Quantum circuit learning
This tutorial shows how to apply qadence for solving a basic quantum
machine learning application: fitting a simple function with the
quantum circuit learning1 (QCL) algorithm.
QCL is a supervised quantum machine learning algorithm that uses a parametrized quantum neural network to learn the behavior of an arbitrary mathematical function using a set of function values as training data. This tutorial shows how to fit the \(sin(x)\) function in the \([-1, 1]\) domain.
In the following, train and test data are defined.
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; the number of feature parameters in the input circuit
determines the number of input features (i.e. the dimensionality of the classical data given as input) whereas
the number of observables determines the number of outputs of the quantum neural network.
Total qubit magnetization is used as observable:
\[
\hat{O} = \sum_i^N \hat{\sigma}_i^z
\]
In the following the observable, quantum circuit and corresponding QNN model are constructed.
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 universal function approximator. Standard PyTorch code is used for training the QNN using a mean-square error loss, Adam optimizer. Training is performend on the GPU if 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: 0.040382579742702956
Epoch 20 - Loss: 0.060943553702643115
Epoch 40 - Loss: 0.02019946175711857
Epoch 60 - Loss: 0.03475947130703542
Epoch 80 - Loss: 0.016117759808504873
Epoch 100 - Loss: 0.35331935672522863
Epoch 120 - Loss: 0.17736054871620344
Epoch 140 - Loss: 0.5524233690614945
Epoch 160 - Loss: 0.07316758575321679
Epoch 180 - Loss: 0.07169735326440273
Epoch 200 - Loss: 0.014875842721625657
Qadence offers some convenience functions to implement this training loop with advanced logging and metrics track features. You can refer to this for more details.
The quantum model is now trained on the training data points. To determine the quality of the results, one can check to 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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