# qml.templates.subroutines.ApproxTimeEvolution¶

class ApproxTimeEvolution(hamiltonian, time, n, do_queue=True)[source]

Applies the Trotterized time-evolution operator for an arbitrary Hamiltonian, expressed in terms of Pauli gates.

The general time-evolution operator for a time-independent 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 Trotter-Suzuki decomposition formula,

$e^{A \ + \ B} \ = \ \lim_{n \to \infty} \Big[ e^{A/n} e^{B/n} \Big]^n,$

to implement an approximation of the time-evolution 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, and Identity). Thus, each term in the Hamiltonian must be expressed this way upon input, or else an error will be raised.

Parameters

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)

 base_name Get base name of the operator. eigvals Eigenvalues of an instantiated operator. generator Generator of the operation. grad_method Gradient computation method. grad_recipe Gradient recipe for the parameter-shift method. inverse Boolean determining if the inverse of the operation was requested. matrix Matrix representation of an instantiated operator in the computational basis. name Get and set the name of the operator. num_params num_wires par_domain parameters Current parameter values. string_for_inverse wires Wires of this operator.
base_name

Get base name of the operator.

eigvals
generator

Generator of the operation.

A length-2 list [generator, scaling_factor], where

• generator is an existing PennyLane operation class or $$2\times 2$$ Hermitian array that acts as the generator of the current operation

• scaling_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 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

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'
wires

Wires of this operator.

Returns

wires

Return type

Wires

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

JacobianTape

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.