# qml.pulse.ParametrizedEvolution¶

class ParametrizedEvolution(H, params=None, t=None, do_queue=True, id=None, **odeint_kwargs)[source]

Parametrized evolution gate, created by passing a ParametrizedHamiltonian to the evolve() function

For a time-dependent Hamiltonian of the form

$H(\{v_j\}, t) = H_\text{drift} + \sum_j f_j(v_j, t) H_j$

it implements the corresponding time-evolution operator $$U(t_1, t_2)$$, which is the solution to the time-dependent Schrodinger equation.

$\frac{d}{dt}U(t) = -i H(\{v_j\}, t) U(t).$

Under the hood, it is using a numerical ordinary differential equation (ODE) solver. It requires jax, and will not work with other machine learning frameworks typically encountered in PennyLane.

Parameters
• H (ParametrizedHamiltonian) – Hamiltonian to evolve

• params (Optional[list]) – trainable parameters, passed as list where each element corresponds to the parameters of a scalar-valued function of the Hamiltonian being evolved.

• t (Union[float, List[float]]) – If a float, it corresponds to the duration of the evolution. If a list of floats, the ODE solver will use all the provided time values, and perform intermediate steps if necessary. It is recommended to just provide a start and end time. Note that such absolute times only have meaning within an instance of ParametrizedEvolution and will not affect other gates.

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

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

Keyword Arguments
• atol (float, optional) – Absolute error tolerance for the ODE solver. Defaults to 1.4e-8.

• rtol (float, optional) – Relative error tolerance for the ODE solver. The error is estimated from comparing a 4th and 5th order Runge-Kutta step in the Dopri5 algorithm. This error is guaranteed to stay below tol = atol + rtol * abs(y) through adaptive step size selection. Defaults to 1.4e-8.

• mxstep (int, optional) – maximum number of steps to take for each timepoint for the ODE solver. Defaults to jnp.inf.

• hmax (float, optional) – maximum step size allowed for the ODE solver. Defaults to jnp.inf.

Warning

The ParametrizedHamiltonian must be Hermitian at all times. This is not explicitly checked when creating a ParametrizedEvolution from the ParametrizedHamiltonian.

Example

To create a ParametrizedEvolution, we first define a ParametrizedHamiltonian describing the system, and then pass it to evolve():

from jax import numpy as jnp

f1 = lambda p, t: jnp.sin(p * t)
H = f1 * qml.PauliY(0)

ev = qml.evolve(H)


The initial ParametrizedEvolution does not have set parameters, and so will not have a matrix defined. To obtain an Operator with a matrix, it must be passed parameters and a time interval:

>>> qml.matrix(ev([1.2], t=[0, 4]))
Array([[ 0.72454906+0.j, -0.6892243 +0.j],
[ 0.6892243 +0.j,  0.72454906+0.j]], dtype=complex64)


The parameters can be updated by calling the ParametrizedEvolution again with different inputs.

When calling the ParametrizedEvolution, keyword arguments can be passed to specify behaviour of the ODE solver.

The ParametrizedEvolution can be implemented in a QNode:

import jax

dev = qml.device("default.qubit.jax", wires=1)
@jax.jit
@qml.qnode(dev, interface="jax")
def circuit(params):
qml.evolve(H)(params, t=[0, 10])
return qml.expval(qml.PauliZ(0))

>>> params = [1.2]
>>> circuit(params)
Array(0.96632576, dtype=float32)

>>> jax.grad(circuit)(params)
[Array(2.3569832, dtype=float32)]


Note

In the example above, the decorator @jax.jit is used to compile this execution just-in-time. This means the first execution will typically take a little longer with the benefit that all following executions will be significantly faster, see the jax docs on jitting. JIT-compiling is optional, and one can remove the decorator when only single executions are of interest.

Warning

The time argument t corresponds to the time window used to compute the scalar-valued functions present in the ParametrizedHamiltonian class. Consequently, executing two ParametrizedEvolution operators using the same time window does not mean both operators are executed simultaneously, but rather that both evaluate their respective scalar-valued functions using the same time window. See Usage Details.

The parameters used when calling the ParametrizedEvolution are expected to have the same order as the functions used to define the ParametrizedHamiltonian. For example:

def f1(p, t):
return jnp.sin(p[0] * t**2) + p[1]

def f2(p, t):
return p * jnp.cos(t)

H = 2 * qml.PauliX(0) + f1 * qml.PauliY(0) + f2 * qml.PauliZ(0)
ev = qml.evolve(H)

>>> params = [[4.6, 2.3], 1.2]
>>> qml.matrix(ev(params, t=0.5))
Array([[-0.18354285-0.26303384j, -0.7271658 -0.606923j  ],
[ 0.7271658 -0.606923j  , -0.18354285+0.26303384j]],      dtype=complex64)


Internally the solver is using f1([4.6, 2.3], t) and f2(1.2, t) at each timestep when finding the matrix.

In the case where we have defined two Hamiltonians, H1 and H2, and we want to find a time evolution where the two are driven simultaneously for some period of time, it is important that both are included in the same call of evolve(). For non-commuting operations, applying qml.evolve(H1)(params, t=[0, 10]) followed by qml.evolve(H2)(params, t=[0, 10]) will not apply the two pulses simultaneously, despite the overlapping time window. Instead, it will execute H1 in the [0, 10] time window, and then subsequently execute H2 using the same time window to calculate the evolution, but without taking into account how the time evolution of H1 affects the evolution of H2 and vice versa.

Consider two non-commuting ParametrizedHamiltonian objects:

from jax import numpy as jnp

ops = [qml.PauliX(0), qml.PauliY(1), qml.PauliZ(2)]
coeffs = [lambda p, t: p for _ in range(3)]
H1 = qml.dot(coeffs, ops)  # time-independent parametrized Hamiltonian

ops = [qml.PauliZ(0), qml.PauliY(1), qml.PauliX(2)]
coeffs = [lambda p, t: p * jnp.sin(t) for _ in range(3)]
H2 = qml.dot(coeffs, ops) # time-dependent parametrized Hamiltonian


The evolutions of the ParametrizedHamiltonian can be used in a QNode.

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

@qml.qnode(dev, interface="jax")
def circuit1(params):
qml.evolve(H1)(params, t=[0, 10])
qml.evolve(H2)(params, t=[0, 10])
return qml.expval(qml.PauliZ(0) @ qml.PauliZ(1) @ qml.PauliZ(2))

@qml.qnode(dev, interface="jax")
def circuit2(params):
qml.evolve(H1 + H2)(params, t=[0, 10])
return qml.expval(qml.PauliZ(0) @ qml.PauliZ(1) @ qml.PauliZ(2))


In circuit1, the two Hamiltonians are evolved over the same time window, but inside different operators. In circuit2, we add the two to form a single ParametrizedHamiltonian. This will combine the two so that the expected parameters will be params1 + params2 (as an addition of list). They can then be included inside a single ParametrizedEvolution.

The resulting evolutions of circuit1 and circuit2 are not identical:

>>> params = jnp.array([1., 2., 3.])
>>> circuit1(params)
Array(-0.01543971, dtype=float32)

>>> params = jnp.concatenate([params, params])  # H1 + H2 requires 6 parameters!
>>> circuit2(params)
Array(-0.78236955, dtype=float32)


Here, circuit1 is not executing the evolution of H1 and H2 simultaneously, but rather executing H1 in the [0, 10] time window and then executing H2 with the same time window, without taking into account how the time evolution of H1 affects the evolution of H2 and vice versa!

One can also provide a list of time values that the ODE solver will use to calculate the evolution of the ParametrizedHamiltonian. Keep in mind that the ODE solver uses an adaptive step size, thus it might use additional intermediate time values.

t = jnp.arange(0., 10.1, 0.1)
@qml.qnode(dev, interface="jax")
def circuit(params):
qml.evolve(H1 + H2)(params, t=t)
return qml.expval(qml.PauliZ(0) @ qml.PauliZ(1) @ qml.PauliZ(2))

>>> circuit(params)
Array(-0.78236955, dtype=float32)
Array([-4.8066125 ,  3.703827  , -1.3297377 , -2.406232  ,  0.6811726 ,
-0.52277344], dtype=float32)


Given that we used the same time window ([0, 10]), the results are the same as before.

 arithmetic_depth Arithmetic depth of the 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. control_wires Control wires of the operator. grad_method Gradient computation method. grad_recipe Gradient recipe for the parameter-shift method. has_adjoint has_decomposition has_diagonalizing_gates 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 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

Arithmetic depth of the 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

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

• '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 = False
has_diagonalizing_gates = False
has_matrix
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.

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.

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(*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]) 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()

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 $$U^{\dagger}$$.

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()

Eigenvalues of the operator in the computational basis.

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.

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.

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 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 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.QueuingManager'>)

Append the operator to the Operator queue.

simplify()

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)

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

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