represents the operator.

Type

int

hyperparameters

Dictionary of non-trainable variables that this operation depends on.

Type

dict

id

Custom string to label a specific operator instance.

inverse

Boolean determining if the inverse of the operation was requested.

is_hermitian

This property determines if an operator is hermitian.

name

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

Number of trainable parameters that the operator depends on.

By default, this property returns as many parameters as were used for the operator creation. If the number of parameters for an operator subclass is fixed, this property can be overwritten to return the fixed value.

Returns

number of parameters

Return type

int

num_wires = -1
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

Wires that the operator acts on.

Returns

wires

Return type

Wires

 Create an operation that is the adjoint of this one. compute_decomposition(time, *coeffs, wires, …) Representation of the operator as a product of other operators. 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). compute_terms(*params, **hyperparams) Representation of the operator as a linear combination of other operators (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 (static method). Returns a tape that has recorded the decomposition of the operator. Generator of an operator that is in single-parameter-form. Multiplier and shift for the given parameter, based on its gradient recipe. Inverts the operator. label([decimals, base_label, cache]) A customizable string representation of the operator. 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. The parameters required to implement a single-qubit gate as an equivalent Rot gate, up to a global phase. 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.

Parameters

do_queue – Whether to add the adjointed gate to the context queue.

Returns

static compute_decomposition(time, *coeffs, wires, hamiltonian, **kwargs)[source]

Representation of the operator as a product of other operators.

$O = O_1 O_2 \dots O_n.$
Parameters
• time_and_coeffs (list[tensor_like or float]) – list of coefficients of the Hamiltonian, prepended by the time variable

• wires (Any or Iterable[Any]) – wires that the operator acts on

• hamiltonian (Hamiltonian) – The commuting Hamiltonian defining the time-evolution operator.

• frequencies (tuple[int or float]) – The unique positive differences between eigenvalues in the spectrum of the Hamiltonian.

• shifts (tuple[int or float]) – The parameter shifts to use in obtaining the generalized parameter shift rules. If unspecified, equidistant shifts are used.

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 $$U$$.

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 $$U$$, 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

static compute_terms(*params, **hyperparams)

Representation of the operator as a linear combination of other operators (static method).

$O = \sum_i c_i O_i$
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

list of coefficients and list of operations

Return type

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

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 $$U$$.

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()

Eigenvalues of the operator in the computational basis (static method).

If diagonalizing_gates are specified and implement a unitary $$U$$, 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.

Note

When eigenvalues are not explicitly defined, they are computed automatically from the matrix representation. Currently, this computation is not differentiable.

A EigvalsUndefinedError is raised if the eigenvalues have not been defined and cannot be inferred from the matrix representation.

Returns

eigenvalues

Return type

tensor_like

expand()

Returns a tape that has recorded 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) [Y0]
+ (1.0) [Z0 X1]


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.

get_parameter_shift(idx)

Multiplier and shift for the given parameter, based on its gradient recipe.

Parameters

idx (int) – parameter index within the operation

Returns

list of multiplier, coefficient, shift for each term in the gradient recipe

Return type

list[[float, float, float]]

Note that the default value for shift is None, which is replaced by the default shift $$\pi/2$$.

inv()

Inverts the operator.

This method concatenates a string to the name of the operation, to indicate that the inverse will be used for computations.

Any subsequent call of this method will toggle between the original operation and the inverse of the operation.

Returns

operation to be inverted

Return type

Operator

label(decimals=None, base_label=None, cache=None)

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 caries 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)"
>>> op.inv()
>>> op.label()
"RX⁻¹"


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.],

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.

Parameters

wire_order (Iterable) – global wire order, must contain all wire labels from the operator’s wires

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.QueuingContext'>)

Append the operator to the Operator queue.

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)

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.

Note

The wire_order argument is currently not implemented, and using it will raise an error.

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]]