Note

Click here to download the full example code

# Quantum transfer learning¶

In this tutorial we apply a machine learning method, known as *transfer learning*, to an
image classifier based on a hybrid classical-quantum network.

This example follows the general structure of the PyTorch tutorial on transfer learning by Sasank Chilamkurthy, with the crucial difference of using a quantum circuit to perform the final classification task.

More details on this topic can be found in the research paper [1] (Mari et al. (2019)).

## Introduction¶

Transfer learning is a well-established technique for training artificial neural networks (see e.g., Ref. [2]), which is based on the general intuition that if a pre-trained network is good at solving a given problem, then, with just a bit of additional training, it can be used to also solve a different but related problem.

As discussed in Ref. [1], this idea can be formalized in terms of two abstract netwoks \(A\) and \(B\), independently from their quantum or classical physical nature.

As sketched in the above figure, one can give the following **general definition of the
transfer learning method**:

- Take a network \(A\) that has been pre-trained on a dataset \(D_A\) and for a given task \(T_A\).
- Remove some of the final layers. In this way, the resulting truncated network \(A'\) can be used as a feature extractor.
- Connect a new trainable network \(B\) at the end of the pre-trained network \(A'\).
- Keep the weights of \(A'\) constant, and train the final block \(B\) with a new dataset \(D_B\) and/or for a new task of interest \(T_B\).

When dealing with hybrid systems, depending on the physical nature (classical or quantum) of the networks \(A\) and \(B\), one can have different implementations of transfer learning as summarized in following table:

Network A | Network B | Tansfer learning scheme |
---|---|---|

Classical | Classical | CC - Standard classical method. See e.g., Ref. [2]. |

Classical | Quantum | CQ - Hybrid model presented in this tutorial. |

Quantum | Classical | QC - Model studied in Ref. [1]. |

Quantum | Quantum | QQ - Model studied in Ref. [1]. |

## Classical-to-quantum transfer learning¶

We focus on the CQ transfer learning scheme discussed in the previous section and we give a specific example.

- As pre-trained network \(A\) we use
**ResNet18**, a deep residual neural network introduced by Microsoft in Ref. [3], which is pre-trained on the*ImageNet*dataset. - After removing its final layer we obtain \(A'\), a pre-processing block which maps any input high-resolution image into 512 abstract features.
- Such features are classified by a 4-qubit “dressed quantum circuit” \(B\), i.e., a variational quantum circuit sandwiched between two classical layers.
- The hybrid model is trained, keeping \(A'\) constant, on the
*Hymenoptera*dataset (a small subclass of ImageNet) containing images of*ants*and*bees*.

A graphical representation of the full data processing pipeline is given in the figure below.

## General setup¶

Note

To use the PyTorch interface in PennyLane, you must first install PyTorch.

In addition to *PennyLane*, we will also need some standard *PyTorch* libraries and the
plotting library *matplotlib*.

```
# Some parts of this code are based on the Python script:
# https://github.com/pytorch/tutorials/blob/master/beginner_source/transfer_learning_tutorial.py
# License: BSD
# Plotting
import matplotlib.pyplot as plt
# PyTorch
import torch
import torch.nn as nn
import torch.optim as optim
from torch.optim import lr_scheduler
import torchvision
from torchvision import datasets, models, transforms
# Pennylane
import pennylane as qml
from pennylane import numpy as np
# Other tools
import time
import os
import copy
# OpenMP: number of parallel threads.
os.environ["OMP_NUM_THREADS"] = "1"
```

## Setting of the main hyper-parameters of the model¶

Note

To reproduce the results of Ref. [1], `num_epochs`

should be set to `30`

which may take a long time.
We suggest to first try with `num_epochs=1`

and, if everything runs smoothly, increase it to a larger value.

```
n_qubits = 4 # Number of qubits
step = 0.0004 # Learning rate
batch_size = 4 # Number of samples for each training step
num_epochs = 1 # Number of training epochs
q_depth = 6 # Depth of the quantum circuit (number of variational layers)
gamma_lr_scheduler = 0.1 # Learning rate reduction applied every 10 epochs.
q_delta = 0.01 # Initial spread of random quantum weights
rng_seed = 0 # Seed for random number generator
start_time = time.time() # Start of the computation timer
```

We initialize a PennyLane device with a `default.qubit`

backend.

```
dev = qml.device("default.qubit", wires=n_qubits)
```

We configure PyTorch to use CUDA only if available. Otherwise the CPU is used.

```
device = torch.device("cuda:0" if torch.cuda.is_available() else "cpu")
```

## Dataset loading¶

Note

The dataset containing images of *ants* and *bees* can be downloaded
here and
should be extracted in the subfolder `../_data/hymenoptera_data`

.

This is a very small dataset (roughly 250 images), too small for training from scratch a
classical or quantum model, however it is enough when using *transfer learning* approach.

The PyTorch packages `torchvision`

and `torch.utils.data`

are used for loading the dataset
and performing standard preliminary image operations: resize, center, crop, normalize, *etc.*

```
data_transforms = {
"train": transforms.Compose(
[
# transforms.RandomResizedCrop(224), # uncomment for data augmentation
# transforms.RandomHorizontalFlip(), # uncomment for data augmentation
transforms.Resize(256),
transforms.CenterCrop(224),
transforms.ToTensor(),
# Normalize input channels using mean values and standard deviations of ImageNet.
transforms.Normalize([0.485, 0.456, 0.406], [0.229, 0.224, 0.225]),
]
),
"val": transforms.Compose(
[
transforms.Resize(256),
transforms.CenterCrop(224),
transforms.ToTensor(),
transforms.Normalize([0.485, 0.456, 0.406], [0.229, 0.224, 0.225]),
]
),
}
data_dir = "../_data/hymenoptera_data"
image_datasets = {
x: datasets.ImageFolder(os.path.join(data_dir, x), data_transforms[x]) for x in ["train", "val"]
}
dataset_sizes = {x: len(image_datasets[x]) for x in ["train", "val"]}
class_names = image_datasets["train"].classes
# Initialize dataloader
dataloaders = {
x: torch.utils.data.DataLoader(image_datasets[x], batch_size=batch_size, shuffle=True)
for x in ["train", "val"]
}
# function to plot images
def imshow(inp, title=None):
"""Display image from tensor."""
inp = inp.numpy().transpose((1, 2, 0))
# Inverse of the initial normalization operation.
mean = np.array([0.485, 0.456, 0.406])
std = np.array([0.229, 0.224, 0.225])
inp = std * inp + mean
inp = np.clip(inp, 0, 1)
plt.imshow(inp)
if title is not None:
plt.title(title)
```

Let us show a batch of the test data, just to have an idea of the classification problem.

```
# Get a batch of training data
inputs, classes = next(iter(dataloaders["val"]))
# Make a grid from batch
out = torchvision.utils.make_grid(inputs)
imshow(out, title=[class_names[x] for x in classes])
# In order to get reproducible results, we set a manual seed for the
# random number generator and re-initialize the dataloaders.
torch.manual_seed(rng_seed)
dataloaders = {
x: torch.utils.data.DataLoader(image_datasets[x], batch_size=batch_size, shuffle=True)
for x in ["train", "val"]
}
```

## Variational quantum circuit¶

We first define some quantum layers that will compose the quantum circuit.

```
def H_layer(nqubits):
"""Layer of single-qubit Hadamard gates.
"""
for idx in range(nqubits):
qml.Hadamard(wires=idx)
def RY_layer(w):
"""Layer of parametrized qubit rotations around the y axis.
"""
for idx, element in enumerate(w):
qml.RY(element, wires=idx)
def entangling_layer(nqubits):
"""Layer of CNOTs followed by another shifted layer of CNOT.
"""
# In other words it should apply something like :
# CNOT CNOT CNOT CNOT... CNOT
# CNOT CNOT CNOT... CNOT
for i in range(0, nqubits - 1, 2): # Loop over even indices: i=0,2,...N-2
qml.CNOT(wires=[i, i + 1])
for i in range(1, nqubits - 1, 2): # Loop over odd indices: i=1,3,...N-3
qml.CNOT(wires=[i, i + 1])
```

Now we define the quantum circuit through the PennyLane qnode decorator .

The structure is that of a typical variational quantum circuit:

**Embedding layer:**All qubits are first initialized in a balanced superposition of*up*and*down*states, then they are rotated according to the input parameters (local embedding).**Variational layers:**A sequence of trainable rotation layers and constant entangling layers is applied.**Measurement layer:**For each qubit, the local expectation value of the \(Z\) operator is measured. This produces a classical output vector, suitable for additional post-processing.

```
@qml.qnode(dev, interface="torch")
def q_net(q_in, q_weights_flat):
# Reshape weights
q_weights = q_weights_flat.reshape(q_depth, n_qubits)
# Start from state |+> , unbiased w.r.t. |0> and |1>
H_layer(n_qubits)
# Embed features in the quantum node
RY_layer(q_in)
# Sequence of trainable variational layers
for k in range(q_depth):
entangling_layer(n_qubits)
RY_layer(q_weights[k])
# Expectation values in the Z basis
exp_vals = [qml.expval(qml.PauliZ(position)) for position in range(n_qubits)]
return tuple(exp_vals)
```

## Dressed quantum circuit¶

We can now define a custom `torch.nn.Module`

representing a *dressed* quantum circuit.

This is a concatenation of:

- A classical pre-processing layer (
`nn.Linear`

). - A classical activation function (
`torch.tanh`

). - A constant
`np.pi/2.0`

scaling. - The previously defined quantum circuit (
`q_net`

). - A classical post-processing layer (
`nn.Linear`

).

The input of the module is a batch of vectors with 512 real parameters (features) and
the output is a batch of vectors with two real outputs (associated with the two classes
of images: *ants* and *bees*).

```
class Quantumnet(nn.Module):
def __init__(self):
super().__init__()
self.pre_net = nn.Linear(512, n_qubits)
self.q_params = nn.Parameter(q_delta * torch.randn(q_depth * n_qubits))
self.post_net = nn.Linear(n_qubits, 2)
def forward(self, input_features):
pre_out = self.pre_net(input_features)
q_in = torch.tanh(pre_out) * np.pi / 2.0
# Apply the quantum circuit to each element of the batch and append to q_out
q_out = torch.Tensor(0, n_qubits)
q_out = q_out.to(device)
for elem in q_in:
q_out_elem = q_net(elem, self.q_params).float().unsqueeze(0)
q_out = torch.cat((q_out, q_out_elem))
return self.post_net(q_out)
```

## Hybrid classical-quantum model¶

We are finally ready to build our full hybrid classical-quantum network.
We follow the *transfer learning* approach:

- First load the classical pre-trained network
*ResNet18*from the`torchvision.models`

zoo. - Freeze all the weights since they should not be trained.
- Replace the last fully connected layer with our trainable dressed quantum circuit (
`Quantumnet`

).

Note

The *ResNet18* model is automatically downloaded by PyTorch and it may take several minutes (only the first time).

```
model_hybrid = torchvision.models.resnet18(pretrained=True)
for param in model_hybrid.parameters():
param.requires_grad = False
# Notice that model_hybrid.fc is the last layer of ResNet18
model_hybrid.fc = Quantumnet()
# Use CUDA or CPU according to the "device" object.
model_hybrid = model_hybrid.to(device)
```

## Training and results¶

Before training the network we need to specify the *loss* function.

We use, as usual in classification problem, the *cross-entropy* which is
directly available within `torch.nn`

.

```
criterion = nn.CrossEntropyLoss()
```

We also initialize the *Adam optimizer* which is called at each training step
in order to update the weights of the model.

```
optimizer_hybrid = optim.Adam(model_hybrid.fc.parameters(), lr=step)
```

We schedule to reduce the learning rate by a factor of `gamma_lr_scheduler`

every 10 epochs.

```
exp_lr_scheduler = lr_scheduler.StepLR(optimizer_hybrid, step_size=10, gamma=gamma_lr_scheduler)
```

What follows is a training function that will be called later. This function should return a trained model that can be used to make predictions (classifications).

```
def train_model(model, criterion, optimizer, scheduler, num_epochs):
since = time.time()
best_model_wts = copy.deepcopy(model.state_dict())
best_acc = 0.0
best_loss = 10000.0 # Large arbitrary number
best_acc_train = 0.0
best_loss_train = 10000.0 # Large arbitrary number
print("Training started:")
for epoch in range(num_epochs):
# Each epoch has a training and validation phase
for phase in ["train", "val"]:
if phase == "train":
scheduler.step()
# Set model to training mode
model.train()
else:
# Set model to evaluate mode
model.eval()
running_loss = 0.0
running_corrects = 0
# Iterate over data.
n_batches = dataset_sizes[phase] // batch_size
it = 0
for inputs, labels in dataloaders[phase]:
since_batch = time.time()
batch_size_ = len(inputs)
inputs = inputs.to(device)
labels = labels.to(device)
optimizer.zero_grad()
# Track/compute gradient and make an optimization step only when training
with torch.set_grad_enabled(phase == "train"):
outputs = model(inputs)
_, preds = torch.max(outputs, 1)
loss = criterion(outputs, labels)
if phase == "train":
loss.backward()
optimizer.step()
# Print iteration results
running_loss += loss.item() * batch_size_
batch_corrects = torch.sum(preds == labels.data).item()
running_corrects += batch_corrects
print(
"Phase: {} Epoch: {}/{} Iter: {}/{} Batch time: {:.4f}".format(
phase,
epoch + 1,
num_epochs,
it + 1,
n_batches + 1,
time.time() - since_batch,
),
end="\r",
flush=True,
)
it += 1
# Print epoch results
epoch_loss = running_loss / dataset_sizes[phase]
epoch_acc = running_corrects / dataset_sizes[phase]
print(
"Phase: {} Epoch: {}/{} Loss: {:.4f} Acc: {:.4f} ".format(
"train" if phase == "train" else "val ",
epoch + 1,
num_epochs,
epoch_loss,
epoch_acc,
)
)
# Check if this is the best model wrt previous epochs
if phase == "val" and epoch_acc > best_acc:
best_acc = epoch_acc
best_model_wts = copy.deepcopy(model.state_dict())
if phase == "val" and epoch_loss < best_loss:
best_loss = epoch_loss
if phase == "train" and epoch_acc > best_acc_train:
best_acc_train = epoch_acc
if phase == "train" and epoch_loss < best_loss_train:
best_loss_train = epoch_loss
# Print final results
model.load_state_dict(best_model_wts)
time_elapsed = time.time() - since
print("Training completed in {:.0f}m {:.0f}s".format(time_elapsed // 60, time_elapsed % 60))
print("Best test loss: {:.4f} | Best test accuracy: {:.4f}".format(best_loss, best_acc))
return model
```

We are ready to perform the actual training process.

```
model_hybrid = train_model(
model_hybrid, criterion, optimizer_hybrid, exp_lr_scheduler, num_epochs=num_epochs
)
```

Out:

```
Training started:
Phase: train Epoch: 1/1 Iter: 1/62 Batch time: 1.0416
Phase: train Epoch: 1/1 Iter: 2/62 Batch time: 1.0628
Phase: train Epoch: 1/1 Iter: 3/62 Batch time: 1.0980
Phase: train Epoch: 1/1 Iter: 4/62 Batch time: 1.0756
Phase: train Epoch: 1/1 Iter: 5/62 Batch time: 1.0353
Phase: train Epoch: 1/1 Iter: 6/62 Batch time: 1.0371
Phase: train Epoch: 1/1 Iter: 7/62 Batch time: 1.0350
Phase: train Epoch: 1/1 Iter: 8/62 Batch time: 1.0460
Phase: train Epoch: 1/1 Iter: 9/62 Batch time: 1.0627
Phase: train Epoch: 1/1 Iter: 10/62 Batch time: 1.0691
Phase: train Epoch: 1/1 Iter: 11/62 Batch time: 1.0452
Phase: train Epoch: 1/1 Iter: 12/62 Batch time: 1.0399
Phase: train Epoch: 1/1 Iter: 13/62 Batch time: 1.0423
Phase: train Epoch: 1/1 Iter: 14/62 Batch time: 1.0627
Phase: train Epoch: 1/1 Iter: 15/62 Batch time: 1.0965
Phase: train Epoch: 1/1 Iter: 16/62 Batch time: 1.0545
Phase: train Epoch: 1/1 Iter: 17/62 Batch time: 1.0739
Phase: train Epoch: 1/1 Iter: 18/62 Batch time: 1.0924
Phase: train Epoch: 1/1 Iter: 19/62 Batch time: 1.2283
Phase: train Epoch: 1/1 Iter: 20/62 Batch time: 1.3366
Phase: train Epoch: 1/1 Iter: 21/62 Batch time: 1.1885
Phase: train Epoch: 1/1 Iter: 22/62 Batch time: 1.2182
Phase: train Epoch: 1/1 Iter: 23/62 Batch time: 1.0855
Phase: train Epoch: 1/1 Iter: 24/62 Batch time: 1.0557
Phase: train Epoch: 1/1 Iter: 25/62 Batch time: 1.1228
Phase: train Epoch: 1/1 Iter: 26/62 Batch time: 1.1052
Phase: train Epoch: 1/1 Iter: 27/62 Batch time: 1.0694
Phase: train Epoch: 1/1 Iter: 28/62 Batch time: 1.1452
Phase: train Epoch: 1/1 Iter: 29/62 Batch time: 1.1402
Phase: train Epoch: 1/1 Iter: 30/62 Batch time: 1.0965
Phase: train Epoch: 1/1 Iter: 31/62 Batch time: 1.1065
Phase: train Epoch: 1/1 Iter: 32/62 Batch time: 1.1216
Phase: train Epoch: 1/1 Iter: 33/62 Batch time: 1.0986
Phase: train Epoch: 1/1 Iter: 34/62 Batch time: 1.1707
Phase: train Epoch: 1/1 Iter: 35/62 Batch time: 1.1692
Phase: train Epoch: 1/1 Iter: 36/62 Batch time: 1.2076
Phase: train Epoch: 1/1 Iter: 37/62 Batch time: 1.1741
Phase: train Epoch: 1/1 Iter: 38/62 Batch time: 1.0879
Phase: train Epoch: 1/1 Iter: 39/62 Batch time: 1.1154
Phase: train Epoch: 1/1 Iter: 40/62 Batch time: 1.0597
Phase: train Epoch: 1/1 Iter: 41/62 Batch time: 1.0976
Phase: train Epoch: 1/1 Iter: 42/62 Batch time: 1.2265
Phase: train Epoch: 1/1 Iter: 43/62 Batch time: 1.0875
Phase: train Epoch: 1/1 Iter: 44/62 Batch time: 1.0673
Phase: train Epoch: 1/1 Iter: 45/62 Batch time: 1.0922
Phase: train Epoch: 1/1 Iter: 46/62 Batch time: 1.1104
Phase: train Epoch: 1/1 Iter: 47/62 Batch time: 1.0512
Phase: train Epoch: 1/1 Iter: 48/62 Batch time: 1.0730
Phase: train Epoch: 1/1 Iter: 49/62 Batch time: 1.0650
Phase: train Epoch: 1/1 Iter: 50/62 Batch time: 1.1813
Phase: train Epoch: 1/1 Iter: 51/62 Batch time: 1.0717
Phase: train Epoch: 1/1 Iter: 52/62 Batch time: 1.1322
Phase: train Epoch: 1/1 Iter: 53/62 Batch time: 1.0649
Phase: train Epoch: 1/1 Iter: 54/62 Batch time: 1.0793
Phase: train Epoch: 1/1 Iter: 55/62 Batch time: 1.0442
Phase: train Epoch: 1/1 Iter: 56/62 Batch time: 1.1201
Phase: train Epoch: 1/1 Iter: 57/62 Batch time: 1.0951
Phase: train Epoch: 1/1 Iter: 58/62 Batch time: 1.1141
Phase: train Epoch: 1/1 Iter: 59/62 Batch time: 1.1266
Phase: train Epoch: 1/1 Iter: 60/62 Batch time: 1.1194
Phase: train Epoch: 1/1 Iter: 61/62 Batch time: 1.1297
Phase: train Epoch: 1/1 Loss: 0.6916 Acc: 0.5164
Phase: val Epoch: 1/1 Iter: 1/39 Batch time: 0.1179
Phase: val Epoch: 1/1 Iter: 2/39 Batch time: 0.1126
Phase: val Epoch: 1/1 Iter: 3/39 Batch time: 0.1387
Phase: val Epoch: 1/1 Iter: 4/39 Batch time: 0.1366
Phase: val Epoch: 1/1 Iter: 5/39 Batch time: 0.1341
Phase: val Epoch: 1/1 Iter: 6/39 Batch time: 0.1329
Phase: val Epoch: 1/1 Iter: 7/39 Batch time: 0.1265
Phase: val Epoch: 1/1 Iter: 8/39 Batch time: 0.1522
Phase: val Epoch: 1/1 Iter: 9/39 Batch time: 0.1357
Phase: val Epoch: 1/1 Iter: 10/39 Batch time: 0.1348
Phase: val Epoch: 1/1 Iter: 11/39 Batch time: 0.1275
Phase: val Epoch: 1/1 Iter: 12/39 Batch time: 0.1266
Phase: val Epoch: 1/1 Iter: 13/39 Batch time: 0.1628
Phase: val Epoch: 1/1 Iter: 14/39 Batch time: 0.1884
Phase: val Epoch: 1/1 Iter: 15/39 Batch time: 0.1469
Phase: val Epoch: 1/1 Iter: 16/39 Batch time: 0.1136
Phase: val Epoch: 1/1 Iter: 17/39 Batch time: 0.1351
Phase: val Epoch: 1/1 Iter: 18/39 Batch time: 0.1665
Phase: val Epoch: 1/1 Iter: 19/39 Batch time: 0.1777
Phase: val Epoch: 1/1 Iter: 20/39 Batch time: 0.1867
Phase: val Epoch: 1/1 Iter: 21/39 Batch time: 0.1790
Phase: val Epoch: 1/1 Iter: 22/39 Batch time: 0.1734
Phase: val Epoch: 1/1 Iter: 23/39 Batch time: 0.1929
Phase: val Epoch: 1/1 Iter: 24/39 Batch time: 0.1432
Phase: val Epoch: 1/1 Iter: 25/39 Batch time: 0.1634
Phase: val Epoch: 1/1 Iter: 26/39 Batch time: 0.1357
Phase: val Epoch: 1/1 Iter: 27/39 Batch time: 0.1374
Phase: val Epoch: 1/1 Iter: 28/39 Batch time: 0.1383
Phase: val Epoch: 1/1 Iter: 29/39 Batch time: 0.1908
Phase: val Epoch: 1/1 Iter: 30/39 Batch time: 0.1146
Phase: val Epoch: 1/1 Iter: 31/39 Batch time: 0.1267
Phase: val Epoch: 1/1 Iter: 32/39 Batch time: 0.1037
Phase: val Epoch: 1/1 Iter: 33/39 Batch time: 0.2038
Phase: val Epoch: 1/1 Iter: 34/39 Batch time: 0.2544
Phase: val Epoch: 1/1 Iter: 35/39 Batch time: 0.1697
Phase: val Epoch: 1/1 Iter: 36/39 Batch time: 0.1494
Phase: val Epoch: 1/1 Iter: 37/39 Batch time: 0.1444
Phase: val Epoch: 1/1 Iter: 38/39 Batch time: 0.1379
Phase: val Epoch: 1/1 Iter: 39/39 Batch time: 0.0692
Phase: val Epoch: 1/1 Loss: 0.6475 Acc: 0.6732
Training completed in 1m 18s
Best test loss: 0.6475 | Best test accuracy: 0.6732
```

## Visualizing the model predictions¶

We first define a visualization function for a batch of test data.

```
def visualize_model(model, num_images=6, fig_name="Predictions"):
images_so_far = 0
_fig = plt.figure(fig_name)
model.eval()
with torch.no_grad():
for _i, (inputs, labels) in enumerate(dataloaders["val"]):
inputs = inputs.to(device)
labels = labels.to(device)
outputs = model(inputs)
_, preds = torch.max(outputs, 1)
for j in range(inputs.size()[0]):
images_so_far += 1
ax = plt.subplot(num_images // 2, 2, images_so_far)
ax.axis("off")
ax.set_title("[{}]".format(class_names[preds[j]]))
imshow(inputs.cpu().data[j])
if images_so_far == num_images:
return
```

Finally, we can run the previous function to see a batch of images with the corresponding predictions.

```
visualize_model(model_hybrid, num_images=batch_size)
plt.show()
```

## References¶

[1] Andrea Mari, Thomas R. Bromley, Josh Izaac, Maria Schuld, and Nathan Killoran.
*Transfer learning in hybrid classical-quantum neural networks*. arXiv:1912.08278 (2019).

[2] Rajat Raina, Alexis Battle, Honglak Lee, Benjamin Packer, and Andrew Y Ng.
*Self-taught learning: transfer learning from unlabeled data*.
Proceedings of the 24th International Conference on Machine Learning*, 759–766 (2007).

[3] Kaiming He, Xiangyu Zhang, Shaoqing ren and Jian Sun. *Deep residual learning for image recognition*.
Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 770-778 (2016).

[4] Ville Bergholm, Josh Izaac, Maria Schuld, Christian Gogolin, Carsten Blank, Keri McKiernan, and Nathan Killoran.
*PennyLane: Automatic differentiation of hybrid quantum-classical computations*. arXiv:1811.04968 (2018).

**Total running time of the script:** ( 1 minutes 19.194 seconds)

## Contents

## Downloads