Hellinger Fidelity

Table of Contents

• Introduction
• In QPanda3
  • Using integer as the random variable value
  • Using large integer as the random variable value
  • Using string as random variable value

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Introduction

Computes the Hellinger fidelity between two counts distributions.

The fidelity is defined as

\[ \left(1-H^{2}\right)^{2} \]

where H is the Hellinger distance Please see Please see Hellinger distance.. This value is bounded in the range \([0, 1]\).

This is equivalent to the standard classical fidelity

\[F(Q, P)=\left(\sum_{i} \sqrt{p_{i} q_{i}}\right)^{2} \]

that in turn is equal to the quantum state fidelity for diagonal density matricess.

In QPanda3

Here is API doc for hellinger_fidelity

Using integer as the random variable value

Python

 
from pyqpanda3.quantum_info import hellinger_fidelity
import random
import numpy as np
def test_hellinger_fidelity_1():
    print("test_hellinger_fidelity_1()")
    # prepare probability distribution
    
    n = random.randint(1, 50)
    
    alpha = np.ones(n)
 
    # Generate two arrays satisfying probability constraints using Dirichlet distribution, p distribution and q distribution
    p_prob_val = np.random.dirichlet(alpha, size=1)[0]
    q_prob_val = np.random.dirichlet(alpha, size=1)[0]
 
    print("sum(p_prob_val):",sum(p_prob_val))
    print("sum(q_prob_val):", sum(q_prob_val))
    
    key = np.random.choice(range(-2*n,2*n), n, replace=False)
    
    p_prob = {i:v for i,v in zip(key,p_prob_val)}
    q_prob = {i: v for i, v in zip(key, q_prob_val)}
    print("p_prob:",p_prob)
    print("q_prob:",q_prob)
    # Calculate the Hellinger fidelity
    hd = hellinger_fidelity(p_prob,q_prob)
    print("hd:",hd)
if __name__ == "__main__":
    test_hellinger_fidelity_1()
 

Output

test_hellinger_fidelity_1()
sum(p_prob_val): 0.9999999999999999
sum(q_prob_val): 0.9999999999999999
p_prob: {5: 0.04854810353110398, 1: 0.009079698162282953, -16: 0.03646928990147027, -26: 0.024478635979747115, -37: 0.012346440874469804, -24: 0.05275981503348704, 23: 0.025130767321854923, 7: 0.01782340035264664, 34: 0.004729714212215611, -14: 0.10819617572813868, 8: 0.051803218470062015, 27: 0.011460093215183802, -1: 0.005134603298097388, 24: 0.009083772900411497, -15: 0.1603100882346676, -18: 0.047817616053414884, -34: 0.036364876495667164, -40: 0.22340527372012, -33: 0.056140353466916704, 26: 0.058918063048041855}
q_prob: {5: 0.09340420649225029, 1: 6.87096659272463e-05, -16: 0.03862071915991972, -26: 0.05399225928912998, -37: 0.07099124608467228, -24: 0.047144812840227035, 23: 0.09267654048044585, 7: 0.04441827322501288, 34: 0.08626934864863559, -14: 0.03915518088648987, 8: 0.042059983612428936, 27: 0.03582921175475073, -1: 0.015179192334639068, 24: 0.09061653931041332, -15: 0.006064973994523014, -18: 0.0738095483130895, -34: 0.04741652216211396, -40: 0.026438659880627324, -33: 0.015171225507386716, 26: 0.0806728463573166}
hd: 0.6316001932101935

Using large integer as the random variable value

Python

 
from pyqpanda3.quantum_info import hellinger_fidelity
import random
import numpy as np
def test_hellinger_fidelity_1():
    print("test_hellinger_fidelity_1()")
    # prepare probability distribution
    
    n = random.randint(1, 50)
    
    alpha = np.ones(n)
 
    # Generate two arrays satisfying probability constraints using Dirichlet distribution, p distribution and q distribution
    p_prob_val = np.random.dirichlet(alpha, size=1)[0]
    q_prob_val = np.random.dirichlet(alpha, size=1)[0]
 
    print("sum(p_prob_val):",sum(p_prob_val))
    print("sum(q_prob_val):", sum(q_prob_val))
    
    key = np.random.choice(range(-2*n,2*n), n, replace=False)
    
    p_prob = {i:v for i,v in zip(key,p_prob_val)}
    q_prob = {i: v for i, v in zip(key, q_prob_val)}
    print("p_prob:",p_prob)
    print("q_prob:",q_prob)
    # Calculate the Hellinger fidelity
    hd = hellinger_fidelity(p_prob,q_prob)
    print("hd:",hd)
if __name__ == "__main__":
    test_hellinger_fidelity_1()
 

Output

test_hellinger_fidelity_2()
p_prob: {1000000000001: 0.12022119026479423, 1000000000002: 0.2806992297584853, 1000000000003: 0.5990795799767205}
q_prob: {1000000000001: 0.09914339060260904, 1000000000002: 0.5537924049116996, 1000000000003: 0.3470642044856914}
hd: 0.9204993678602602

Using string as random variable value

Python

 
from pyqpanda3.quantum_info import hellinger_fidelity
import random
import numpy as np
def test_hellinger_fidelity_1():
    print("test_hellinger_fidelity_1()")
    # prepare probability distribution
    
    n = random.randint(1, 50)
    
    alpha = np.ones(n)
 
    # Generate two arrays satisfying probability constraints using Dirichlet distribution, p distribution and q distribution
    p_prob_val = np.random.dirichlet(alpha, size=1)[0]
    q_prob_val = np.random.dirichlet(alpha, size=1)[0]
 
    print("sum(p_prob_val):",sum(p_prob_val))
    print("sum(q_prob_val):", sum(q_prob_val))
    
    key = np.random.choice(range(-2*n,2*n), n, replace=False)
    
    p_prob = {i:v for i,v in zip(key,p_prob_val)}
    q_prob = {i: v for i, v in zip(key, q_prob_val)}
    print("p_prob:",p_prob)
    print("q_prob:",q_prob)
    # Calculate the Hellinger fidelity
    hd = hellinger_fidelity(p_prob,q_prob)
    print("hd:",hd)
if __name__ == "__main__":
    test_hellinger_fidelity_1()
 

Output

test_hellinger_fidelity_3()
p_prob: {'ab': 0.8238217892709107, 'bc': 0.17617821072908926}
q_prob: {'ab': 0.5325124974169014, 'bc': 0.46748750258309846}
hd: 0.9012219516940595