qml.pulse.ParametrizedEvolution¶

class
ParametrizedEvolution
(H, params=None, t=None, do_queue=True, id=None, **odeint_kwargs)[source]¶ Bases:
pennylane.operation.Operation
Parametrized evolution gate, created by passing a
ParametrizedHamiltonian
to theevolve()
functionFor a timedependent Hamiltonian of the form
\[H(\{v_j\}, t) = H_\text{drift} + \sum_j f_j(v_j, t) H_j\]it implements the corresponding timeevolution operator \(U(t_1, t_2)\), which is the solution to the timedependent 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 scalarvalued 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
1.4e8
.rtol (float, optional) – Relative error tolerance for the ODE solver. The error is estimated from comparing a 4th and 5th order RungeKutta step in the Dopri5 algorithm. This error is guaranteed to stay below
tol = atol + rtol * abs(y)
through adaptive step size selection. Defaults to 1.4e8.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 aParametrizedEvolution
from theParametrizedHamiltonian
.Example
To create a
ParametrizedEvolution
, we first define aParametrizedHamiltonian
describing the system, and then pass it toevolve()
: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 justintime. 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. JITcompiling 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 scalarvalued functions present in theParametrizedHamiltonian
class. Consequently, executing twoParametrizedEvolution
operators using the same time window does not mean both operators are executed simultaneously, but rather that both evaluate their respective scalarvalued functions using the same time window. See Usage Details.Usage Details
The parameters used when calling the
ParametrizedEvolution
are expected to have the same order as the functions used to define theParametrizedHamiltonian
. 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([[0.183542850.26303384j, 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)
andf2(1.2, t)
at each timestep when finding the matrix.In the case where we have defined two Hamiltonians,
H1
andH2
, 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 ofevolve()
. For noncommuting operations, applyingqml.evolve(H1)(params, t=[0, 10])
followed byqml.evolve(H2)(params, t=[0, 10])
will not apply the two pulses simultaneously, despite the overlapping time window. Instead, it will executeH1
in the[0, 10]
time window, and then subsequently executeH2
using the same time window to calculate the evolution, but without taking into account how the time evolution ofH1
affects the evolution ofH2
and vice versa.Consider two noncommuting
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) # timeindependent 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) # timedependent 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. Incircuit2
, we add the two to form a singleParametrizedHamiltonian
. This will combine the two so that the expected parameters will beparams1 + params2
(as an addition oflist
). They can then be included inside a singleParametrizedEvolution
.The resulting evolutions of
circuit1
andcircuit2
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 ofH1
andH2
simultaneously, but rather executingH1
in the[0, 10]
time window and then executingH2
with the same time window, without taking into account how the time evolution ofH1
affects the evolution ofH2
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) >>> jax.grad(circuit)(params) 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.Attributes
Arithmetic depth of the operator.
Holdover from when inplace inversion changed then name.
The basis of an operation, or for controlled gates, of the target operation.
Batch size of the operator if it is used with broadcasted parameters.
Control wires of the operator.
Gradient computation method.
Gradient recipe for the parametershift method.
bool(x) > bool
Integer hash that uniquely represents the operator.
Dictionary of nontrainable variables that this operation depends on.
Custom string to label a specific operator instance.
This property determines if an 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.
Returns the frequencies for each operator parameter with respect to an expectation value of the form \(\langle \psi  U(\mathbf{p})^\dagger \hat{O} U(\mathbf{p})\psi\rangle\).
Trainable parameters that the operator depends on.
Wires that the operator acts on.

arithmetic_depth
¶ Arithmetic depth of the operator.

base_name
¶ Holdover from when inplace 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
andCNOT
havebasis = "X"
, whereasControlledPhaseShift
andRZ
havebasis = "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 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

control_wires
¶ Control wires of the operator.
For operations that are not controlled, this is an empty
Wires
object of length0
. Returns
The control wires of the operation.
 Return type

grad_method
¶ Gradient computation method.
'A'
: analytic differentiation using the parametershift method.'F'
: finite difference numerical differentiation.None
: the operation may not be differentiated.
Default is
'F'
, orNone
if the Operation has zero parameters.

grad_recipe
= None¶ Gradient recipe for the parametershift 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 nontrainable 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 nonnegative 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) >>> qml.gradients.eigvals_to_frequencies(tuple(gen_eigvals)) (1.0,)
For more details on this relationship, see
eigvals_to_frequencies()
.

parameters
¶ Trainable parameters that the operator depends on.
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).
Representation of the operator as a product of other operators.
Sequence of gates that diagonalize the operator in the computational basis.
eigvals
()Eigenvalues of the operator in the computational basis.
expand
()Returns a tape that has recorded the decomposition of the operator.
Generator of an operator that is in singleparameterform.
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.
pow
(z)A list of new operators equal to this one raised to the given power.
queue
([context])Append the operator to the Operator queue.
simplify
()Reduce the depth of nested operators to the minimum.
The parameters required to implement a singlequbit 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.
terms
()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
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.
 Parameters
params (list) – trainable parameters of the operator, as stored in the
parameters
attributehyperparams (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.
 Parameters
params (list) – trainable parameters of the operator, as stored in the
parameters
attributehyperparams (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
attributehyperparams (dict) – nontrainable 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.See also
 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.See also
 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.See also
 Returns
eigenvalues
 Return type
tensor_like

expand
()¶ Returns a tape that has recorded 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) [Y0] + (1.0) [Z0 X1]
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)¶ 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 nonparameter 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 matrixvalued 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.], [0., 1.]], requires_grad=True)] >>> op3 = qml.QubitUnitary(np.eye(4), wires=(0,1)) >>> op3.label(cache=cache) 'U(M1)' >>> cache['matrices'] [tensor([[1., 0.], [0., 1.]], requires_grad=True), tensor([[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]], requires_grad=True)]

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.
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.See also
 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

single_qubit_rot_angles
()¶ The parameters required to implement a singlequbit 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.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
()¶ 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