from pennylane import math
from ..postprocessing import even_parity_ids
from ..qnodes import joint_probs_qnode, local_parity_expval_qnode
[docs]def post_process_I_3322_joint_probs(probs_vec):
"""Applies post-processing to multi-qubit probabilities in order to coarse-grain
them into the dichotomic parity observables required by the :math:`I_{3322}` inequality.
An :math:`N`-qubit circuit has :math:`2^N` measurement outcomes.
To construct the joint probabilitye :math:`P(00|xy)` for binary outputs, the joint
probabilities can be partitioned into two sets, :math:`\\{Even\\}` and :math`\\{Odd\\}` which
denote the set of *Even* and *Odd* parity bit strings.
The :math:`2^N` joint probabilities are expressed as :math:`P(\\vec{a},\\vec{b}|x,y)` where
:math:`\\vec{a}` and :math:`\\vec{b}` are each :math:`N`-bit strings.
Since the :math:`I_{3322}` inequality only requires dichotomic probabilities :math:`P(00|xy)`,
our post-processing only needs to calculate this value.
To reduce the joint probabilities :math:`P(\\vec{a},\\vec{b}|x,y)` to dichotomic probabilities
:math:`P(00|x,y)` we aggregate the probabilities of even parity bit strings with
.. math::
P(00|xy) = \\{Even\\}_A\\{Even\\}_B
= \\sum_{\\vec{a}\\in \\{Even\\}} \\sum_{\\vec{b}\\in\\{Even\\}} P(\\vec{a},\\vec{b}|x,y).
:param n_qubits: The number of wires measured to obtained the joint probabilites.
:type n_qubits: int
:param probs_vec: A probability vector obtained by measuring all wires in the
computational basis.
:type probs_vec: list[float]
:returns: The dichotomic probability :math:`P(00|xy)`.
"""
n_local_qubits = int(math.log2(len(probs_vec)) / 2)
probs = math.reshape(probs_vec, (2**n_local_qubits, 2**n_local_qubits))
even_ids = even_parity_ids(n_local_qubits)
return sum([sum([probs[a, b] for b in even_ids]) for a in even_ids])
[docs]def I_3322_bell_inequality_cost_fn(network_ansatz, **qnode_kwargs):
"""Constructs a cost function that maximizes the score of the :math:`I_{3322}` Bell inequality.
:param network_ansatz: A ``NetworkAnsatz`` class specifying the quantum network simulation.
:type network_ansatz: NetworkAnsatz
:returns: A cost function evaluated as ``cost(*network_settings)`` where
the ``network_settings`` are obtained from the provided
``network_ansatz`` class.
"""
I_3322_joint_probs_qnode = joint_probs_qnode(network_ansatz, **qnode_kwargs)
I_3322_local_expval_qnode = local_parity_expval_qnode(network_ansatz, **qnode_kwargs)
static_prep_inputs = [[0] * len(layer_nodes) for layer_nodes in network_ansatz.layers[0:-1]]
def cost(*network_settings):
score = 0
for x, y, mult in [
(0, 0, 1),
(0, 1, 1),
(0, 2, 1),
(1, 0, 1),
(1, 1, 1),
(1, 2, -1),
(2, 0, 1),
(2, 1, -1),
]:
settings = network_ansatz.qnode_settings(
network_settings, static_prep_inputs + [[x, y]]
)
probs_vec_xy = I_3322_joint_probs_qnode(settings)
prob00_xy = post_process_I_3322_joint_probs(probs_vec_xy)
score += mult * prob00_xy
settings_00 = network_ansatz.qnode_settings(network_settings, static_prep_inputs + [[0, 0]])
settings_11 = network_ansatz.qnode_settings(network_settings, static_prep_inputs + [[1, 1]])
expval_00 = I_3322_local_expval_qnode(settings_00)
expval_11 = I_3322_local_expval_qnode(settings_11)
# - P_A(0|0)
score += -1 * (expval_00[0] + 1) / 2
# - 2 * P_B(0|0)
score += -2 * (expval_00[1] + 1) / 2
# - P_B(0|1)
score += -1 * (expval_11[1] + 1) / 2
return -(score)
return cost