Note
This page was generated from docs/tutorials/03_quantum_kernel.ipynb.
Quantum Kernel Machine Learning¶
The general task of machine learning is to find and study patterns in data. For many datasets, the datapoints are better understood in a higher dimensional feature space, through the use of a kernel function: \(k(\vec{x}_i, \vec{x}_j) = \langle f(\vec{x}_i), f(\vec{x}_j) \rangle\) where \(k\) is the kernel function, \(\vec{x}_i, \vec{x}_j\) are \(n\) dimensional inputs, \(f\) is a map from \(n\)-dimension to \(m\)-dimension space and \(\langle a,b \rangle\) denotes the dot product. When considering finite data, a kernel function can be represented as a matrix: \(K_{ij} = k(\vec{x}_i,\vec{x}_j)\).
In quantum kernel machine learning, a quantum feature map \(\phi(\vec{x})\) is used to map a classical feature vector \(\vec{x}\) to a quantum Hilbert space, \(| \phi(\vec{x})\rangle \langle \phi(\vec{x})|\), such that \(K_{ij} = \left| \langle \phi^\dagger(\vec{x}_j)| \phi(\vec{x}_i) \rangle \right|^{2}\). See Supervised learning with quantum enhanced feature spaces for more details.
In this notebook, we use qiskit
to calculate a kernel matrix using a quantum feature map, then use this kernel matrix in scikit-learn
classification and clustering algorithms.
[1]:
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
Classification¶
For our classification example, we will use the ad hoc dataset as described in Supervised learning with quantum enhanced feature spaces, and the scikit-learn
support vector machine classification (svc
) algorithm.
[2]:
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.
[3]:
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
.
The following code gives the kernel as a callable function:
[4]:
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
The following code precomputes and plots the training and testing kernel matrices before providing them to the scikit-learn
svc
algorithm:
[5]:
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:
[6]:
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¶
For our clustering example, we will again use the ad hoc dataset as described in Supervised learning with quantum enhanced feature spaces, and the scikit-learn
spectral
clustering algorithm.
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.
[7]:
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.
[8]:
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.
The following code precomputes and plots the kernel matrices before providing it to the scikit-learn spectral clustering algorithm, and scoring the labels using normalized mutual information, since we a priori know the class labels.
[9]:
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: 0.7287008798015752
scikit-learn
has other algorithms that can use a precomputed kernel matrix, here are a few:
[10]:
import qiskit.tools.jupyter
%qiskit_version_table
%qiskit_copyright
Version Information
Qiskit Software | Version |
---|---|
qiskit-terra | 0.23.1 |
qiskit-aer | 0.11.2 |
qiskit-machine-learning | 0.5.0 |
System information | |
Python version | 3.8.16 |
Python compiler | GCC 11.3.0 |
Python build | default, Jan 11 2023 00:28:51 |
OS | Linux |
CPUs | 2 |
Memory (Gb) | 6.781219482421875 |
Tue Feb 07 03:48:02 2023 UTC |
This code is a part of Qiskit
© Copyright IBM 2017, 2023.
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
of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
Any modifications or derivative works of this code must retain this
copyright notice, and modified files need to carry a notice indicating
that they have been altered from the originals.