Hellinger Distance

Table of Contents

• Introduction
  • Definitions and Formulas
  • Nature and Characteristics
    • Range restriction
    • Stability
    • Symmetry
    • Application and Significance
      • Quantum information processing
      • Classification and identification of quantum states
      • Quantum machine learning
• 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

In the quantum field, Hellinger Distance is mainly used to measure the similarity between two probability distributions. This concept originated from probability statistics and information theory and is widely used in machine learning. The following is a detailed introduction to the Hailinger distance in the quantum field:

Definitions and Formulas

Hellinger distance is a special distance measure used to evaluate the similarity between two discrete probability distributions. Given two discrete probability distributions \( P = {p1, p2, …, pn}\) and \(Q = {q1, q2, …, qn}\), their Hellinger distance can be calculated using the following formula

\[ D_{\text {Hellinger }}(P, Q)=\sqrt{\frac{1}{2} \sum_{i=1}^{n}\left(\sqrt{p_{i}}-\sqrt{q_{i}}\right)^{2}} \]

This formula calculates the distance after square root transformation of two probability distributions, and ensures that the result remains within the interval 0,1 through square root operation. Among them, 0 indicates that the two distributions are completely identical, and 1 indicates that the two distributions are completely different.

Nature and Characteristics

Range restriction

The value of Hellinger distance is between 0 and 1, which makes it have special properties in the probability space, making it easy to compare and interpret.

Stability

Even if the element values in the probability distribution vary greatly, the Hellinger distance can maintain a certain degree of stability and accuracy.

Symmetry

The Hellinger distance is a symmetric distance measure, that is, \(D_{Hellinger}(P,Q) = D_{Hellinger}(Q,P)\) .

Application and Significance

In the quantum field, the application of Hellinger distance is mainly focused on the following aspects:

Quantum information processing

In the fields of quantum communication, quantum computing, and other quantum information processing, Hellinger distance can be used to evaluate the similarity between different quantum states, thereby guiding the encoding, transmission, and decoding of quantum information.

Classification and identification of quantum states

In the task of classification and identification of quantum states, Hellinger distance can be used as a part of the classifier to compare the similarity between the quantum states to be classified and the known quantum states, thus achieving accurate classification and identification.

Quantum machine learning

In the field of quantum machine learning, Hellinger distance can be used as a distance metric tool to construct quantum machine learning models based on probability distributions, such as quantum support vector machines and quantum clustering.

In QPanda3

Here is API doc for hellinger_distance

Using integer as the random variable value

Python

 
from pyqpanda3.quantum_info import hellinger_distance
import random
import numpy as np
def test_hellinger_distance_1():
    print("test_hellinger_distance_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 distance
    hd = hellinger_distance(p_prob,q_prob)
    print("hd:",hd)
 
if __name__ == "__main__":
    test_hellinger_distance_1()
 

Output

test_hellinger_distance_1()
sum(p_prob_val): 0.9999999999999999
sum(q_prob_val): 0.9999999999999998
p_prob: {-34: 0.07712188633486798, -43: 0.0021738588008518066, -40: 0.06545344411436894, 52: 0.027455777672370986, -63: 0.017321484561736326, -18: 0.0061523729552665565, -68: 0.028955428941571864, -45: 0.005067773803139212, -53: 0.019347483061192116, -19: 0.038422474076999065, 11: 0.013807373687689771, 80: 0.012652870142195793, -38: 0.0059116186641265025, 74: 0.007793372442072249, -74: 0.00087742930744125, -23: 0.028886736653718742, 6: 0.022981557641975476, 12: 0.012157852967685802, -46: 0.050276053184079836, -70: 0.033016719337137596, -17: 0.06567815571113278, -44: 0.024115235881328936, 33: 0.00237716615149921, -10: 0.00783214565197405, 10: 0.009112240694593808, -4: 0.009145320973744145, 65: 0.03519111895150703, 64: 0.0029129102416369287, -6: 0.0022656764245848637, -80: 0.0007735326730851091, -27: 0.04077993504560989, 30: 0.050661191547952795, 31: 0.04024798452226477, 58: 0.028390297693020355, 63: 0.023446227630413336, -81: 0.012809640115083416, -76: 0.00415933935876831, -56: 0.06537671148628461, 61: 0.0029169671069667496, -54: 0.08823235220634285, 59: 0.007742281581718114}
q_prob: {-34: 0.03457068849138363, -43: 0.004772839775182649, -40: 0.05876174923023205, 52: 0.004514882467470146, -63: 0.0023376071699338892, -18: 0.031166679017488225, -68: 0.010315123790310512, -45: 0.05216095388830002, -53: 0.012815596062048464, -19: 0.019876476216471567, 11: 0.016646038345262078, 80: 0.005707215422743231, -38: 0.03447117527893066, 74: 0.037502737027731846, -74: 0.0018889327122107945, -23: 0.0009203140064451261, 6: 0.009228622921980043, 12: 0.013669293668669537, -46: 0.0017679876752143108, -70: 0.017638212047885776, -17: 0.007484458247633026, -44: 0.029601534194591692, 33: 0.024829564558624628, -10: 0.012842045650639709, 10: 0.0006956768320232853, -4: 0.007677693551464498, 65: 0.051461955064759514, 64: 0.006251227334274152, -6: 0.012987213419061098, -80: 0.05062656204554687, -27: 0.014650982148159889, 30: 0.022886363985379892, 31: 0.012025384934912836, 58: 0.026762884176056614, 63: 0.024403218630415755, -81: 0.005035368039735155, -76: 0.025210681937574014, -56: 0.0009270713478883147, 61: 0.02000229669587882, -54: 0.23884368054853133, 59: 0.034061011440954234}
hd: 0.43507275019757485

Using large integer as the random variable value

Python

 
from pyqpanda3.quantum_info import hellinger_distance
import random
import numpy as np
def test_hellinger_distance_2():
    print("test_hellinger_distance_2()")
    # prepare probability distribution
    
    n = random.randint(1, 5)
    
    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]
    
    key = [1000000000001,1000000000002,1000000000003,10000000000004,1000000000005]
    
    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 distance
    hd = hellinger_distance(p_prob,q_prob)
    print("hd:",hd)
 
if __name__ == "__main__":
    test_hellinger_distance_2()
 

Output

test_hellinger_distance_2()
p_prob: {1000000000001: 0.03777468613192854, 1000000000002: 0.31747765049945736, 1000000000003: 0.45123869480136053, 10000000000004: 0.19350896856725355}
q_prob: {1000000000001: 0.024525210045130204, 1000000000002: 0.06444511900799894, 1000000000003: 0.18338277734850766, 10000000000004: 0.7276468935983633}
hd: 0.4045010669051521

Using string as random variable value

Python

 
import random
import numpy as np
from pyqpanda3.quantum_info import hellinger_distance
def test_hellinger_distance_3():
    print("test_hellinger_distance_3()")
    # prepare probability distribution
    
    n = random.randint(1, 5)
    
    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]
    
    key = ['ab','bc','cd','de','ef']
    
    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 distance
    hd = hellinger_distance(p_prob,q_prob)
    print("hd:",hd)
if __name__ == "__main__":
    test_hellinger_distance_3()
 

Output

test_hellinger_distance_3()
p_prob: {'ab': 0.7808567524866675, 'bc': 0.21914324751333236}
q_prob: {'ab': 0.6427825171953377, 'bc': 0.3572174828046623}
hd: 0.10838483574009858