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Questa pagina è stata generata da docs/tutorials/05_torch_connector.ipynb.

Torch Connector e QNNs Ibride

Questo tutorial introduce la classe di Qiskit TorchConnector e dimostra come essa permetta una facile integrazione di qualsiasi NeuralNetwork da Qiskit Machine Learning in un flusso di lavoro in PyTorch. TorchConnector rende disponibile una NeuralNetwork di Qiskit come un Module di PyTorch. Il modulo che ne risulta può essere perfettamente incorporato in una classica architettura PyTorch e allenato senza considerazioni addizionali, permettendo quindi lo sviluppo ed il testing di nuove architetture di machine learning ibride quantistiche-classiche.

Contenuto:

Parte 1: Semplice Classificazione & Regressione

La prima parte di questo tutorial dimostra come, per semplici compiti di classificazione e regressione, si possa allenare una quantum neural network usando un motore di differenziazione automatica di PyTorch (torch.autograd, link).

  1. Classificazione

    1. Classification with PyTorch and EstimatorQNN

    2. Classification with PyTorch and SamplerQNN

  2. Regressione

    1. Regression with PyTorch and SamplerQNN

Parte 2: Classificazione su MNIST, QNNs Ibride

La seconda parte di questo tutorial mostra come incorporare una (Quantum) NeuralNetwork in un flusso di lavoro specifico di PyTorch (in questo caso una tipica architettura di CNN) al il fine di classificare i dati MNIST in modalità ibrida quantistica-classica.


[1]:
# Necessary imports

import numpy as np
import matplotlib.pyplot as plt

from torch import Tensor
from torch.nn import Linear, CrossEntropyLoss, MSELoss
from torch.optim import LBFGS

from qiskit import QuantumCircuit
from qiskit.utils import algorithm_globals
from qiskit.circuit import Parameter
from qiskit.circuit.library import RealAmplitudes, ZZFeatureMap
from qiskit_machine_learning.neural_networks import SamplerQNN, EstimatorQNN
from qiskit_machine_learning.connectors import TorchConnector

# Set seed for random generators
algorithm_globals.random_seed = 42

Parte 1: Semplice Classificazione & Regressione

1. Classificazione

Per prima cosa, mostriamo come TorchConnector permetta di allenare una Quantum NeuralNetwork per risolvere i compiti di classificazione usando il motore di differenziazione automatica in PyTorch. Per illustrare ciò, eseguiremo una classificazione binaria su un dataset generato in modo casuale.

[2]:
# Generate random dataset

# Select dataset dimension (num_inputs) and size (num_samples)
num_inputs = 2
num_samples = 20

# Generate random input coordinates (X) and binary labels (y)
X = 2 * algorithm_globals.random.random([num_samples, num_inputs]) - 1
y01 = 1 * (np.sum(X, axis=1) >= 0)  # in { 0,  1}, y01 will be used for SamplerQNN example
y = 2 * y01 - 1  # in {-1, +1}, y will be used for EstimatorQNN example

# Convert to torch Tensors
X_ = Tensor(X)
y01_ = Tensor(y01).reshape(len(y)).long()
y_ = Tensor(y).reshape(len(y), 1)

# Plot dataset
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()
../_images/tutorials_05_torch_connector_4_0.png

A. Classification with PyTorch and EstimatorQNN

Linking an EstimatorQNN to PyTorch is relatively straightforward. Here we illustrate this by using the EstimatorQNN constructed from a feature map and an ansatz.

[3]:
# Set up a circuit
feature_map = ZZFeatureMap(num_inputs)
ansatz = RealAmplitudes(num_inputs)
qc = QuantumCircuit(num_inputs)
qc.compose(feature_map, inplace=True)
qc.compose(ansatz, inplace=True)
qc.draw("mpl")
[3]:
../_images/tutorials_05_torch_connector_6_0.png
[4]:
# Setup QNN
qnn1 = EstimatorQNN(
    circuit=qc, input_params=feature_map.parameters, weight_params=ansatz.parameters
)

# Set up PyTorch module
# Note: If we don't explicitly declare the initial weights
# they are chosen uniformly at random from [-1, 1].
initial_weights = 0.1 * (2 * algorithm_globals.random.random(qnn1.num_weights) - 1)
model1 = TorchConnector(qnn1, initial_weights=initial_weights)
print("Initial weights: ", initial_weights)
Initial weights:  [-0.01256962  0.06653564  0.04005302 -0.03752667  0.06645196  0.06095287
 -0.02250432 -0.04233438]
[5]:
# Test with a single input
model1(X_[0, :])
[5]:
tensor([-0.3285], grad_fn=<_TorchNNFunctionBackward>)
Optimizer

La scelta dell’optimizer può essere cruciale per il successo dell’addestramento di qualsiasi modello di machine learning. Quando usiamo TorchConnector, abbiamo accesso a tutti gli algoritmi di ottimizzazione definiti nel package [torch.optim] (link). Alcuni degli algoritmi più famosi usati in popolari architetture di machine learning includono Adam, SGD, o Adagrad. Ad ogni modo, in questo tutorial useremo l’algoritmo L-BFGS (torch.optim.LBFGS), che è uno uno dei più conosciuti algoritmi di ottimizzazione di secondo-ordine per compiti di ottimizzazione numerica.

Funzione di Costo

Per quanto riguarda la funzioni di costo (loss function), possiamo sfruttare i moduli predefiniti di PyTorch che si posso trovare in torch.nn, come ad esempio le loss Cross-Entropy o Mean Squared Error.

💡 Clarification : In classical machine learning, the general rule of thumb is to apply a Cross-Entropy loss to classification tasks, and MSE loss to regression tasks. However, this recommendation is given under the assumption that the output of the classification network is a class probability value in the \([0, 1]\) range (usually this is achieved through a Softmax layer). Because the following example for EstimatorQNN does not include such layer, and we don’t apply any mapping to the output (the following section shows an example of application of parity mapping with SamplerQNNs), the QNN’s output can take any value in the range \([-1, 1]\). In case you were wondering, this is the reason why this particular example uses MSELoss for classification despite it not being the norm (but we encourage you to experiment with different loss functions and see how they can impact training results).

[6]:
# Define optimizer and loss
optimizer = LBFGS(model1.parameters())
f_loss = MSELoss(reduction="sum")

# Start training
model1.train()  # set model to training mode


# Note from (https://pytorch.org/docs/stable/optim.html):
# Some optimization algorithms such as LBFGS need to
# reevaluate the function multiple times, so you have to
# pass in a closure that allows them to recompute your model.
# The closure should clear the gradients, compute the loss,
# and return it.
def closure():
    optimizer.zero_grad()  # Initialize/clear gradients
    loss = f_loss(model1(X_), y_)  # Evaluate loss function
    loss.backward()  # Backward pass
    print(loss.item())  # Print loss
    return loss


# Run optimizer step4
optimizer.step(closure)
25.535646438598633
22.696760177612305
20.039228439331055
19.687908172607422
19.267208099365234
19.025373458862305
18.154708862304688
17.337854385375977
19.082578659057617
17.073287963867188
16.21839141845703
14.992582321166992
14.929339408874512
14.914533615112305
14.907636642456055
14.902364730834961
14.902134895324707
14.90211009979248
14.902111053466797
[6]:
tensor(25.5356, grad_fn=<MseLossBackward0>)
[7]:
# Evaluate model and compute accuracy
y_predict = []
for x, y_target in zip(X, y):
    output = model1(Tensor(x))
    y_predict += [np.sign(output.detach().numpy())[0]]

print("Accuracy:", sum(y_predict == y) / len(y))

# 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()
Accuracy: 0.8
../_images/tutorials_05_torch_connector_11_1.png

I cerchi rossi indicano dei dati che sono stati classificati in modo errato.

B. Classification with PyTorch and SamplerQNN

Linking a SamplerQNN to PyTorch requires a bit more attention than EstimatorQNN. Without the correct setup, backpropagation is not possible.

In particolare, dobbiamo assicurarci di restituire un vettore di probabilità denso nel pass forward della rete (sparse=False). Questo parametro è impostato a False di default, quindi dobbiamo solo essere sicuri che non venga modificato.

⚠️ Attention: If we define a custom interpret function ( in the example: parity), we must remember to explicitly provide the desired output shape ( in the example: 2). For more info on the initial parameter setup for SamplerQNN, please check out the official qiskit documentation.

[8]:
# Define feature map and ansatz
feature_map = ZZFeatureMap(num_inputs)
ansatz = RealAmplitudes(num_inputs, entanglement="linear", reps=1)

# Define quantum circuit of num_qubits = input dim
# Append feature map and ansatz
qc = QuantumCircuit(num_inputs)
qc.compose(feature_map, inplace=True)
qc.compose(ansatz, inplace=True)

# Define SamplerQNN and initial setup
parity = lambda x: "{:b}".format(x).count("1") % 2  # optional interpret function
output_shape = 2  # parity = 0, 1
qnn2 = SamplerQNN(
    circuit=qc,
    input_params=feature_map.parameters,
    weight_params=ansatz.parameters,
    interpret=parity,
    output_shape=output_shape,
)

# Set up PyTorch module
# Reminder: If we don't explicitly declare the initial weights
# they are chosen uniformly at random from [-1, 1].
initial_weights = 0.1 * (2 * algorithm_globals.random.random(qnn2.num_weights) - 1)
print("Initial weights: ", initial_weights)
model2 = TorchConnector(qnn2, initial_weights)
Initial weights:  [ 0.0364991  -0.0720495  -0.06001836 -0.09852755]

Per un ripasso su come scegliere gli optimizer e le funzione di costo, puoi andare indietro a questa sezione.

[9]:
# Define model, optimizer, and loss
optimizer = LBFGS(model2.parameters())
f_loss = CrossEntropyLoss()  # Our output will be in the [0,1] range

# Start training
model2.train()

# Define LBFGS closure method (explained in previous section)
def closure():
    optimizer.zero_grad(set_to_none=True)  # Initialize gradient
    loss = f_loss(model2(X_), y01_)  # Calculate loss
    loss.backward()  # Backward pass

    print(loss.item())  # Print loss
    return loss


# Run optimizer (LBFGS requires closure)
optimizer.step(closure);
0.6925069093704224
0.6881508231163025
0.6516683101654053
0.6485998034477234
0.6394743919372559
0.7057444453239441
0.669085681438446
0.766187310218811
0.7188469171524048
0.7919709086418152
0.7598814964294434
0.7028256058692932
0.7486447095870972
0.6890242695808411
0.7760348916053772
0.7892935276031494
0.7556288242340088
0.7058126330375671
0.7203161716461182
0.7030722498893738
[10]:
# Evaluate model and compute accuracy
y_predict = []
for x in X:
    output = model2(Tensor(x))
    y_predict += [np.argmax(output.detach().numpy())]

print("Accuracy:", sum(y_predict == y01) / len(y01))

# plot results
# red == wrongly classified
for x, y_target, y_ 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_:
        plt.scatter(x[0], x[1], s=200, facecolors="none", edgecolors="r", linewidths=2)
plt.plot([-1, 1], [1, -1], "--", color="black")
plt.show()
Accuracy: 0.5
../_images/tutorials_05_torch_connector_17_1.png

I cerchi rossi indicano dei dati che sono stati classificati in modo errato.

2. Regressione

We use a model based on the EstimatorQNN to also illustrate how to perform a regression task. The chosen dataset in this case is randomly generated following a sine wave.

[11]:
# Generate random dataset

num_samples = 20
eps = 0.2
lb, ub = -np.pi, np.pi
f = lambda x: np.sin(x)

X = (ub - lb) * algorithm_globals.random.random([num_samples, 1]) + lb
y = f(X) + eps * (2 * algorithm_globals.random.random([num_samples, 1]) - 1)
plt.plot(np.linspace(lb, ub), f(np.linspace(lb, ub)), "r--")
plt.plot(X, y, "bo")
plt.show()
../_images/tutorials_05_torch_connector_20_0.png

A. Regression with PyTorch and EstimatorQNN

The network definition and training loop will be analogous to those of the classification task using EstimatorQNN. In this case, we define our own feature map and ansatz, but let’s do it a little different.

[12]:
# Construct simple feature map
param_x = Parameter("x")
feature_map = QuantumCircuit(1, name="fm")
feature_map.ry(param_x, 0)

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

qc = QuantumCircuit(1)
qc.compose(feature_map, inplace=True)
qc.compose(ansatz, inplace=True)

# Construct QNN
qnn3 = EstimatorQNN(circuit=qc, input_params=[param_x], weight_params=[param_y])

# Set up PyTorch module
# Reminder: If we don't explicitly declare the initial weights
# they are chosen uniformly at random from [-1, 1].
initial_weights = 0.1 * (2 * algorithm_globals.random.random(qnn3.num_weights) - 1)
model3 = TorchConnector(qnn3, initial_weights)

Per un ripasso su come scegliere gli optimizer e le funzione di costo, puoi andare indietro a questa sezione.

[13]:
# Define optimizer and loss function
optimizer = LBFGS(model3.parameters())
f_loss = MSELoss(reduction="sum")

# Start training
model3.train()  # set model to training mode

# Define objective function
def closure():
    optimizer.zero_grad(set_to_none=True)  # Initialize gradient
    loss = f_loss(model3(Tensor(X)), Tensor(y))  # Compute batch loss
    loss.backward()  # Backward pass
    print(loss.item())  # Print loss
    return loss


# Run optimizer
optimizer.step(closure)
14.947757720947266
2.948650360107422
8.952412605285645
0.37905153632164
0.24995625019073486
0.2483610212802887
0.24835753440856934
[13]:
tensor(14.9478, grad_fn=<MseLossBackward0>)
[14]:
# Plot target function
plt.plot(np.linspace(lb, ub), f(np.linspace(lb, ub)), "r--")

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

# Plot fitted line
y_ = []
for x in np.linspace(lb, ub):
    output = model3(Tensor([x]))
    y_ += [output.detach().numpy()[0]]
plt.plot(np.linspace(lb, ub), y_, "g-")
plt.show()
../_images/tutorials_05_torch_connector_26_0.png

Part 2: Classificazione su MNIST, QNNs Ibride

In questa seconda parte, mostriamo come sfruttare una neural network ibrida quantistica-classica, usando TorchConnector, per eseguire una complessa classificazione sul dataset MNIST che contiene immagini di numeri scritti a mano.

Puoi leggere la sezione dedicata nel Qiskit Textbook. per una più dettagliata spiegazione sulle neural network ibride quantistico-classiche (pre-TorchConnector).

[15]:
# Additional torch-related imports
import torch
from torch import cat, no_grad, manual_seed
from torch.utils.data import DataLoader
from torchvision import datasets, transforms
import torch.optim as optim
from torch.nn import (
    Module,
    Conv2d,
    Linear,
    Dropout2d,
    NLLLoss,
    MaxPool2d,
    Flatten,
    Sequential,
    ReLU,
)
import torch.nn.functional as F

Step 1: Definire i Data-loader per train e test

Utilizziamo la API torchvision per scaricare direttamente un sottoinsieme del MNIST dataset e definire un torch DataLoader (link) per train e test.

[16]:
# Train Dataset
# -------------

# Set train shuffle seed (for reproducibility)
manual_seed(42)

batch_size = 1
n_samples = 100  # We will concentrate on the first 100 samples

# Use pre-defined torchvision function to load MNIST train data
X_train = datasets.MNIST(
    root="./data", train=True, download=True, transform=transforms.Compose([transforms.ToTensor()])
)

# Filter out labels (originally 0-9), leaving only labels 0 and 1
idx = np.append(
    np.where(X_train.targets == 0)[0][:n_samples], np.where(X_train.targets == 1)[0][:n_samples]
)
X_train.data = X_train.data[idx]
X_train.targets = X_train.targets[idx]

# Define torch dataloader with filtered data
train_loader = DataLoader(X_train, batch_size=batch_size, shuffle=True)

Con una rapida visualizzazione, possiamo osservare che il train dataset consiste di immagini di 0 e 1 scritti a mano.

[17]:
n_samples_show = 6

data_iter = iter(train_loader)
fig, axes = plt.subplots(nrows=1, ncols=n_samples_show, figsize=(10, 3))

while n_samples_show > 0:
    images, targets = data_iter.__next__()

    axes[n_samples_show - 1].imshow(images[0, 0].numpy().squeeze(), cmap="gray")
    axes[n_samples_show - 1].set_xticks([])
    axes[n_samples_show - 1].set_yticks([])
    axes[n_samples_show - 1].set_title("Labeled: {}".format(targets[0].item()))

    n_samples_show -= 1
../_images/tutorials_05_torch_connector_33_0.png
[18]:
# Test Dataset
# -------------

# Set test shuffle seed (for reproducibility)
# manual_seed(5)

n_samples = 50

# Use pre-defined torchvision function to load MNIST test data
X_test = datasets.MNIST(
    root="./data", train=False, download=True, transform=transforms.Compose([transforms.ToTensor()])
)

# Filter out labels (originally 0-9), leaving only labels 0 and 1
idx = np.append(
    np.where(X_test.targets == 0)[0][:n_samples], np.where(X_test.targets == 1)[0][:n_samples]
)
X_test.data = X_test.data[idx]
X_test.targets = X_test.targets[idx]

# Define torch dataloader with filtered data
test_loader = DataLoader(X_test, batch_size=batch_size, shuffle=True)

Step 2: Definire la QNN ed il Modello Ibrido

This second step shows the power of the TorchConnector. After defining our quantum neural network layer (in this case, a EstimatorQNN), we can embed it into a layer in our torch Module by initializing a torch connector as TorchConnector(qnn).

⚠️ Attenzione: Per avere una adeguate backpropagation dei gradienti dei modelli ibridi DOBBIAMO impostare il parametro iniziale input_gradients a TRUE durante l’inializzazione della qnn.

[19]:
# Define and create QNN
def create_qnn():
    feature_map = ZZFeatureMap(2)
    ansatz = RealAmplitudes(2, reps=1)
    qc = QuantumCircuit(2)
    qc.compose(feature_map, inplace=True)
    qc.compose(ansatz, inplace=True)

    # REMEMBER TO SET input_gradients=True FOR ENABLING HYBRID GRADIENT BACKPROP
    qnn = EstimatorQNN(
        circuit=qc,
        input_params=feature_map.parameters,
        weight_params=ansatz.parameters,
        input_gradients=True,
    )
    return qnn


qnn4 = create_qnn()
[20]:
# Define torch NN module


class Net(Module):
    def __init__(self, qnn):
        super().__init__()
        self.conv1 = Conv2d(1, 2, kernel_size=5)
        self.conv2 = Conv2d(2, 16, kernel_size=5)
        self.dropout = Dropout2d()
        self.fc1 = Linear(256, 64)
        self.fc2 = Linear(64, 2)  # 2-dimensional input to QNN
        self.qnn = TorchConnector(qnn)  # Apply torch connector, weights chosen
        # uniformly at random from interval [-1,1].
        self.fc3 = Linear(1, 1)  # 1-dimensional output from QNN

    def forward(self, x):
        x = F.relu(self.conv1(x))
        x = F.max_pool2d(x, 2)
        x = F.relu(self.conv2(x))
        x = F.max_pool2d(x, 2)
        x = self.dropout(x)
        x = x.view(x.shape[0], -1)
        x = F.relu(self.fc1(x))
        x = self.fc2(x)
        x = self.qnn(x)  # apply QNN
        x = self.fc3(x)
        return cat((x, 1 - x), -1)


model4 = Net(qnn4)

Step 3: Addestramento (Training)

[21]:
# Define model, optimizer, and loss function
optimizer = optim.Adam(model4.parameters(), lr=0.001)
loss_func = NLLLoss()

# Start training
epochs = 10  # Set number of epochs
loss_list = []  # Store loss history
model4.train()  # Set model to training mode

for epoch in range(epochs):
    total_loss = []
    for batch_idx, (data, target) in enumerate(train_loader):
        optimizer.zero_grad(set_to_none=True)  # Initialize gradient
        output = model4(data)  # Forward pass
        loss = loss_func(output, target)  # Calculate loss
        loss.backward()  # Backward pass
        optimizer.step()  # Optimize weights
        total_loss.append(loss.item())  # Store loss
    loss_list.append(sum(total_loss) / len(total_loss))
    print("Training [{:.0f}%]\tLoss: {:.4f}".format(100.0 * (epoch + 1) / epochs, loss_list[-1]))
Training [10%]  Loss: -1.1630
Training [20%]  Loss: -1.5294
Training [30%]  Loss: -1.7855
Training [40%]  Loss: -1.9863
Training [50%]  Loss: -2.2257
Training [60%]  Loss: -2.4513
Training [70%]  Loss: -2.6758
Training [80%]  Loss: -2.8832
Training [90%]  Loss: -3.1006
Training [100%] Loss: -3.3061
[22]:
# Plot loss convergence
plt.plot(loss_list)
plt.title("Hybrid NN Training Convergence")
plt.xlabel("Training Iterations")
plt.ylabel("Neg. Log Likelihood Loss")
plt.show()
../_images/tutorials_05_torch_connector_41_0.png

Now we’ll save the trained model, just to show how a hybrid model can be saved and re-used later for inference. To save and load hybrid models, when using the TorchConnector, follow the PyTorch recommendations of saving and loading the models.

[23]:
torch.save(model4.state_dict(), "model4.pt")

Step 4: Valutazione del modello (Evaluation)

We start from recreating the model and loading the state from the previously saved file. You create a QNN layer using another simulator or a real hardware. So, you can train a model on real hardware available on the cloud and then for inference use a simulator or vice verse. For a sake of simplicity we create a new quantum neural network in the same way as above.

[24]:
qnn5 = create_qnn()
model5 = Net(qnn5)
model5.load_state_dict(torch.load("model4.pt"))
[24]:
<All keys matched successfully>
[25]:
model5.eval()  # set model to evaluation mode
with no_grad():

    correct = 0
    for batch_idx, (data, target) in enumerate(test_loader):
        output = model5(data)
        if len(output.shape) == 1:
            output = output.reshape(1, *output.shape)

        pred = output.argmax(dim=1, keepdim=True)
        correct += pred.eq(target.view_as(pred)).sum().item()

        loss = loss_func(output, target)
        total_loss.append(loss.item())

    print(
        "Performance on test data:\n\tLoss: {:.4f}\n\tAccuracy: {:.1f}%".format(
            sum(total_loss) / len(total_loss), correct / len(test_loader) / batch_size * 100
        )
    )
Performance on test data:
        Loss: -3.3585
        Accuracy: 100.0%
[26]:
# Plot predicted labels

n_samples_show = 6
count = 0
fig, axes = plt.subplots(nrows=1, ncols=n_samples_show, figsize=(10, 3))

model5.eval()
with no_grad():
    for batch_idx, (data, target) in enumerate(test_loader):
        if count == n_samples_show:
            break
        output = model5(data[0:1])
        if len(output.shape) == 1:
            output = output.reshape(1, *output.shape)

        pred = output.argmax(dim=1, keepdim=True)

        axes[count].imshow(data[0].numpy().squeeze(), cmap="gray")

        axes[count].set_xticks([])
        axes[count].set_yticks([])
        axes[count].set_title("Predicted {}".format(pred.item()))

        count += 1
../_images/tutorials_05_torch_connector_48_0.png

🎉🎉🎉🎉 Ora sei in grado di sperimentare con architetture ibride personalizzate e con nuovi dataset usando Qiskit Machine Learning. Buona Fortuna!

[27]:
import qiskit.tools.jupyter

%qiskit_version_table
%qiskit_copyright

Version Information

Qiskit SoftwareVersion
qiskit-terra0.22.0
qiskit-aer0.11.1
qiskit-ignis0.7.0
qiskit0.33.0
qiskit-machine-learning0.5.0
System information
Python version3.7.9
Python compilerMSC v.1916 64 bit (AMD64)
Python builddefault, Aug 31 2020 17:10:11
OSWindows
CPUs4
Memory (Gb)31.837730407714844
Thu Nov 03 09:57:38 2022 GMT Standard Time

This code is a part of Qiskit

© Copyright IBM 2017, 2022.

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.