# Computing the CDF with QAE in Qiskit

I want to load a lognormal distribution and then use an IntegerComparator to flip a qubit ($$|0\rangle$$ to $$|1\rangle$$) if its value is less than a threshold. Then I want to use an Quantum Amplitude Estimation algorithm to calculate the probability of measuring $$|1\rangle$$.

My code so far is:

import matplotlib.pyplot as plt
import numpy as np

from qiskit import Aer, QuantumCircuit, QuantumRegister, execute
from qiskit.aqua.algorithms import IterativeAmplitudeEstimation
from qiskit.circuit.library import LogNormalDistribution, IntegerComparator

num_uncertainty_qubits = 3

S = 100
vol = 0.4
r = 0.04
T = 3*(30/365)

mu = np.log(S) + (r-0.5*vol**2)*T
sigma = vol*np.sqrt(T)

mean = np.exp(mu - 0.5*sigma**2)
variance = (np.exp(sigma**2)-1)*np.exp(2*mu + sigma**2)
stddev = np.sqrt(variance)

low = np.maximum(0, mean-3*stddev)
high = mean + 3*stddev

uncertainty_model = LogNormalDistribution(num_uncertainty_qubits, mu=mu, sigma=sigma**2, bounds=(low, high))

# 3 qubit LogNormalDistribution model
uncertainty_model = LogNormalDistribution(3, mu=mu, sigma=sigma, bounds=(low, high))

# function to create the quantum circuit of the IntegerComparator and the uncertainty model
# x_eval is the threshold below which the qubit should be flipped
def get_cdf_circuit(x_eval):
qr_state = QuantumRegister(uncertainty_model.num_qubits, 'state')
qr_obj = QuantumRegister(1, 'obj')
qr_comp = QuantumRegister(2, 'compare')
state_preparation = QuantumCircuit(qr_state, qr_obj, qr_comp)

state_preparation.append(uncertainty_model, qr_state)
comparator = IntegerComparator(uncertainty_model.num_qubits, x_eval, geq=False)

state_preparation.append(comparator, qr_state[:]+qr_obj[:]+qr_comp[:])

return state_preparation

# Function to implement the Amplitude Estimation algorithm

def run_ae_for_cdf(x_eval, epsilon=0.01, alpha=0.05, simulator='qasm_simulator'):

state_preparation = get_cdf_circuit(x_eval)

ae_var = IterativeAmplitudeEstimation(state_preparation=state_preparation,
epsilon=epsilon, alpha=alpha,
objective_qubits=[len(qr_state)])
result_var = ae_var.run(quantum_instance=Aer.get_backend(simulator), shots=100)

return result_var['estimation']

Broadly speaking, I want to flip the objective qubit to $$|1\rangle$$ if the state is less than or equal to x_eval.

On running this, I'm getting the same amplitude estimation every time, even when I use different threshold values.

PS: https://i.sstatic.net/tHyDM.jpg This is the link to the QuantumCircuit. P(X) is the LogNormalDistribution and Cmp is the comparator.

PPS: Using LinearAmplitudeFunction to compare floating numbers:

def get_comparator(threshold, num_qubits, low, high):
breakpoints = [low, threshold]
offsets = [0,0]
slopes = [0,0]
f_min = 1
f_max = 0
objective = LinearAmplitudeFunction(
num_qubits,
slopes,
offsets,
domain=(low, high),
image=(f_min, f_max),
breakpoints=breakpoints
)
return objective
• For me your code works, though I had to change len(qr_state) to 3 because qr_state is not globally defined. If I run run_ae_for_cdf for the values 2, 3 and 4 I obtain 0.129, 0.309 and 0.489, respectively. Can you post the entire code, including imports and how you call the function? Dec 17, 2020 at 13:56
• I got rid of the individual quantum registers so that there isn't any confusion. And I changed len(qr_state) to 3. However, for all different x_eval values, I'm getting 1 as the output which shouldn't happen. Dec 18, 2020 at 4:25
• @Cryoris I've uploaded the entire code. I just call run_ae_for_cdf(70) to check the code. PS: 70 is just an example, can be any integer value. Dec 18, 2020 at 12:09
• For me your code works, see the output in the answer below :) Dec 20, 2020 at 12:03

I think it would be also useful to see your Quantum Circuit using %matplotlib inline
and qc.draw('mpl') to see whether all gates are correctly connected. I had a similar problem with QAE and figured out by using this method that the order of qubits I tried to append the IntegerComparator onto was wrong. Cheers

• I've added a link to the quantum circuit. Let me know what you think. Dec 18, 2020 at 4:41
• What values are you selecting for mu=mu, sigma=sigma, bounds=(low, high)? If I use the default values and do not define particular values I get 0.113, 0.284, 0.464 for x_eval = [2, 3, 4]. Dec 18, 2020 at 11:31
• I updated the link to the decomposed quantum circuit diagram. I've also added the entire code. Let me know. Dec 18, 2020 at 12:10
• Please note that with num_uncertainty_qubits = 3 you will only be able to reflect values up to 7. Your lower and upper bound, as well as the x_eval=70 are not within that limit. You either need to increase the number of uncertainty qubits or scale down your problem to be within that limit. Dec 18, 2020 at 12:24
• This solves it. Thanks a lot for your help! Dec 19, 2020 at 10:22

I am also working with IterativeAmplitudeEstimation in qiskit. I was able to run QAE for 'Credit Analysis with Quantum Computing' as described in

https://qiskit.org/documentation/tutorials/finance/09_credit_risk_analysis.html

by preparing the quantum circuits from scratch. But when using the libraries as described on the page I got many errors (see below). I guess some of the libraries have simply just moved?

from qiskit.circuit.library import IntegerComparator
from qiskit.aqua.algorithms.amplitude_estimators.iqae import IterativeAmplitudeEstimation
from qiskit.aqua.components.uncertainty_models import NormalDistribution
from qiskit.aqua.components.uncertainty_models import LogNormalDistribution

I for example get the following error for LogNormalDistribution when I load the model from qiskit.aqua.components.uncertainty_models

__init__() got an unexpected keyword argument 'bounds'

or 'LogNormalDistribution' object has no attribute 'num_qubits'

• Thanks for your answer. I imported the LogNormalDistribution from qiskit.circuit.library Dec 18, 2020 at 4:27
• I reinstalled qiskit and now all libraries work perfectly. I use from qiskit.circuit.library import LogNormalDistribution from qiskit.circuit.library import WeightedAdder from qiskit.circuit.library import IntegerComparator from qiskit.aqua.algorithms import IterativeAmplitudeEstimation Dec 18, 2020 at 11:33

I had to exchange len(qr_state) with num_uncertainty_qubits in the code you posted

# in run_ae_for_cdf
ae_var = IterativeAmplitudeEstimation(state_preparation=state_preparation,
epsilon=epsilon, alpha=alpha,
objective_qubits=[num_uncertainty_qubits])

but then for me it seems to work just fine.

The output is

# executing your script with python -i and then:
>>> run_ae_for_cdf(1)
0.03904972139433151
>>> run_ae_for_cdf(3)
0.35743729093195115
>>> run_ae_for_cdf(6)
0.8437041835749457
>>> run_ae_for_cdf(70)
0.9999743710958622

You have 3 qubits in your system so the largest integer you can represent is 7 ($$= 2^3 - 1$$) so for values $$\geq 8$$ the CDF should be 1. That's because the CDF $$F(x)$$ is the probability of measuring a state smaller than $$x$$: $$F(x) = \mathrm{Pr}(\text{measure any state } |i\rangle \text{ with } i \leq x)$$ Since $$i$$ can reach 7 at most, we should see $$F(x) = \begin{cases} \text{some increasing function, if } x \leq 7 \\ 1 \text{, if } x > 7 \end{cases}$$

With your code you can generate the following plot

>>> from matplotlib import pyplot as plt
>>> i = list(range(20))
>>> f = [run_ae_for_cdf(val) for val in i]
>>> plt.axhline(1, color="k", linestyle="-")  # the max we reach
>>> plt.axvline(8, color="k", linestyle=":")  # from here on it should be 1
>>> plt.xlabel("x")
>>> plt.ylabel("CDF, $F(x)$")
>>> plt.show()

which seems to meet what we expected.

• This is perfect. Also, is there any way to compare floating numbers in qiskit? I mean instead of the 7 that was taken in this example, can I use a float? I've added a code block where I tried implementing the same using LinearAmplitudeFunction with threshold as a floating number. Dec 21, 2020 at 4:57