qml.templates.layers.SimplifiedTwoDesign

class SimplifiedTwoDesign(initial_layer_weights, weights, wires, do_queue=True)

Bases: pennylane.operation.Operation

Layers consisting of a simplified 2-design architecture of Pauli-Y rotations and controlled-Z entanglers proposed in Cerezo et al. (2020).

A 2-design is an ensemble of unitaries whose statistical properties are the same as sampling random unitaries with respect to the Haar measure up to the first 2 moments.

The template is not a strict 2-design, since it does not consist of universal 2-qubit gates as building blocks, but has been shown in Cerezo et al. (2020) to exhibit important properties to study “barren plateaus” in quantum optimization landscapes.

The template starts with an initial layer of single qubit Pauli-Y rotations, before the main \(L\) layers are applied. The basic building block of the main layers are controlled-Z entanglers followed by a pair of Pauli-Y rotation gates (one for each wire). Each layer consists of an “even” part whose entanglers start with the first qubit, and an “odd” part that starts with the second qubit.

This is an example of two layers, including the initial layer:

The argument initial_layer_weights contains the rotation angles of the initial layer of Pauli-Y rotations, while weights contains the pairs of Pauli-Y rotation angles of the respective layers. Each layer takes \(\lfloor M/2 \rfloor + \lfloor (M-1)/2 \rfloor = M-1\) pairs of angles, where \(M\) is the number of wires. The number of layers \(L\) is derived from the first dimension of weights.

Parameters

• initial_layer_weights (tensor_like) – weight tensor for the initial rotation block, shape (M,)

• weights (tensor_like) – tensor of rotation angles for the layers, shape (L, M-1, 2)

• wires (Iterable) – wires that the template acts on

Usage Details

template - here shown for two layers - is used inside a QNode:

import pennylane as qml
from pennylane.templates import SimplifiedTwoDesign
from math import pi

n_wires = 3
dev = qml.device('default.qubit', wires=n_wires)

@qml.qnode(dev)
def circuit(init_weights, weights):
    SimplifiedTwoDesign(initial_layer_weights=init_weights, weights=weights, wires=range(n_wires))
    return [qml.expval(qml.PauliZ(wires=i)) for i in range(n_wires)]

init_weights = [pi, pi, pi]
weights_layer1 = [[0., pi],
                  [0., pi]]
weights_layer2 = [[pi, 0.],
                  [pi, 0.]]
weights = [weights_layer1, weights_layer2]

>>> circuit(init_weights, weights)
[1., -1., 1.]

Parameter shapes

A list of shapes for the two weights arguments can be computed with the static method shape() and used when creating randomly initialised weight tensors:

shapes = SimplifiedTwoDesign.shape(n_layers=2, n_wires=2)
weights = [np.random.random(size=shape) for shape in shapes]

Attributes

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 = 2

num_wires = -1

par_domain = 'A'

parameters

Current parameter values.

string_for_inverse = '.inv'

wires

Wires of this operator.

Returns

wires

Return type

Wires

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.
shape(n_layers, n_wires)	Returns a list of shapes for the 2 parameter tensors.

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()

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.

static shape(n_layers, n_wires)

Returns a list of shapes for the 2 parameter tensors.

Parameters

• n_layers (int) – number of layers

• n_wires (int) – number of wires

Returns

list of shapes

Return type

list[tuple[int]]