qcr:2606.44918.1

Quantum Neural Network Classifier and Regressor

This Qiskit Machine Learning tutorial shows how to use quantum neural networks for supervised learning through two high-level model classes: the NeuralNetworkClassifier and the NeuralNetworkRegressor. These wrap a quantum neural network (QNN) in a scikit-learn-style interface with fit and predict methods, so a parameterized quantum circuit can be trained on labeled data just like a classical model. The tutorial walks through building classifiers and regressors on top of both the EstimatorQNN and SamplerQNN, choosing a data-encoding feature map and a trainable ansatz, selecting a loss function and classical optimizer, and training the model end to end on small datasets. It demonstrates binary and multi-class classification as well as regression, shows how to interpret the trained model's predictions and decision boundaries, and covers convenience variants such as the variational quantum classifier (VQC) and regressor (VQR). By presenting the familiar fit/predict workflow on quantum models, the tutorial makes quantum machine learning approachable to anyone with a classical ML background and provides a reusable template for applying QNNs to supervised-learning problems in Qiskit.

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In [ ]:

# --- Setup cell added by QCR (not part of the original tutorial) ---
# Source: qiskit-community/qiskit-machine-learning @ 0.9.0, Apache License 2.0.
# Installs the example's dependencies. If a later cell still reports a missing
# package, restart the runtime/kernel and run again from the top.
%pip install -q qiskit-machine-learning==0.9.0 matplotlib pylatexenc scikit-learn

Neural Network Classifier & Regressor

In this tutorial we show how the NeuralNetworkClassifier and NeuralNetworkRegressor are used. Both take as an input a (Quantum) NeuralNetwork and leverage it in a specific context. In both cases we also provide a pre-configured variant for convenience, the Variational Quantum Classifier (VQC) and Variational Quantum Regressor (VQR). The tutorial is structured as follows:

Classification
- Classification with an EstimatorQNN
- Classification with a SamplerQNN
- Variational Quantum Classifier (VQC)
Regression
- Regression with an EstimatorQNN
- Variational Quantum Regressor (VQR)

In [1]:

import matplotlib.pyplot as plt
import numpy as np
from IPython.display import clear_output
from qiskit import QuantumCircuit
from qiskit.circuit import Parameter
from qiskit.circuit.library import real_amplitudes, zz_feature_map
from qiskit_machine_learning.optimizers import COBYLA, L_BFGS_B
from qiskit_machine_learning.utils import algorithm_globals

from qiskit_machine_learning.algorithms.classifiers import NeuralNetworkClassifier, VQC
from qiskit_machine_learning.algorithms.regressors import NeuralNetworkRegressor, VQR
from qiskit_machine_learning.neural_networks import SamplerQNN, EstimatorQNN
from qiskit_machine_learning.circuit.library import qnn_circuit

algorithm_globals.random_seed = 42

Classification

We prepare a simple classification dataset to illustrate the following algorithms.

In [2]:

num_inputs = 2
num_samples = 20
X = 2 * algorithm_globals.random.random([num_samples, num_inputs]) - 1
y01 = 1 * (np.sum(X, axis=1) >= 0)  # in { 0,  1}
y = 2 * y01 - 1  # in {-1, +1}
y_one_hot = np.zeros((num_samples, 2))
for i in range(num_samples):
    y_one_hot[i, y01[i]] = 1

for x, y_target in zip(X, y):
    if y_target == 1:
        plt.plot(x[0], x[1], "bo")
    else:
        plt.plot(x[0], x[1], "go")
plt.plot([-1, 1], [1, -1], "--", color="black")
plt.show()

Classification with an `EstimatorQNN`

First we show how an EstimatorQNN can be used for classification within a NeuralNetworkClassifier. In this context, the EstimatorQNN is expected to return one-dimensional output in . This only works for binary classification and we assign the two classes to . To simplify the composition of parameterized quantum circuit from a feature map and an ansatz we can use the qnn_circuit.

In [3]:

# construct QNN with the qnn_circuit's default zz_feature_map feature map and real_amplitudes ansatz.
qc, fm_params, anz_params = qnn_circuit(num_qubits=2)
qc.draw("mpl", style="clifford")

Create a quantum neural network. As we are performing a local statevector simulation, we will set the estimator parameter from qiskit.primitives.StatevectorEstimator.

In [4]:

from qiskit.primitives import StatevectorEstimator as Estimator

estimator = Estimator()
estimator_qnn = EstimatorQNN(
    circuit=qc, input_params=fm_params, weight_params=anz_params, estimator=estimator
)

No gradient function provided, creating a gradient function. If your Estimator requires transpilation, please provide a pass manager.

In [5]:

# QNN maps inputs to [-1, +1]
estimator_qnn.forward(X[0, :], algorithm_globals.random.random(estimator_qnn.num_weights))

array([[0.24998863]])

We will add a callback function called callback_graph. This will be called for each iteration of the optimizer and will be passed two parameters: the current weights and the value of the objective function at those weights. For our function, we append the value of the objective function to an array so we can plot iteration versus objective function value and update the graph with each iteration. However, you can do whatever you want with a callback function as long as it gets the two parameters mentioned passed.

In [6]:

# callback function that draws a live plot when the .fit() method is called
def callback_graph(weights, obj_func_eval):
    clear_output(wait=True)
    objective_func_vals.append(obj_func_eval)
    plt.title("Objective function value against iteration")
    plt.xlabel("Iteration")
    plt.ylabel("Objective function value")
    plt.plot(range(len(objective_func_vals)), objective_func_vals)
    plt.show()

In [7]:

# construct neural network classifier
estimator_classifier = NeuralNetworkClassifier(
    estimator_qnn, optimizer=COBYLA(maxiter=60), callback=callback_graph
)

In [8]:

# create empty array for callback to store evaluations of the objective function
objective_func_vals = []
plt.rcParams["figure.figsize"] = (12, 6)

# fit classifier to data
estimator_classifier.fit(X, y)

# return to default figsize
plt.rcParams["figure.figsize"] = (6, 4)

# score classifier
estimator_classifier.score(X, y)

0.8

In [9]:

# evaluate data points
y_predict = estimator_classifier.predict(X)

# plot results
# red == wrongly classified
for x, y_target, y_p in zip(X, y, y_predict):
    if y_target == 1:
        plt.plot(x[0], x[1], "bo")
    else:
        plt.plot(x[0], x[1], "go")
    if y_target != y_p:
        plt.scatter(x[0], x[1], s=200, facecolors="none", edgecolors="r", linewidths=2)
plt.plot([-1, 1], [1, -1], "--", color="black")
plt.show()

Now, when the model is trained, we can explore the weights of the neural network. Please note, the number of weights is defined by ansatz.

In [10]:

estimator_classifier.weights

array([ 0.61473052, -0.81218415,  0.3983311 , -0.18685675,  1.22468766,
        0.97744221,  1.85731519, -1.2402504 ])

Classification with a `SamplerQNN`

Next we show how a SamplerQNN can be used for classification within a NeuralNetworkClassifier. In this context, the SamplerQNN is expected to return -dimensional probability vector as output, where denotes the number of classes. The underlying Sampler primitive returns quasi-distributions of bit strings and we just need to define a mapping from the measured bitstrings to the different classes. For binary classification we use the parity mapping. Again we can use the qnn_circuit function to set up a parameterized quantum circuit from a feature map and ansatz of our choice. Please note that, once again, we are setting the StatevectorSampler instance from qiskit.primitives to the QNN and relying on statevector.

In [11]:

# construct a quantum circuit from the default zz_feature_map feature map and a customized real_amplitudes ansatz
qc, fm_params, anz_params = qnn_circuit(ansatz=real_amplitudes(num_inputs, reps=1))
qc.draw("mpl", style="clifford")

In [12]:

# parity maps bitstrings to 0 or 1
def parity(x):
    return "{:b}".format(x).count("1") % 2


output_shape = 2  # corresponds to the number of classes, possible outcomes of the (parity) mapping.

In [13]:

from qiskit.primitives import StatevectorSampler as Sampler

sampler = Sampler()
# construct QNN
sampler_qnn = SamplerQNN(
    circuit=qc,
    input_params=fm_params,
    weight_params=anz_params,
    interpret=parity,
    output_shape=output_shape,
    sampler=sampler,
)

No gradient function provided, creating a gradient function. If your Sampler requires transpilation, please provide a pass manager.

In [14]:

# construct classifier
sampler_classifier = NeuralNetworkClassifier(
    neural_network=sampler_qnn, optimizer=COBYLA(maxiter=30), callback=callback_graph
)

In [15]:

# create empty array for callback to store evaluations of the objective function
objective_func_vals = []
plt.rcParams["figure.figsize"] = (12, 6)

# fit classifier to data
sampler_classifier.fit(X, y01)

# return to default figsize
plt.rcParams["figure.figsize"] = (6, 4)

# score classifier
sampler_classifier.score(X, y01)

0.55

In [16]:

# evaluate data points
y_predict = sampler_classifier.predict(X)

# plot results
# red == wrongly classified
for x, y_target, y_p in zip(X, y01, y_predict):
    if y_target == 1:
        plt.plot(x[0], x[1], "bo")
    else:
        plt.plot(x[0], x[1], "go")
    if y_target != y_p:
        plt.scatter(x[0], x[1], s=200, facecolors="none", edgecolors="r", linewidths=2)
plt.plot([-1, 1], [1, -1], "--", color="black")
plt.show()

Again, once the model is trained we can take a look at the weights. As we set reps=1 explicitly in our ansatz, we can see less parameters than in the previous model.

In [17]:

sampler_classifier.weights

array([-0.2391    ,  1.09296819, -0.33722594, -0.23161501])

Variational Quantum Classifier (`VQC`)

The VQC is a special variant of the NeuralNetworkClassifier with a SamplerQNN. It applies a parity mapping (or extensions to multiple classes) to map from the bitstring to the classification, which results in a probability vector, which is interpreted as a one-hot encoded result. By default, it applies this the CrossEntropyLoss function that expects labels given in one-hot encoded format and will return predictions in that format too.

In [18]:

# construct feature map, ansatz, and optimizer
feature_map = zz_feature_map(num_inputs)
ansatz = real_amplitudes(num_inputs, reps=1)

# construct variational quantum classifier
vqc = VQC(
    feature_map=feature_map,
    ansatz=ansatz,
    loss="cross_entropy",
    optimizer=COBYLA(maxiter=30),
    callback=callback_graph,
    sampler=sampler,
)

No gradient function provided, creating a gradient function. If your Sampler requires transpilation, please provide a pass manager.

In [19]:

# create empty array for callback to store evaluations of the objective function
objective_func_vals = []
plt.rcParams["figure.figsize"] = (12, 6)

# fit classifier to data
vqc.fit(X, y_one_hot)

# return to default figsize
plt.rcParams["figure.figsize"] = (6, 4)

# score classifier
vqc.score(X, y_one_hot)

0.7

In [20]:

# evaluate data points
y_predict = vqc.predict(X)

# plot results
# red == wrongly classified
for x, y_target, y_p in zip(X, y_one_hot, y_predict):
    if y_target[0] == 1:
        plt.plot(x[0], x[1], "bo")
    else:
        plt.plot(x[0], x[1], "go")
    if not np.all(y_target == y_p):
        plt.scatter(x[0], x[1], s=200, facecolors="none", edgecolors="r", linewidths=2)
plt.plot([-1, 1], [1, -1], "--", color="black")
plt.show()

Multiple classes with VQC

In this section we generate an artificial dataset that contains samples of three classes and show how to train a model to classify this dataset. This example shows how to tackle more interesting problems in machine learning. Of course, for a sake of short training time we prepare a tiny dataset. We employ make_classification from SciKit-Learn to generate a dataset. There 10 samples in the dataset, 2 features, that means we can still have a nice plot of the dataset, as well as no redundant features, these are features are generated as a combinations of the other features. Also, we have 3 different classes in the dataset, each classes one kind of centroid and we set class separation to 2.0, a slight increase from the default value of 1.0 to ease the classification problem.

Once the dataset is generated we scale the features into the range [0, 1].

In [21]:

from sklearn.datasets import make_classification
from sklearn.preprocessing import MinMaxScaler

X, y = make_classification(
    n_samples=10,
    n_features=2,
    n_classes=3,
    n_redundant=0,
    n_clusters_per_class=1,
    class_sep=2.0,
    random_state=algorithm_globals.random_seed,
)
X = MinMaxScaler().fit_transform(X)

Let's see how our dataset looks like.

In [22]:

plt.scatter(X[:, 0], X[:, 1], c=y)

<matplotlib.collections.PathCollection at 0x130d3dd9df0>

We also transform labels and make them categorical.

In [23]:

y_cat = np.empty(y.shape, dtype=str)
y_cat[y == 0] = "A"
y_cat[y == 1] = "B"
y_cat[y == 2] = "C"
print(y_cat)

['A' 'A' 'B' 'C' 'C' 'A' 'B' 'B' 'A' 'C']

We create an instance of VQC similar to the previous example, but in this case we pass a minimal set of parameters. Instead of feature map and ansatz we pass just the number of qubits that is equal to the number of features in the dataset, an optimizer with a low number of iteration to reduce training time, a quantum instance, and a callback to observe progress.

In [24]:

vqc = VQC(
    num_qubits=2,
    optimizer=COBYLA(maxiter=30),
    callback=callback_graph,
    sampler=sampler,
)

No gradient function provided, creating a gradient function. If your Sampler requires transpilation, please provide a pass manager.

Start the training process in the same way as in previous examples.

In [25]:

# create empty array for callback to store evaluations of the objective function
objective_func_vals = []
plt.rcParams["figure.figsize"] = (12, 6)

# fit classifier to data
vqc.fit(X, y_cat)

# return to default figsize
plt.rcParams["figure.figsize"] = (6, 4)

# score classifier
vqc.score(X, y_cat)

0.7

Despite we had the low number of iterations, we achieved quite a good score. Let see the output of the predict method and compare the output with the ground truth.

In [26]:

predict = vqc.predict(X)
print(f"Predicted labels: {predict}")
print(f"Ground truth:     {y_cat}")

Predicted labels: ['A' 'A' 'B' 'A' 'A' 'A' 'B' 'B' 'A' 'B']
Ground truth:     ['A' 'A' 'B' 'C' 'C' 'A' 'B' 'B' 'A' 'C']

Regression

We prepare a simple regression dataset to illustrate the following algorithms.

In [27]:

num_samples = 20
eps = 0.2
lb, ub = -np.pi, np.pi
X_ = np.linspace(lb, ub, num=50).reshape(50, 1)
f = lambda x: np.sin(x)

X = (ub - lb) * algorithm_globals.random.random([num_samples, 1]) + lb
y = f(X[:, 0]) + eps * (2 * algorithm_globals.random.random(num_samples) - 1)

plt.plot(X_, f(X_), "r--")
plt.plot(X, y, "bo")
plt.show()

Regression with an `EstimatorQNN`

Here we restrict to regression with an EstimatorQNN that returns values in . More complex and also multi-dimensional models could be constructed, also based on SamplerQNN but that exceeds the scope of this tutorial.

In [28]:

# construct simple feature map
param_x = Parameter("x")
feature_map = QuantumCircuit(1, name="fm")
feature_map.ry(param_x, 0)

# construct simple ansatz
param_y = Parameter("y")
ansatz = QuantumCircuit(1, name="vf")
ansatz.ry(param_y, 0)

# construct a circuit
qc, fm_params, anz_params = qnn_circuit(feature_map=feature_map, ansatz=ansatz)

# construct QNN
regression_estimator_qnn = EstimatorQNN(
    circuit=qc, input_params=fm_params, weight_params=anz_params, estimator=estimator
)

No gradient function provided, creating a gradient function. If your Estimator requires transpilation, please provide a pass manager.

In [29]:

# construct the regressor from the neural network
regressor = NeuralNetworkRegressor(
    neural_network=regression_estimator_qnn,
    loss="squared_error",
    optimizer=L_BFGS_B(maxiter=5),
    callback=callback_graph,
)

In [30]:

# create empty array for callback to store evaluations of the objective function
objective_func_vals = []
plt.rcParams["figure.figsize"] = (12, 6)

# fit to data
regressor.fit(X, y)

# return to default figsize
plt.rcParams["figure.figsize"] = (6, 4)

# score the result
regressor.score(X, y)

0.9735416717503363

In [31]:

# plot target function
plt.plot(X_, f(X_), "r--")

# plot data
plt.plot(X, y, "bo")

# plot fitted line
y_ = regressor.predict(X_)
plt.plot(X_, y_, "g-")
plt.show()

Similarly to the classification models, we can obtain an array of trained weights by querying a corresponding property of the model. In this model we have only one parameter defined as param_y above.

In [32]:

regressor.weights

array([-1.53376712])

Regression with the Variational Quantum Regressor (`VQR`)

Similar to the VQC for classification, the VQR is a special variant of the NeuralNetworkRegressor with a EstimatorQNN. By default it considers the L2Loss function to minimize the mean squared error between predictions and targets.

In [33]:

vqr = VQR(
    feature_map=feature_map,
    ansatz=ansatz,
    optimizer=L_BFGS_B(maxiter=5),
    callback=callback_graph,
    estimator=estimator,
)

No gradient function provided, creating a gradient function. If your Estimator requires transpilation, please provide a pass manager.

In [34]:

# create empty array for callback to store evaluations of the objective function
objective_func_vals = []
plt.rcParams["figure.figsize"] = (12, 6)

# fit regressor
vqr.fit(X, y)

# return to default figsize
plt.rcParams["figure.figsize"] = (6, 4)

# score result
vqr.score(X, y)

0.9781336848314651

In [35]:

# plot target function
plt.plot(X_, f(X_), "r--")

# plot data
plt.plot(X, y, "bo")

# plot fitted line
y_ = vqr.predict(X_)
plt.plot(X_, y_, "g-")
plt.show()

In [36]:

import tutorial_magics

%qiskit_version_table
%qiskit_copyright

Version Information

Software	Version
`qiskit`	1.4.3
`qiskit_machine_learning`	0.9.0
System information
Python version	3.12.9
OS	Windows
Thu Jul 17 16:45:26 2025 Eastern Daylight Time

This code is a part of a Qiskit project

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.

In [ ]:

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Jun 17, 2026

qcr:2606.44918.1

Cite all versions? Use the base QCR ID to always reference the latest version of this entry.

Tools used

Qiskit

Keywords

classifier

regressor

quantum-machine-learning

qiskit

supervised-learning