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Quantum Kernel Machine Learning

L’obiettivo generale del machine learning è quello di trovare e studiare pattern nei dati. Per molti dataset, i dati sono meglio interpretati in uno spazio delle feature di dimensione più alta rispetto a quello di partenza, raggiunto attraverso l’uso di una funzione kernel: \(k(\vec{x}_i, \vec{x}_j) = \langle f(\vec{x}_i), f(\vec{x}_j) \rangle\), dove \(k\) è la funzione kernel, \(\vec{x}_i, \vec{x}_j\) sono input a \(n\) dimensioni, \(f\) è una mappa da uno spazio \(n\)-dimensionale ad uno spazio \(m\)-dimensionale, e \(\langle a, \rangle\) indica il prodotto scalare. Quando si considerano dati finiti, una funzione kernel può essere rappresentata come una matrice: \(K_{ij} = k(\vec{x}_i,\vec{x}_j)\).

Nel quantum kernel machine learning, si utilizza una feature map quantistica \(\phi(\vec{x})\) per mappare un vettore classico \(\vec{x}\) in uno spazio di Hilbert quantistico, \(| \phi(\vec{x})\rangle \langle \phi(\vec{x})|\), tale che \(K_{ij} = \left| \langle \phi^\dagger(\vec{x}_j)| \phi(\vec{x}_i) \rangle \right|^{2}\). Per maggiori dettagli, si può fare riferimento a Supervised learning with quantum enhanced feature spaces .

In questo notebook usiamo qiskit per calcolare una matrice di kernel usando una feature map quantistica, poi utilizziamo questa matrice di kernel negli algoritmi di classificazione e di clustering in scikit-learn.

import matplotlib.pyplot as plt
import numpy as np

from sklearn.svm import SVC
from sklearn.cluster import SpectralClustering
from sklearn.metrics import normalized_mutual_info_score

from qiskit import BasicAer
from qiskit.algorithms.state_fidelities import ComputeUncompute
from qiskit.circuit.library import ZZFeatureMap
from qiskit.primitives import Sampler
from qiskit.utils import algorithm_globals
from qiskit_machine_learning.algorithms import QSVC
from qiskit_machine_learning.kernels import FidelityQuantumKernel
from qiskit_machine_learning.datasets import ad_hoc_data

seed = 12345
algorithm_globals.random_seed = seed

Classificazione

Seguendo Supervised learning with quantum enhanced feature spaces, nel nostro esempio di classificazione, utilizzeremo un dataset ad hoc e, come algoritmo di classificazione, useremo la support vector machine (svc) di scikit-learn.

adhoc_dimension = 2
train_features, train_labels, test_features, test_labels, adhoc_total = ad_hoc_data(
    training_size=20,
    test_size=5,
    n=adhoc_dimension,
    gap=0.3,
    plot_data=False,
    one_hot=False,
    include_sample_total=True,
)

plt.figure(figsize=(5, 5))
plt.ylim(0, 2 * np.pi)
plt.xlim(0, 2 * np.pi)
plt.imshow(
    np.asmatrix(adhoc_total).T,
    interpolation="nearest",
    origin="lower",
    cmap="RdBu",
    extent=[0, 2 * np.pi, 0, 2 * np.pi],
)

plt.scatter(
    train_features[np.where(train_labels[:] == 0), 0],
    train_features[np.where(train_labels[:] == 0), 1],
    marker="s",
    facecolors="w",
    edgecolors="b",
    label="A train",
)
plt.scatter(
    train_features[np.where(train_labels[:] == 1), 0],
    train_features[np.where(train_labels[:] == 1), 1],
    marker="o",
    facecolors="w",
    edgecolors="r",
    label="B train",
)
plt.scatter(
    test_features[np.where(test_labels[:] == 0), 0],
    test_features[np.where(test_labels[:] == 0), 1],
    marker="s",
    facecolors="b",
    edgecolors="w",
    label="A test",
)
plt.scatter(
    test_features[np.where(test_labels[:] == 1), 0],
    test_features[np.where(test_labels[:] == 1), 1],
    marker="o",
    facecolors="r",
    edgecolors="w",
    label="B test",
)

plt.legend(bbox_to_anchor=(1.05, 1), loc="upper left", borderaxespad=0.0)
plt.title("Ad hoc dataset for classification")

plt.show()

With our training and testing datasets ready, we set up the FidelityQuantumKernel class to calculate a kernel matrix using the ZZFeatureMap. We use the reference implementation of the Sampler primitive and the ComputeUncompute fidelity that computes overlaps between states. These are the default values and if you don’t pass a Sampler or Fidelity instance, the same objects will be created automatically for you.

adhoc_feature_map = ZZFeatureMap(feature_dimension=adhoc_dimension, reps=2, entanglement="linear")
sampler = Sampler()
fidelity = ComputeUncompute(sampler=sampler)
adhoc_kernel = FidelityQuantumKernel(fidelity=fidelity, feature_map=adhoc_feature_map)

The scikit-learn SVC algorithm allows us to define a custom kernel in two ways: by providing the kernel as a callable function or by precomputing the kernel matrix. We can do either of these using the FidelityQuantumKernel class in qiskit.

Il seguente codice fornisce il kernel come funzione richiamabile:

adhoc_svc = SVC(kernel=adhoc_kernel.evaluate)
adhoc_svc.fit(train_features, train_labels)
adhoc_score = adhoc_svc.score(test_features, test_labels)

print(f"Callable kernel classification test score: {adhoc_score}")

Callable kernel classification test score: 1.0

Il seguente codice precalcola e mostra le matrici kernel realative ai dati di training e di test prima di fornirle all’algoritmo svc di scikit-learn:

adhoc_matrix_train = adhoc_kernel.evaluate(x_vec=train_features)
adhoc_matrix_test = adhoc_kernel.evaluate(x_vec=test_features, y_vec=train_features)

fig, axs = plt.subplots(1, 2, figsize=(10, 5))
axs[0].imshow(
    np.asmatrix(adhoc_matrix_train), interpolation="nearest", origin="upper", cmap="Blues"
)
axs[0].set_title("Ad hoc training kernel matrix")
axs[1].imshow(np.asmatrix(adhoc_matrix_test), interpolation="nearest", origin="upper", cmap="Reds")
axs[1].set_title("Ad hoc testing kernel matrix")
plt.show()

adhoc_svc = SVC(kernel="precomputed")
adhoc_svc.fit(adhoc_matrix_train, train_labels)
adhoc_score = adhoc_svc.score(adhoc_matrix_test, test_labels)

print(f"Precomputed kernel classification test score: {adhoc_score}")

Precomputed kernel classification test score: 1.0

Qiskit Machine Learning also contains the QSVC class that extends the SVC class from scikit-learn, that can be used as follows:

qsvc = QSVC(quantum_kernel=adhoc_kernel)
qsvc.fit(train_features, train_labels)
qsvc_score = qsvc.score(test_features, test_labels)

print(f"QSVC classification test score: {qsvc_score}")

QSVC classification test score: 1.0

Clustering

Seguendo Supervised learning with quantum enhanced feature spaces, nel nostro esempio di clustering, utilizzeremo un dataset ad hoc e, l’algoritmo di clustering spectral di scikit-learn.

We will regenerate the dataset with a larger gap between the two classes, and as clustering is an unsupervised machine learning task, we don’t need a test sample.

adhoc_dimension = 2
train_features, train_labels, test_features, test_labels, adhoc_total = ad_hoc_data(
    training_size=25,
    test_size=0,
    n=adhoc_dimension,
    gap=0.6,
    plot_data=False,
    one_hot=False,
    include_sample_total=True,
)

plt.figure(figsize=(5, 5))
plt.ylim(0, 2 * np.pi)
plt.xlim(0, 2 * np.pi)
plt.imshow(
    np.asmatrix(adhoc_total).T,
    interpolation="nearest",
    origin="lower",
    cmap="RdBu",
    extent=[0, 2 * np.pi, 0, 2 * np.pi],
)
plt.scatter(
    train_features[np.where(train_labels[:] == 0), 0],
    train_features[np.where(train_labels[:] == 0), 1],
    marker="s",
    facecolors="w",
    edgecolors="b",
    label="A",
)
plt.scatter(
    train_features[np.where(train_labels[:] == 1), 0],
    train_features[np.where(train_labels[:] == 1), 1],
    marker="o",
    facecolors="w",
    edgecolors="r",
    label="B",
)

plt.legend(bbox_to_anchor=(1.05, 1), loc="upper left", borderaxespad=0.0)
plt.title("Ad hoc dataset for clustering")

plt.show()

We again set up the FidelityQuantumKernel class to calculate a kernel matrix using the ZZFeatureMap, and the default values this time.

adhoc_feature_map = ZZFeatureMap(feature_dimension=adhoc_dimension, reps=2, entanglement="linear")

adhoc_kernel = FidelityQuantumKernel(feature_map=adhoc_feature_map)

The scikit-learn spectral clustering algorithm allows us to define a custom kernel in two ways: by providing the kernel as a callable function or by precomputing the kernel matrix. Using the FidelityQuantumKernel class in Qiskit Machine Learning, we can only use the latter.

Il seguente codice pre-calcola e mostra le matrici di kernel prima di fornirle all’algoritmo di clustering spectral di scikit-learning, e prima di valutare i label utilizzando l’informazione reciproca normalizzata, dato che conosciamo a priori i label delle classi.

adhoc_matrix = adhoc_kernel.evaluate(x_vec=train_features)

plt.figure(figsize=(5, 5))
plt.imshow(np.asmatrix(adhoc_matrix), interpolation="nearest", origin="upper", cmap="Greens")
plt.title("Ad hoc clustering kernel matrix")
plt.show()

adhoc_spectral = SpectralClustering(2, affinity="precomputed")
cluster_labels = adhoc_spectral.fit_predict(adhoc_matrix)
cluster_score = normalized_mutual_info_score(cluster_labels, train_labels)

print(f"Clustering score: {cluster_score}")

Clustering score: 0.7287008798015754

scikit-learn fornisce anche altri algoritmi che possono utilizzare una matrice di kernel pre-calcolata, eccone alcuni:

• Agglomerative clustering

• Support vector regression

• Ridge regression

• Gaussian process regression

• Principal component analysis

import qiskit.tools.jupyter

%qiskit_version_table
%qiskit_copyright

Version Information

Qiskit Software	Version
qiskit-terra	0.22.0
qiskit-aer	0.11.0
qiskit-ignis	0.7.0
qiskit	0.33.0
qiskit-machine-learning	0.5.0
System information
Python version	3.7.9
Python compiler	MSC v.1916 64 bit (AMD64)
Python build	default, Aug 31 2020 17:10:11
OS	Windows
CPUs	4
Memory (Gb)	31.837730407714844
Mon Oct 10 12:01:53 2022 GMT Daylight Time

This code is a part of Qiskit

© Copyright IBM 2017, 2022.

This code is licensed under the Apache License, Version 2.0. You may
obtain a copy of this license in the LICENSE.txt file in the root directory
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Any modifications or derivative works of this code must retain this
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that they have been altered from the originals.