How can I load a probability distribution using a quantum circuit in Qiskit?

Several quantum circuit representations for common distributions are given in uncertainty models. For generic probability distributions, you can train a quantum circuit representation using quantum generative adversarial networks. For a respective tutorial, please see here.

There are practical examples in Qiskit on how to use common probability distributions with uncertainty models. Let us refer to the following example from Qiskit-AQUA (Algorithms for QUantum computing Applications) on using amplitude estimation algorithm to evaluate a fixed income asset with uncertain interest rates.

import numpy as np
from qiskit import BasicAer
from qiskit.aqua.algorithms import AmplitudeEstimation
from qiskit.aqua.components.uncertainty_models import MultivariateNormalDistribution
from qiskit.finance.components.uncertainty_problems import FixedIncomeExpectedValue

# Create a suitable multivariate distribution
mvnd = MultivariateNormalDistribution(num_qubits=[2, 2],
                                      low=[0, 0], high=[0.12, 0.24],
                                      mu=[0.12, 0.24], sigma=0.01 * np.eye(2))

# Create fixed income component
fixed_income = FixedIncomeExpectedValue(mvnd, np.eye(2), np.zeros(2),
                                        cash_flow=[1.0, 2.0], c_approx=0.125)

# Set number of evaluation qubits (samples)
num_eval_qubits = 5

# Construct and run amplitude estimation
algo = AmplitudeEstimation(num_eval_qubits, fixed_income)
result = algo.run(BasicAer.get_backend('statevector_simulator'))

print('Estimated value:t%.4f' % result['estimation'])
print('Probability:    t%.4f' % result['max_probability'])