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简介
在量子领域，海林格距离（Hellinger）主要用于衡量两个概率分布之间的相似性。这个概念来源于概率统计和信息理论，并且在机器学习中得到了广泛应用。以下是海林格距离在量子领域中的详细介绍：
定义与公式
海林格距离是一种特殊的距离度量，用于评估两个离散概率分布之间的相似性。给定两个离散概率分布 \( P = {p_1, p_2, …, p_n}\) 和 \( Q = {q_1, q_2, …, q_n}\)，它们的 Hellinger 距离可以通过以下公式计算：
\[D_{\text {Hellinger }}(P, Q)=\sqrt{\frac{1}{2} \sum_{i=1}^{n}\left(\sqrt{p_{i}}-\sqrt{q_{i}}\right)^{2}}
\]
这个公式通过对两个概率分布进行平方根变换来计算距离，并通过平方根操作确保结果保持在区间 0 到 1 之间。其中，0 表示两个分布完全相同，1 表示两个分布完全不同。
性质与特征
范围限制
海林格距离的值介于 0 和 1 之间，这使得它在概率空间中具有特殊的性质，易于进行比较和解释。
稳定性
即使概率分布中的元素值差异很大，海林格距离也能保持一定程度的稳定性和准确性。
对称性
海林格距离是对称的距离度量，即 \(D_{\text{Hellinger}}(P, Q) = D_{\text{Hellinger}}(Q, P)\)。
应用与意义
在量子领域，海林格距离的应用主要集中在以下几个方面：
量子信息处理
在量子通信、量子计算等量子信息处理领域，海林格距离可以用来评估不同量子态之间的相似性，从而指导量子信息的编码、传输和解码。
量子态的分类与识别
在量子态分类与识别任务中，海林格距离可以作为分类器的一部分，用于比较待分类量子态与已知量子态之间的相似性，从而实现准确的分类与识别。
量子机器学习
在量子机器学习领域，海林格距离可以作为一种距离度量工具，用于基于概率分布构建量子机器学习模型，如量子支持向量机和量子聚类。
在 QPanda3 中
海林格距离的API文档链接
使用整数作为随机变量值
Python
import random
import numpy as np
def test_hellinger_distance_1():
print("test_hellinger_distance_1()")
n = random.randint(1, 50)
alpha = np.ones(n)
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)
hd = hellinger_distance(p_prob,q_prob)
print("hd:",hd)
if __name__ == "__main__":
test_hellinger_distance_1()
执行结果如下所示：
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
使用大整数作为随机变量值
Python
import random
import numpy as np
def test_hellinger_distance_2():
print("test_hellinger_distance_2()")
n = random.randint(1, 5)
alpha = np.ones(n)
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)
hd = hellinger_distance(p_prob,q_prob)
print("hd:",hd)
if __name__ == "__main__":
test_hellinger_distance_2()
执行结果如下所示：
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
使用字符串作为随机变量值
Python
import random
import numpy as np
def test_hellinger_distance_3():
print("test_hellinger_distance_3()")
n = random.randint(1, 5)
alpha = np.ones(n)
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)
hd = hellinger_distance(p_prob,q_prob)
print("hd:",hd)
if __name__ == "__main__":
test_hellinger_distance_3()
执行结果如下所示：
test_hellinger_distance_3()
p_prob: {'ab': 0.7808567524866675, 'bc': 0.21914324751333236}
q_prob: {'ab': 0.6427825171953377, 'bc': 0.3572174828046623}
hd: 0.10838483574009858
