Quantum Circuit Learning
In this tutorial, we show how to apply qadence for solving a basic quantum
machine learning application: fitting a simple function with the
quantum circuit learning (QCL) algorithm.
Quantum circuit learning 1 is a supervised quantum machine learning algorithm that uses parametrized quantum neural networks to learn the behavior of an arbitrary mathematical function starting from some training data extracted from it. We choose the function
For this tutorial, we show how to fit the \(sin(x)\) function in the domain \([-1, 1]\).
Let's start with defining training and test data.
from typing import Callable
import torch
# make sure all tensors are kept on the same device
# only available from PyTorch 2.0
device = "cuda" if torch.cuda.is_available() else "cpu"
torch.set_default_device(device)
# notice that the domain does not include 1 and -1
# this avoids a singularity in the rotation angles when
# when encoding the domain points into the quantum circuit
# with a non-linear transformation (see below)
def qcl_training_data(
domain: tuple = (-0.99, 0.99), n_points: int = 100
) -> tuple[torch.Tensor, torch.Tensor]:
start, end = domain
x_rand, _ = torch.sort(torch.DoubleTensor(n_points).uniform_(start, end))
y_rand = torch.sin(x_rand)
return x_rand, y_rand
test_frac = 0.25
x, y = qcl_training_data()
n_test = int(len(x) * test_frac)
x_train, y_train = x[0:n_test-len(x)], y[0:n_test-len(x)]
x_test, y_test = x[n_test-len(x):], y[n_test-len(x):]
Train the QCL model
Qadence provides the QNN convenience constructor to build a quantum neural network.
The QNN class needs a circuit and a list of observables; both the number of feature parameters and the number
of observables in the list must be equal to the number of desired outputs of the quantum neural network.
As observable, we use the total qubit magnetization leveraging a convenience constructor provided by qadence:
\[
\hat{O} = \sum_i^N \hat{\sigma}_i^z
\]
import sympy
import qadence as qd
from qadence.operations import RX
n_qubits = 8
# create a simple feature map with a non-linear parameter transformation
feature_param = qd.FeatureParameter("phi")
feature_map = qd.kron(RX(i, feature_param) for i in range(n_qubits))
featre_map = qd.tag(feature_map, "feature_map")
# create a digital-analog variational ansatz using Qadence convenience constructors
ansatz = qd.hea(n_qubits, depth=n_qubits, strategy=qd.Strategy.SDAQC)
ansatz = qd.tag(ansatz, "ansatz")
# total magnetization observable
observable = qd.hamiltonian_factory(n_qubits, detuning = qd.Z)
circuit = qd.QuantumCircuit(n_qubits, feature_map, ansatz)
model = qd.QNN(circuit, [observable])
expval = model(values=torch.rand(10))
print(expval)
The QCL algorithm uses the output of the quantum neural network as a tunable function approximator. We can use standard PyTorch code for training the QNN using a mean-square error loss, the Adam optimizer and also train on the GPU if any is available:
# train the model
n_epochs = 200
lr = 0.5
input_values = {"phi": x_train}
mse_loss = torch.nn.MSELoss() # standard PyTorch loss function
optimizer = torch.optim.Adam(model.parameters(), lr=lr) # standard PyTorch Adam optimizer
print(f"Initial loss: {mse_loss(model(input_values), y_train)}")
y_pred_initial = model({"phi": x_test})
running_loss = 0.0
for i in range(n_epochs):
optimizer.zero_grad()
loss = mse_loss(model(input_values), y_train)
loss.backward()
optimizer.step()
if (i+1) % 20 == 0:
print(f"Epoch {i+1} - Loss: {loss.item()}")
Initial loss: 0.04038257974270301
Epoch 20 - Loss: 0.060943553558947074
Epoch 40 - Loss: 0.02019946267299006
Epoch 60 - Loss: 0.03476874387797955
Epoch 80 - Loss: 0.10615164456464049
Epoch 100 - Loss: 1.2905890122975225
Epoch 120 - Loss: 0.21929946455255525
Epoch 140 - Loss: 0.013236293097701446
Epoch 160 - Loss: 0.016120784698980737
Epoch 180 - Loss: 0.023339611950154646
Epoch 200 - Loss: 0.03365795073769631
The quantum model is now trained on the training data points. Let's see how well it fits the function on the test set.
import matplotlib.pyplot as plt
y_pred = model({"phi": x_test})
# convert all the results to numpy arrays for plotting
x_train_np = x_train.cpu().detach().numpy().flatten()
y_train_np = y_train.cpu().detach().numpy().flatten()
x_test_np = x_test.cpu().detach().numpy().flatten()
y_pred_initial_np = y_pred_initial.cpu().detach().numpy().flatten()
y_pred_np = y_pred.cpu().detach().numpy().flatten()
fig, _ = plt.subplots()
plt.scatter(x_train_np, y_train_np, label="Training points", marker="o", color="orange")
plt.plot(x_test_np, y_pred_initial_np, label="Initial prediction", color="green", alpha=0.5)
plt.plot(x_test_np, y_pred_np, label="Final prediction")
plt.legend()
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