量子ソフトウェア勉強会:「量子インスパイアード古典アルゴリズム」ハンズオン資料 4
2023年9月20日(水)
高比良宗一（名城大学情報工学部情報工学科）

\[\newcommand{\norm}[1]{\lVert#1\rVert} \renewcommand{\top}{\mathsf{T}} \newcommand{\SQ}{\mathrm{SQ}}\]

2クラス問題に対する量子インスパイアード古典アルゴリズム#

このノートブックでは，2クラス問題について考察します．具体的にはIRISデータセットを例にして，線形識別関数の係数ベクトル\(\theta\)に関する\(\SQ(\theta)\)を考察します．そして，テスト用のデータがどちらのクラスであるかどうか，\(\theta\)をそのまま構成することなく，判別することについて考えます．

[1]:

import numpy as np
import scipy as sp
import numpy.linalg as la
import random
import cmath, math
import matplotlib.pyplot as plt
import pprint
from sklearn.datasets import load_iris
from sklearn.model_selection import train_test_split

データセットを読み込む#

IRISデータセットはPythonのライブラリscikit-learnに含まれています．このライブラリにあるload_irisを使うことで読み込むことができます．次のセルでデータをロードしてみます．

[2]:

from sklearn.datasets import load_iris
iris = load_iris()
pprint.pprint(iris)

{'DESCR': '.. _iris_dataset:\n'
          '\n'
          'Iris plants dataset\n'
          '--------------------\n'
          '\n'
          '**Data Set Characteristics:**\n'
          '\n'
          '    :Number of Instances: 150 (50 in each of three classes)\n'
          '    :Number of Attributes: 4 numeric, predictive attributes and the '
          'class\n'
          '    :Attribute Information:\n'
          '        - sepal length in cm\n'
          '        - sepal width in cm\n'
          '        - petal length in cm\n'
          '        - petal width in cm\n'
          '        - class:\n'
          '                - Iris-Setosa\n'
          '                - Iris-Versicolour\n'
          '                - Iris-Virginica\n'
          '                \n'
          '    :Summary Statistics:\n'
          '\n'
          '    ============== ==== ==== ======= ===== ====================\n'
          '                    Min  Max   Mean    SD   Class Correlation\n'
          '    ============== ==== ==== ======= ===== ====================\n'
          '    sepal length:   4.3  7.9   5.84   0.83    0.7826\n'
          '    sepal width:    2.0  4.4   3.05   0.43   -0.4194\n'
          '    petal length:   1.0  6.9   3.76   1.76    0.9490  (high!)\n'
          '    petal width:    0.1  2.5   1.20   0.76    0.9565  (high!)\n'
          '    ============== ==== ==== ======= ===== ====================\n'
          '\n'
          '    :Missing Attribute Values: None\n'
          '    :Class Distribution: 33.3% for each of 3 classes.\n'
          '    :Creator: R.A. Fisher\n'
          '    :Donor: Michael Marshall (MARSHALL%PLU@io.arc.nasa.gov)\n'
          '    :Date: July, 1988\n'
          '\n'
          'The famous Iris database, first used by Sir R.A. Fisher. The '
          'dataset is taken\n'
          "from Fisher's paper. Note that it's the same as in R, but not as in "
          'the UCI\n'
          'Machine Learning Repository, which has two wrong data points.\n'
          '\n'
          'This is perhaps the best known database to be found in the\n'
          "pattern recognition literature.  Fisher's paper is a classic in the "
          'field and\n'
          'is referenced frequently to this day.  (See Duda & Hart, for '
          'example.)  The\n'
          'data set contains 3 classes of 50 instances each, where each class '
          'refers to a\n'
          'type of iris plant.  One class is linearly separable from the other '
          '2; the\n'
          'latter are NOT linearly separable from each other.\n'
          '\n'
          '.. topic:: References\n'
          '\n'
          '   - Fisher, R.A. "The use of multiple measurements in taxonomic '
          'problems"\n'
          '     Annual Eugenics, 7, Part II, 179-188 (1936); also in '
          '"Contributions to\n'
          '     Mathematical Statistics" (John Wiley, NY, 1950).\n'
          '   - Duda, R.O., & Hart, P.E. (1973) Pattern Classification and '
          'Scene Analysis.\n'
          '     (Q327.D83) John Wiley & Sons.  ISBN 0-471-22361-1.  See page '
          '218.\n'
          '   - Dasarathy, B.V. (1980) "Nosing Around the Neighborhood: A New '
          'System\n'
          '     Structure and Classification Rule for Recognition in Partially '
          'Exposed\n'
          '     Environments".  IEEE Transactions on Pattern Analysis and '
          'Machine\n'
          '     Intelligence, Vol. PAMI-2, No. 1, 67-71.\n'
          '   - Gates, G.W. (1972) "The Reduced Nearest Neighbor Rule".  IEEE '
          'Transactions\n'
          '     on Information Theory, May 1972, 431-433.\n'
          '   - See also: 1988 MLC Proceedings, 54-64.  Cheeseman et al"s '
          'AUTOCLASS II\n'
          '     conceptual clustering system finds 3 classes in the data.\n'
          '   - Many, many more ...',
 'data': array([[5.1, 3.5, 1.4, 0.2],
       [4.9, 3. , 1.4, 0.2],
       [4.7, 3.2, 1.3, 0.2],
       [4.6, 3.1, 1.5, 0.2],
       [5. , 3.6, 1.4, 0.2],
       [5.4, 3.9, 1.7, 0.4],
       [4.6, 3.4, 1.4, 0.3],
       [5. , 3.4, 1.5, 0.2],
       [4.4, 2.9, 1.4, 0.2],
       [4.9, 3.1, 1.5, 0.1],
       [5.4, 3.7, 1.5, 0.2],
       [4.8, 3.4, 1.6, 0.2],
       [4.8, 3. , 1.4, 0.1],
       [4.3, 3. , 1.1, 0.1],
       [5.8, 4. , 1.2, 0.2],
       [5.7, 4.4, 1.5, 0.4],
       [5.4, 3.9, 1.3, 0.4],
       [5.1, 3.5, 1.4, 0.3],
       [5.7, 3.8, 1.7, 0.3],
       [5.1, 3.8, 1.5, 0.3],
       [5.4, 3.4, 1.7, 0.2],
       [5.1, 3.7, 1.5, 0.4],
       [4.6, 3.6, 1. , 0.2],
       [5.1, 3.3, 1.7, 0.5],
       [4.8, 3.4, 1.9, 0.2],
       [5. , 3. , 1.6, 0.2],
       [5. , 3.4, 1.6, 0.4],
       [5.2, 3.5, 1.5, 0.2],
       [5.2, 3.4, 1.4, 0.2],
       [4.7, 3.2, 1.6, 0.2],
       [4.8, 3.1, 1.6, 0.2],
       [5.4, 3.4, 1.5, 0.4],
       [5.2, 4.1, 1.5, 0.1],
       [5.5, 4.2, 1.4, 0.2],
       [4.9, 3.1, 1.5, 0.2],
       [5. , 3.2, 1.2, 0.2],
       [5.5, 3.5, 1.3, 0.2],
       [4.9, 3.6, 1.4, 0.1],
       [4.4, 3. , 1.3, 0.2],
       [5.1, 3.4, 1.5, 0.2],
       [5. , 3.5, 1.3, 0.3],
       [4.5, 2.3, 1.3, 0.3],
       [4.4, 3.2, 1.3, 0.2],
       [5. , 3.5, 1.6, 0.6],
       [5.1, 3.8, 1.9, 0.4],
       [4.8, 3. , 1.4, 0.3],
       [5.1, 3.8, 1.6, 0.2],
       [4.6, 3.2, 1.4, 0.2],
       [5.3, 3.7, 1.5, 0.2],
       [5. , 3.3, 1.4, 0.2],
       [7. , 3.2, 4.7, 1.4],
       [6.4, 3.2, 4.5, 1.5],
       [6.9, 3.1, 4.9, 1.5],
       [5.5, 2.3, 4. , 1.3],
       [6.5, 2.8, 4.6, 1.5],
       [5.7, 2.8, 4.5, 1.3],
       [6.3, 3.3, 4.7, 1.6],
       [4.9, 2.4, 3.3, 1. ],
       [6.6, 2.9, 4.6, 1.3],
       [5.2, 2.7, 3.9, 1.4],
       [5. , 2. , 3.5, 1. ],
       [5.9, 3. , 4.2, 1.5],
       [6. , 2.2, 4. , 1. ],
       [6.1, 2.9, 4.7, 1.4],
       [5.6, 2.9, 3.6, 1.3],
       [6.7, 3.1, 4.4, 1.4],
       [5.6, 3. , 4.5, 1.5],
       [5.8, 2.7, 4.1, 1. ],
       [6.2, 2.2, 4.5, 1.5],
       [5.6, 2.5, 3.9, 1.1],
       [5.9, 3.2, 4.8, 1.8],
       [6.1, 2.8, 4. , 1.3],
       [6.3, 2.5, 4.9, 1.5],
       [6.1, 2.8, 4.7, 1.2],
       [6.4, 2.9, 4.3, 1.3],
       [6.6, 3. , 4.4, 1.4],
       [6.8, 2.8, 4.8, 1.4],
       [6.7, 3. , 5. , 1.7],
       [6. , 2.9, 4.5, 1.5],
       [5.7, 2.6, 3.5, 1. ],
       [5.5, 2.4, 3.8, 1.1],
       [5.5, 2.4, 3.7, 1. ],
       [5.8, 2.7, 3.9, 1.2],
       [6. , 2.7, 5.1, 1.6],
       [5.4, 3. , 4.5, 1.5],
       [6. , 3.4, 4.5, 1.6],
       [6.7, 3.1, 4.7, 1.5],
       [6.3, 2.3, 4.4, 1.3],
       [5.6, 3. , 4.1, 1.3],
       [5.5, 2.5, 4. , 1.3],
       [5.5, 2.6, 4.4, 1.2],
       [6.1, 3. , 4.6, 1.4],
       [5.8, 2.6, 4. , 1.2],
       [5. , 2.3, 3.3, 1. ],
       [5.6, 2.7, 4.2, 1.3],
       [5.7, 3. , 4.2, 1.2],
       [5.7, 2.9, 4.2, 1.3],
       [6.2, 2.9, 4.3, 1.3],
       [5.1, 2.5, 3. , 1.1],
       [5.7, 2.8, 4.1, 1.3],
       [6.3, 3.3, 6. , 2.5],
       [5.8, 2.7, 5.1, 1.9],
       [7.1, 3. , 5.9, 2.1],
       [6.3, 2.9, 5.6, 1.8],
       [6.5, 3. , 5.8, 2.2],
       [7.6, 3. , 6.6, 2.1],
       [4.9, 2.5, 4.5, 1.7],
       [7.3, 2.9, 6.3, 1.8],
       [6.7, 2.5, 5.8, 1.8],
       [7.2, 3.6, 6.1, 2.5],
       [6.5, 3.2, 5.1, 2. ],
       [6.4, 2.7, 5.3, 1.9],
       [6.8, 3. , 5.5, 2.1],
       [5.7, 2.5, 5. , 2. ],
       [5.8, 2.8, 5.1, 2.4],
       [6.4, 3.2, 5.3, 2.3],
       [6.5, 3. , 5.5, 1.8],
       [7.7, 3.8, 6.7, 2.2],
       [7.7, 2.6, 6.9, 2.3],
       [6. , 2.2, 5. , 1.5],
       [6.9, 3.2, 5.7, 2.3],
       [5.6, 2.8, 4.9, 2. ],
       [7.7, 2.8, 6.7, 2. ],
       [6.3, 2.7, 4.9, 1.8],
       [6.7, 3.3, 5.7, 2.1],
       [7.2, 3.2, 6. , 1.8],
       [6.2, 2.8, 4.8, 1.8],
       [6.1, 3. , 4.9, 1.8],
       [6.4, 2.8, 5.6, 2.1],
       [7.2, 3. , 5.8, 1.6],
       [7.4, 2.8, 6.1, 1.9],
       [7.9, 3.8, 6.4, 2. ],
       [6.4, 2.8, 5.6, 2.2],
       [6.3, 2.8, 5.1, 1.5],
       [6.1, 2.6, 5.6, 1.4],
       [7.7, 3. , 6.1, 2.3],
       [6.3, 3.4, 5.6, 2.4],
       [6.4, 3.1, 5.5, 1.8],
       [6. , 3. , 4.8, 1.8],
       [6.9, 3.1, 5.4, 2.1],
       [6.7, 3.1, 5.6, 2.4],
       [6.9, 3.1, 5.1, 2.3],
       [5.8, 2.7, 5.1, 1.9],
       [6.8, 3.2, 5.9, 2.3],
       [6.7, 3.3, 5.7, 2.5],
       [6.7, 3. , 5.2, 2.3],
       [6.3, 2.5, 5. , 1.9],
       [6.5, 3. , 5.2, 2. ],
       [6.2, 3.4, 5.4, 2.3],
       [5.9, 3. , 5.1, 1.8]]),
 'data_module': 'sklearn.datasets.data',
 'feature_names': ['sepal length (cm)',
                   'sepal width (cm)',
                   'petal length (cm)',
                   'petal width (cm)'],
 'filename': 'iris.csv',
 'frame': None,
 'target': array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
       0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
       0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
       1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
       1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
       2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
       2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]),
 'target_names': array(['setosa', 'versicolor', 'virginica'], dtype='<U10')}

分類問題#

scikit-learnに含まれているtrain_test_splitを使うと，データを訓練用とテスト用の配列に分割することができます．

[3]:

from sklearn.model_selection import train_test_split

iris = load_iris()
x_train, x_test, y_train, y_test = train_test_split(iris.data[50:], iris.target[50:], random_state=1)
print("訓練用データ\n", x_train)
print("訓練用ラベル\n", y_train)
print("テスト用データ\n", x_test)
print("テスト用ラベル\n", y_test)

訓練用データ
 [[6.  3.4 4.5 1.6]
 [6.7 3.3 5.7 2.5]
 [6.7 3.  5.  1.7]
 [5.7 2.9 4.2 1.3]
 [5.6 3.  4.1 1.3]
 [7.7 3.8 6.7 2.2]
 [5.9 3.  5.1 1.8]
 [6.5 3.  5.8 2.2]
 [6.7 3.  5.2 2.3]
 [6.  3.  4.8 1.8]
 [5.5 2.6 4.4 1.2]
 [5.1 2.5 3.  1.1]
 [7.2 3.6 6.1 2.5]
 [6.1 2.8 4.7 1.2]
 [5.4 3.  4.5 1.5]
 [6.3 3.4 5.6 2.4]
 [6.3 2.9 5.6 1.8]
 [6.1 3.  4.9 1.8]
 [6.7 3.1 4.4 1.4]
 [6.3 2.8 5.1 1.5]
 [6.1 3.  4.6 1.4]
 [5.7 3.  4.2 1.2]
 [6.9 3.1 5.1 2.3]
 [6.8 2.8 4.8 1.4]
 [6.2 3.4 5.4 2.3]
 [5.  2.3 3.3 1. ]
 [7.6 3.  6.6 2.1]
 [6.4 2.9 4.3 1.3]
 [6.5 2.8 4.6 1.5]
 [6.7 2.5 5.8 1.8]
 [5.7 2.8 4.1 1.3]
 [6.1 2.8 4.  1.3]
 [6.4 3.1 5.5 1.8]
 [5.5 2.3 4.  1.3]
 [6.7 3.3 5.7 2.1]
 [5.5 2.4 3.8 1.1]
 [6.5 3.  5.5 1.8]
 [6.9 3.2 5.7 2.3]
 [5.8 2.6 4.  1.2]
 [6.2 2.9 4.3 1.3]
 [6.9 3.1 5.4 2.1]
 [6.6 2.9 4.6 1.3]
 [6.5 3.2 5.1 2. ]
 [7.  3.2 4.7 1.4]
 [6.7 3.1 5.6 2.4]
 [7.3 2.9 6.3 1.8]
 [6.3 2.5 4.9 1.5]
 [6.4 2.7 5.3 1.9]
 [5.7 2.5 5.  2. ]
 [4.9 2.4 3.3 1. ]
 [6.3 2.5 5.  1.9]
 [6.1 2.9 4.7 1.4]
 [7.7 2.6 6.9 2.3]
 [7.7 3.  6.1 2.3]
 [5.6 2.9 3.6 1.3]
 [5.7 2.6 3.5 1. ]
 [6.  2.9 4.5 1.5]
 [5.9 3.  4.2 1.5]
 [6.2 2.2 4.5 1.5]
 [5.9 3.2 4.8 1.8]
 [6.3 3.3 6.  2.5]
 [6.6 3.  4.4 1.4]
 [6.3 3.3 4.7 1.6]
 [5.6 2.8 4.9 2. ]
 [6.2 2.8 4.8 1.8]
 [6.4 3.2 4.5 1.5]
 [5.6 3.  4.5 1.5]
 [5.8 2.8 5.1 2.4]
 [7.2 3.  5.8 1.6]
 [5.7 2.8 4.5 1.3]
 [7.2 3.2 6.  1.8]
 [5.2 2.7 3.9 1.4]
 [7.7 2.8 6.7 2. ]
 [6.  2.2 4.  1. ]
 [6.3 2.3 4.4 1.3]]
訓練用ラベル
 [1 2 1 1 1 2 2 2 2 2 1 1 2 1 1 2 2 2 1 2 1 1 2 1 2 1 2 1 1 2 1 1 2 1 2 1 2
 2 1 1 2 1 2 1 2 2 1 2 2 1 2 1 2 2 1 1 1 1 1 1 2 1 1 2 2 1 1 2 2 1 2 1 2 1
 1]
テスト用データ
 [[7.4 2.8 6.1 1.9]
 [6.1 2.6 5.6 1.4]
 [6.  2.7 5.1 1.6]
 [7.9 3.8 6.4 2. ]
 [6.8 3.2 5.9 2.3]
 [5.8 2.7 4.1 1. ]
 [6.7 3.1 4.7 1.5]
 [6.4 2.8 5.6 2.2]
 [6.  2.2 5.  1.5]
 [6.4 3.2 5.3 2.3]
 [5.8 2.7 5.1 1.9]
 [5.5 2.5 4.  1.3]
 [4.9 2.5 4.5 1.7]
 [7.1 3.  5.9 2.1]
 [5.8 2.7 5.1 1.9]
 [5.8 2.7 3.9 1.2]
 [5.5 2.4 3.7 1. ]
 [5.6 2.7 4.2 1.3]
 [6.4 2.8 5.6 2.1]
 [5.  2.  3.5 1. ]
 [6.9 3.1 4.9 1.5]
 [6.3 2.7 4.9 1.8]
 [6.5 3.  5.2 2. ]
 [6.8 3.  5.5 2.1]
 [5.6 2.5 3.9 1.1]]
テスト用ラベル
 [2 2 1 2 2 1 1 2 2 2 2 1 2 2 2 1 1 1 2 1 1 2 2 2 1]

今回はsepal length(がく片の長さ)とpetal length(花びらの長さ)から，versicolorか，virginicaのどちらであるか判断することを考えます．IRISデータセットの50行目以降がversicolor及びvirginicaのデータで，sepal length, petal lengthが0列目,2列目であることを踏まえると，行列\(A\), \(b\)は次のセルのように定義できます．

[4]:

from sklearn.datasets import load_iris

# IRISデータセットを読み込み，versicolorとvirginicaに分ける．
iris = load_iris()

x_train, x_test, y_train, y_test = train_test_split(iris.data[50:], iris.target[50:] , random_state=1)
size = len(x_train)

# Aとbを定義する．
A = np.zeros((size, 3))
A[:,0] = np.ones(size)
A[:,1] = x_train[:,0]
A[:,2] = x_train[:,2]
# A = A/np.linalg.norm(A)
b   = np.array([1 if t==1 else -1  for t in y_train])

擬似逆行列#

分類する直線に関するパラメータ\(\theta\)は，\(\theta = A^+ b\)のようにして求めることができます．この\(\theta\)によって，分類する直線は，\(\theta_0 + \theta_1 x + \theta_2 y = 0\)のようにして表されます．

[5]:

exact_theta = np.linalg.pinv(A) @ b # or np.linalg.lstsq(A, b)[0]
exact_theta

[5]:

array([ 2.64301213,  0.66176535, -1.37653241])

量子インスパイアード古典アルゴリズム#

\(\SQ(A)\)を構成し，3_SQ(x).ipynbでまとめたSQxを用いて，\(\mathrm{SQ}(\theta)\)を定義します．

[6]:

from quantum_inspired import MatrixBasedDataStructure, SQx
SQA = MatrixBasedDataStructure(A)
r, c = 1000, 1000
sq_theta = SQx(SQA, b, r, c, 2, 1000)

\(\mathrm{SQ}(\theta)\)の\(\mathrm{Query}(i)\)を用いて\(\tilde{\theta}\)を構成してみます．

[7]:

m, n = A.shape
tilde_theta = np.zeros(n)
for i in range(n):
    tilde_theta[i] = sq_theta.query(i)
tilde_theta

[7]:

array([-6.29859579e-05, -1.72154583e-04,  2.36267084e-04])

図を描画する#

上記で求めたNumPyによる解\(\theta\)と，量子インスパイアード古典アルゴリズムによる近似解\(\tilde{\theta}\)について，直線を引いてみて，その様子を見てみます．

[8]:

def line(theta, x):
    return -(theta[0] + theta[1]*x)/theta[2]

[9]:

plt.figure(figsize=(8,6))

# 点を描画する
class1_xs, class1_ys = [], []
class2_xs, class2_ys = [], []
for xt, yt in zip(x_train, y_train):
    if yt == 1:
        class1_xs.append(xt[0])
        class1_ys.append(xt[2])
    else:
        class2_xs.append(xt[0])
        class2_ys.append(xt[2])

plt.scatter(class1_xs, class1_ys, label=iris.target_names[1], color='tab:orange')
plt.scatter(class2_xs, class2_ys, label=iris.target_names[2], color='tab:green')

# 直線を描画する
xs         = np.linspace(4,8)
line_exact = [line(exact_theta, x) for x in xs] # NumPy
line_qi    = [line(tilde_theta, x) for x in xs] # 量子インスパイアード

plt.plot(xs, line_exact, color='blue', label="exact")
plt.plot(xs, line_qi   , color='red',  label="quantum inspired")

# その他
plt.legend(loc='lower right')
plt.xlabel(iris.feature_names[0])
plt.ylabel(iris.feature_names[2])
plt.show()

../_images/notebooks_08_04_quantum_inspired_binary_classification_18_0.png

サンプリング回数を増やした時の様子#

[10]:

from quantum_inspired import MatrixBasedDataStructure, SQx
SQA = MatrixBasedDataStructure(A)
sq_theta100  = SQx(SQA, b, 100, 100, 2, 100)
sq_theta200  = SQx(SQA, b, 200, 200, 2, 200)
sq_theta400  = SQx(SQA, b, 400, 400, 2, 400)
sq_theta800  = SQx(SQA, b, 800, 800, 2, 800)
sq_theta1600 = SQx(SQA, b, 1600, 1600, 2, 1600)

[11]:

tilde_theta100  = np.array([sq_theta100.query(i) for i in range(n)])
tilde_theta200  = np.array([sq_theta200.query(i) for i in range(n)])
tilde_theta400  = np.array([sq_theta400.query(i) for i in range(n)])
tilde_theta800  = np.array([sq_theta800.query(i) for i in range(n)])
tilde_theta1600 = np.array([sq_theta1600.query(i) for i in range(n)])

[12]:

plt.figure(figsize=(8,6))

# 点を描画する
class1_xs, class1_ys = [], []
class2_xs, class2_ys = [], []
for xt, yt in zip(x_train, y_train):
    if yt == 1:
        class1_xs.append(xt[0])
        class1_ys.append(xt[2])
    else:
        class2_xs.append(xt[0])
        class2_ys.append(xt[2])

plt.scatter(class1_xs, class1_ys, label=iris.target_names[1], color='tab:orange')
plt.scatter(class2_xs, class2_ys, label=iris.target_names[2], color='tab:green')

# 直線を描画する
xs         = np.linspace(4,8)
line_exact = [line(exact_theta, x) for x in xs]       # NumPy

# 量子インスパイアード
line_qi100 = [line(tilde_theta100, x) for x in xs]
line_qi200 = [line(tilde_theta200, x) for x in xs]
line_qi400 = [line(tilde_theta400, x) for x in xs]
line_qi800 = [line(tilde_theta800, x) for x in xs]
line_qi1600 = [line(tilde_theta1600, x) for x in xs]

plt.plot(xs, line_exact,  color='blue', label="exact")
plt.plot(xs, line_qi100,  label="quantum inspired r=c=100")
plt.plot(xs, line_qi200,  label="quantum inspired r=c=200")
plt.plot(xs, line_qi400,  label="quantum inspired r=c=400")
plt.plot(xs, line_qi800,  label="quantum inspired r=c=800")
plt.plot(xs, line_qi1600, label="quantum inspired r=c=1600")

# その他
plt.legend(loc='lower right')
plt.xlabel(iris.feature_names[0])
plt.ylabel(iris.feature_names[2])
plt.show()

../_images/notebooks_08_04_quantum_inspired_binary_classification_22_0.png

予測#

新しいデータ\((x, y)\)が来たときに，どちらのクラス(versicolor(1)かvirginica(2))に属するかどうかは，\(f(x,y) = \theta_0 + \theta_1 x + \theta_2 y = (\theta_0,\theta_1,\theta_2)^\top (1, x, y)\)の正負によって決定します．

[13]:

from sklearn.metrics import accuracy_score

def f(theta, x, y):
    fx = np.vdot(theta, np.array([1, x, y]))
    if fx > 0:
        return 1
    else:
        return 2

テスト用データに対する予測値を計算します．

[14]:

prediction = []
for xt in x_test:
    c = f(exact_theta, xt[0], xt[2])
    prediction.append(c)
prediction

[14]:

[2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1]

上記のpredictionがy_testと同様であるか確認してみます．

[15]:

y_test

[15]:

array([2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 2,
       2, 2, 1])

正解率を計算してみます．

[16]:

accuracy_score(y_test, prediction)

[16]:

0.92

次に量子インスパイアード古典アルゴリズムについて考察します．内積をそのまま計算すると，行列のサイズ分だけ計算量を必要とします．そこで，内積の推定を用いて，\(f(x,y)\)の値を推定し，その推定値に基づいて，どちらのクラスであるか識別します：

[18]:

def qi_vdot(SQa, Qb, sample_size=100):
    val = 0
    for _ in range(sample_size):
        i  = SQa.sample()
        zi = Qb[i]/SQa.query(i)
        val += zi
    return val/sample_size * SQa.norm()**2

def qi_f(SQtheta, x, y, sample_size=100):
    fx = qi_vdot(SQtheta, np.array([1, x, y]), sample_size)
    if fx > 0:
        return 1
    else:
        return 2

xt = x_test[0]
v = np.array([1, xt[0], xt[2]])
qi_vdot(sq_theta, v, 100)

[18]:

0.0