# Normally distributed random numbers using quantum circuits **URL:** \ **Category:** PennyLane Help\ **Created:** [June 17, 2023, 8:53am UTC](https://discuss.pennylane.ai/t/normally-distributed-random-numbers-using-quantum-circuits/3063 "2023-06-17T08:53:50Z")\ **Posts on this page:** 8\ **Page:** 1
**Author:** ![Solomon](https://yyz2.discourse-cdn.com/flex012/user_avatar/discuss.pennylane.ai/solomon/32/1706_2.png) [@Solomon](https://discuss.pennylane.ai/u/Solomon)\ **Post date:** [June 17, 2023, 8:53am UTC](https://discuss.pennylane.ai/t/normally-distributed-random-numbers-using-quantum-circuits/3063/1 "2023-06-17T08:53:50Z")
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: ```auto 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? ![image](https://canada1.discourse-cdn.com/flex012/uploads/pennylane/original/2X/a/adc1ab188486cc4709c0484eb5a1c391a6a7642a.png) ---
**Author:** ![isaacdevlugt](https://yyz2.discourse-cdn.com/flex012/user_avatar/discuss.pennylane.ai/isaacdevlugt/32/1159_2.png) [@isaacdevlugt](https://discuss.pennylane.ai/u/isaacdevlugt)\ **Post date:** [June 19, 2023, 2:45pm UTC](https://discuss.pennylane.ai/t/normally-distributed-random-numbers-using-quantum-circuits/3063/2 "2023-06-19T14:45:16Z")
Hey @Solomon, I’m not super familiar with this transformation, but a couple things about your implementation stand out as being slightly questionable for me: 1. `u1` and `u2` need to be uniformly sampled in the interval (0, 1). In your `generate_normal_distribution_quantum` function, `u1` and `u2` don’t appear to be that 🤔. 2. Related to the first point, but it makes more sense to me to have a quantum circuit that directly generates uniformly distributed samples in (0, 1) (or -1 to 1 with appropriate post-processing). Best of luck! ---
**Author:** ![Solomon](https://yyz2.discourse-cdn.com/flex012/user_avatar/discuss.pennylane.ai/solomon/32/1706_2.png) [@Solomon](https://discuss.pennylane.ai/u/Solomon)\ **Post date:** [June 20, 2023, 11:02am UTC](https://discuss.pennylane.ai/t/normally-distributed-random-numbers-using-quantum-circuits/3063/3 "2023-06-20T11:02:34Z")
Issac thanks, Yes … this was late night programming 🙂 My goal here is to implement a Quantum Generative Adversarial Network (QGAN). A critical component of this endeavor is the ability to sample from a normal distribution. However, given the constraints of the available quantum random number generator, which only produces 1 and -1, there’s a necessity to create continuous random numbers between -1 and 1 (or 0 and 1). The question at hand is whether it’s feasible to train and optimize a Parametrized Quantum Circuit (PQC) to generate a Gaussian distribution with a zero mean. If this is possible, the corresponding code would be very beneficial. I saw this example: [Quantum generative adversarial networks with Cirq + TensorFlow](https://pennylane.ai/qml/demos/tutorial_QGAN) but, as far as I can see, it is NOT drawing samples from a true gaussian distribution. ---
**Author:** ![Solomon](https://yyz2.discourse-cdn.com/flex012/user_avatar/discuss.pennylane.ai/solomon/32/1706_2.png) [@Solomon](https://discuss.pennylane.ai/u/Solomon)\ **Post date:** [June 20, 2023, 12:24pm UTC](https://discuss.pennylane.ai/t/normally-distributed-random-numbers-using-quantum-circuits/3063/4 "2023-06-20T12:24:48Z")
I found this which was very informative: [https://arxiv.org/pdf/1904.00043.pdf](https://arxiv.org/pdf/1904.00043.pdf) But still requires an involved training scheme. ---
**Author:** ![isaacdevlugt](https://yyz2.discourse-cdn.com/flex012/user_avatar/discuss.pennylane.ai/isaacdevlugt/32/1159_2.png) [@isaacdevlugt](https://discuss.pennylane.ai/u/isaacdevlugt)\ **Post date:** [June 20, 2023, 1:35pm UTC](https://discuss.pennylane.ai/t/normally-distributed-random-numbers-using-quantum-circuits/3063/5 "2023-06-20T13:35:18Z")
> A critical component of this endeavor is the ability to sample from a normal distribution. Are you talking about how the parameters are initialized? If so, I think `np.random.normal` should suffice 🙂. Maybe getting a quantum circuit to replicate this behaviour is a little overkill! ---
**Author:** ![Solomon](https://yyz2.discourse-cdn.com/flex012/user_avatar/discuss.pennylane.ai/solomon/32/1706_2.png) [@Solomon](https://discuss.pennylane.ai/u/Solomon)\ **Post date:** [June 20, 2023, 1:39pm UTC](https://discuss.pennylane.ai/t/normally-distributed-random-numbers-using-quantum-circuits/3063/6 "2023-06-20T13:39:42Z")
Hello Issac, No, I’m referring to the distribution Z, from which the generator of the GAN pulls its input samples. ---
**Author:** ![Solomon](https://yyz2.discourse-cdn.com/flex012/user_avatar/discuss.pennylane.ai/solomon/32/1706_2.png) [@Solomon](https://discuss.pennylane.ai/u/Solomon)\ **Post date:** [June 21, 2023, 1:45pm UTC](https://discuss.pennylane.ai/t/normally-distributed-random-numbers-using-quantum-circuits/3063/7 "2023-06-21T13:45:18Z")
I found this in Julia, which does exactly what I want to do with pennylane: [https://docs.yaoquantum.org/dev/generated/examples/6.quantum-circuit-born-machine/index.html](https://docs.yaoquantum.org/dev/generated/examples/6.quantum-circuit-born-machine/index.html) Do you know of any successful attempt at this using PennyLane? Thanks. ---
**Author:** ![isaacdevlugt](https://yyz2.discourse-cdn.com/flex012/user_avatar/discuss.pennylane.ai/isaacdevlugt/32/1159_2.png) [@isaacdevlugt](https://discuss.pennylane.ai/u/isaacdevlugt)\ **Post date:** [June 21, 2023, 2:53pm UTC](https://discuss.pennylane.ai/t/normally-distributed-random-numbers-using-quantum-circuits/3063/8 "2023-06-21T14:53:17Z")
> No, I’m referring to the distribution Z, from which the generator of the GAN pulls its input samples. I’m not sure I completely understand, but if it’s the samples that the generator _generates_ and then feeds into the discriminator, then the underlying distribution that defines how the generator generates samples won’t be a normal distribution in general. The generator itself can be thought of as some complex, multi-dimensional probability distribution 🙂. > I found this in Julia, which does exactly what I want to do with pennylane: > [Quantum Circuit Born Machine · Documentation | Yao](https://docs.yaoquantum.org/dev/generated/examples/6.quantum-circuit-born-machine/index.html) The circuit used here (screenshot) can most certainly be implemented in PennyLane 🙂. It was simply trained to learn a gaussian distribution. You’d be able to follow pretty much exactly what’s outlined there in Yao, but in PennyLane! ![Screenshot 2023-06-21 at 10.51.04 AM](https://canada1.discourse-cdn.com/flex012/uploads/pennylane/original/2X/8/87c55a9223222ca484a023a8679d24a72a43a2eb.png)