I understand that creating normally distributed random numbers using quantum circuits is not straightforward due to the inherent probabilistic nature of quantum measurements, which follows a Born rule distribution rather than a normal distribution.

However, I used the well-known technique called Box-Muller transformation, which generates normally distributed numbers from uniformly distributed ones. Here’s a simple example using PennyLane to generate uniformly distributed random numbers:

```
import pennylane as qml
from pennylane import numpy as np
import matplotlib.pyplot as plt
from scipy.special import kl_div
n_qubits = 10
shots = 10000
dev = qml.device('default.qubit', wires=n_qubits, shots=shots)
@qml.qnode(dev)
def qnode():
for i in range(n_qubits):
qml.Hadamard(wires=i)
return [qml.sample(qml.PauliZ(wires=i)) for i in range(n_qubits)]
def generate_normal_distribution_quantum():
samples = qnode() # this gives you a list of numpy arrays
samples = np.concatenate(samples) # concatenate arrays into one
samples = (samples + 1) / 2 # map the {-1, 1} outcomes to {0, 1}
# the random number generated from each qubit is now uniformly distributed in {0, 1}
uniform_random_numbers = samples.flatten()
u1 = uniform_random_numbers[::2]
u2 = uniform_random_numbers[1::2]
# if u1 contains zero, replace it with another random number
u1 = np.where(u1 == 0, np.random.uniform(1e-9, 1.0, size=len(u1)), u1)
# create normally distributed numbers from uniformly distributed ones using Box-Muller transformation
normal_random_numbers_quantum = np.sqrt(-2 * np.log(u1)) * np.cos(2 * np.pi * u2)
# remove NaN values if any
normal_random_numbers_quantum = normal_random_numbers_quantum[~np.isnan(normal_random_numbers_quantum)]
return normal_random_numbers_quantum
def generate_normal_distribution_numpy(size):
normal_random_numbers_numpy = np.random.normal(0, 1, size)
return normal_random_numbers_numpy
def calculate_kl_divergence(dist1, dist2, bins=50):
hist1, bins1 = np.histogram(dist1, bins=bins, density=True)
hist2, _ = np.histogram(dist2, bins=bins1, density=True)
# Calculate KL divergence
kl_divergence = kl_div(hist1+1e-8, hist2+1e-8).sum()
return kl_divergence
def plot_distributions(dist1, dist2, bins=50):
plt.figure(figsize=(10, 6))
plt.hist(dist1, bins=bins, density=True, alpha=0.5, label='Quantum Generated')
plt.hist(dist2, bins=bins, density=True, alpha=0.5, label='Numpy Generated')
plt.title('Comparison of normal distributions')
plt.legend()
plt.show()
def main():
quantum_dist = generate_normal_distribution_quantum()
numpy_dist = generate_normal_distribution_numpy(len(quantum_dist))
kl_divergence = calculate_kl_divergence(numpy_dist, quantum_dist)
print(f"KL Divergence: {kl_divergence}")
plot_distributions(quantum_dist, numpy_dist)
if __name__ == "__main__":
main()
```

However, the plots do not make sense, can anyone take a look?