qml.templates.subroutines.ApproxTimeEvolution¶

class
ApproxTimeEvolution
(hamiltonian, time, n, do_queue=True)[source]¶ Bases:
pennylane.operation.Operation
Applies the Trotterized timeevolution operator for an arbitrary Hamiltonian, expressed in terms of Pauli gates.
The general timeevolution operator for a timeindependent Hamiltonian is given by
\[U(t) \ = \ e^{i H t},\]for some Hamiltonian of the form:
\[H \ = \ \displaystyle\sum_{j} H_j.\]Implementing this unitary with a set of quantum gates is difficult, as the terms \(H_j\) don’t necessarily commute with one another. However, we are able to exploit the TrotterSuzuki decomposition formula,
\[e^{A \ + \ B} \ = \ \lim_{n \to \infty} \Big[ e^{A/n} e^{B/n} \Big]^n,\]to implement an approximation of the timeevolution operator as
\[U \ \approx \ \displaystyle\prod_{k \ = \ 1}^{n} \displaystyle\prod_{j} e^{i H_j t / n},\]with the approximation becoming better for larger \(n\). The circuit implementing this unitary is of the form:
It is also important to note that this decomposition is exact for any value of \(n\) when each term of the Hamiltonian commutes with every other term.
Note
This template uses the
PauliRot
operation in order to implement exponentiated terms of the input Hamiltonian. This operation only takes terms that are explicitly written in terms of products of Pauli matrices (PauliX
,PauliY
,PauliZ
, andIdentity
). Thus, each term in the Hamiltonian must be expressed this way upon input, or else an error will be raised. Parameters
hamiltonian (Hamiltonian) – The Hamiltonian defining the timeevolution operator. The Hamiltonian must be explicitly written in terms of products of Pauli gates (
PauliX
,PauliY
,PauliZ
, andIdentity
).time (int or float) – The time of evolution, namely the parameter \(t\) in \(e^{ i H t}\).
n (int) – The number of Trotter steps used when approximating the timeevolution operator.
Usage Details
The template is used inside a qnode:
import pennylane as qml from pennylane.templates import ApproxTimeEvolution n_wires = 2 wires = range(n_wires) dev = qml.device('default.qubit', wires=n_wires) coeffs = [1, 1] obs = [qml.PauliX(0), qml.PauliX(1)] hamiltonian = qml.Hamiltonian(coeffs, obs) @qml.qnode(dev) def circuit(time): ApproxTimeEvolution(hamiltonian, time, 1) return [qml.expval(qml.PauliZ(wires=i)) for i in wires]
>>> circuit(1) tensor([0.41614684 0.41614684], requires_grad=True)
Attributes
Get base name of the operator.
Eigenvalues of an instantiated operator.
Generator of the operation.
Gradient computation method.
Gradient recipe for the parametershift method.
Boolean determining if the inverse of the operation was requested.
Matrix representation of an instantiated operator in the computational basis.
Get and set the name of the operator.
Current parameter values.
Wires of this operator.

base_name
¶ Get base name of the operator.

eigvals
¶

generator
¶ Generator of the operation.
A length2 list
[generator, scaling_factor]
, wheregenerator
is an existing PennyLane operation class or \(2\times 2\) Hermitian array that acts as the generator of the current operationscaling_factor
represents a scaling factor applied to the generator operation
For example, if \(U(\theta)=e^{i0.7\theta \sigma_x}\), then \(\sigma_x\), with scaling factor \(s\), is the generator of operator \(U(\theta)\):
generator = [PauliX, 0.7]
Default is
[None, 1]
, indicating the operation has no generator.

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

inverse
¶ Boolean determining if the inverse of the operation was requested.

matrix
¶

name
¶ Get and set the name of the operator.

num_params
= 3¶

num_wires
= 1¶

par_domain
= 'A'¶

parameters
¶ Current parameter values.

string_for_inverse
= '.inv'¶
Methods
adjoint
([do_queue])Create an operation that is the adjoint of this one.
decomposition
(*params, wires)Returns a template decomposing the operation into other quantum operations.
expand
()Returns a tape containing the decomposed operations, rather than a list.
get_parameter_shift
(idx[, shift])Multiplier and shift for the given parameter, based on its gradient recipe.
inv
()Inverts the operation, such that the inverse will be used for the computations by the specific device.
queue
()Append the operator to the Operator queue.

adjoint
(do_queue=False)¶ 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
The adjointed operation.

static
decomposition
(*params, wires)¶ Returns a template decomposing the operation into other quantum operations.

expand
()[source]¶ Returns a tape containing the decomposed operations, rather than a list.
 Returns
Returns a quantum tape that contains the operations decomposition, or if not implemented, simply the operation itself.
 Return type

get_parameter_shift
(idx, shift=1.5707963267948966)¶ Multiplier and shift for the given parameter, based on its gradient recipe.
 Parameters
idx (int) – parameter index
 Returns
multiplier, shift
 Return type
float, float

inv
()¶ Inverts the operation, such that the inverse will be used for the computations by the specific device.
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

queue
()¶ Append the operator to the Operator queue.
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