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#  $\langle$QonFusion$\rangle$\- Quantum Approaches to Gaussian Random
Variables: Applications in Stable Diffusion and Brownian Motion.

Shlomo Kashani $\Phi$ Applied Physics, Johns Hopkins University, Maryland
U.S.A<EMAIL_ADDRESS>
###### Abstract

In the present study, we delineate a strategy focused on non-parametric
quantum circuits for the generation of Gaussian random variables (GRVs). This
quantum-centric approach serves as a substitute for conventional pseudorandom
number generators (PRNGs), such as the torch.rand function in PyTorch. The
principal theme of our research is the incorporation of Quantum Random Number
Generators (QRNGs) into classical models of diffusion. Notably, our Quantum
Gaussian Random Variable Generator fulfills dual roles, facilitating
simulations in both Stable Diffusion (SD) and Brownian Motion (BM). This
diverges markedly from prevailing methods that utilize parametric quantum
circuits (PQCs), often in conjunction with variational quantum eigensolvers
(VQEs). Although conventional techniques can accurately approximate ground
states in complex systems or model elaborate probability distributions, they
require a computationally demanding optimization process to tune parameters.
Our non-parametric strategy obviates this necessity. To facilitate
assimilating our methodology into existing computational frameworks, we put
forward QonFusion, a Python library congruent with both PyTorch and PennyLane,
functioning as a bridge between classical and quantum computational paradigms.
We validate QonFusion through extensive statistical testing, including tests
which confirm the statistical equivalence of the Gaussian samples from our
quantum approach to classical counterparts within defined significance limits.
QonFusion is available at
https://boltzmannentropy.github.io/qonfusion.github.io/ to reproduce all
findings here.

###### Contents

  1. 1 Introduction
    1. 1.1 The Convergence of Quantum Computing and Generative AI models
    2. 1.2 A general overview of our pipeline
    3. 1.3 Diffusion Models
      1. 1.3.1 Stable Diffusion
      2. 1.3.2 Brownian Motion
  2. 2 The Fundamentals of Classical Pseudo-Random Number Generation
    1. 2.1 The Marsaglia Polar Method Explained
  3. 3 Quantum Random Number Generators (QRNGs)
    1. 3.1 Parametric Quantum Circuits (PQCs)
    2. 3.2 Non-Parametric Quantum Circuits
    3. 3.3 Generation of Uniform Random Numbers
    4. 3.4 Sampling vs. Measuring Expectation Values
    5. 3.5 Generating Uniformly Distributed Rotations
    6. 3.6 Generating GRV’s
  4. 4 Results
    1. 4.1 Statistical Evaluation of Quantum-Generated Distributions
    2. 4.2 Quantum-Augmented Gaussian Noise in Forward Diffusion Processes
    3. 4.3 Quantum-Enhanced Gaussian Noise in Brownian Motion Simulations
  5. 5 Discussion
  6. 6 Acknowledgements.

##  1 Introduction

###  1.1 The Convergence of Quantum Computing and Generative AI models

Our investigations into QRNGs reveal a broad spectrum of potential
applications. Chief among these is the ability for QRNGs to serve as direct
replacements for traditional random number generators like torch.rand or
numpy.rand, which are routinely employed in the realm of deep learning.
Furthermore, QRNGs are effective in introducing Gaussian noise during the
forward pass of SD models. They also offer valuable perspectives for
simulating reversible Stochastic Differential Equations (SDEs), such as BM. A
noteworthy discovery, referenced in section (1.3.2), is the finding that even
a minimal deviation from a zero mean in the Gaussian Probability Density
Function (PDF) during a BM simulation via QRNG results in irreversibility.
This utility is especially relevant in the current scientific landscape where
deep learning intersects extensively with quantum many-body systems,
particularly in high-energy physics [45]. Simultaneously, the interface
between quantum computing and quantum neural networks is attracting
substantial academic interest.

While these research directions might initially appear disparate, they
inherently exhibit complementary attributes, particularly in the realm of
generative SD models [21]. Foundational contributions from Dhariwal et al.
[14] and Moghadam et al. [34] have highlighted the efficacy of diffusion
models in the creation of both general-purpose and medical images. Such models
present a viable substitute for the labor-intensive and economically taxing
process of manual tumor annotation.

Given these insights, an imperative question arises: Is it feasible to
substitute all elements of a classical Machine Learning (ML) diffusion model
with their quantum equivalents? The answer, although layered, is primarily
negative. As a case in point, consider the U-Net architecture [40], renowned
for its original purpose in image segmentation. It has since been adapted to
perform denoising functions within diffusion models. This dual-functionality
architecture, comprising both an encoder and a decoder, can generate noise-
highlighting masks that can be integrated into specific denoising algorithms.
Yet, when one peels back the layers of the quantum realm, it quickly becomes
apparent that the field is less developed than its classical counterpart.

To bridge our initial foray into QRNGs with the more nuanced world of Quantum
Machine Learning (QML), it’s crucial to establish the state-of-play in both
the classical and quantum settings. QML, unlike its classical ML analogue,
lacks well-defined quantum versions of key architectural elements such as
encoders and decoders. This shortfall presents a significant barrier to the
complete transition from classical to quantum frameworks in diffusion models.
Consequently, our current research aims are situated in the early stages of
classical diffusion models, specifically focusing on incorporating and
simulating basic quantum circuits for QRNGs, as illustrated in Figure 1 [18,
11]. Though this focus might appear limiting, it serves as a gateway to a
broader, yet largely unexplored, landscape of opportunities.

Most of our experiments are conducted on classical computing platforms,
harnessing the power of PennyLane’s quantum simulator backends to serve as the
computational engines for our quantum circuits. Optimised for computational
efficiency, these backends are well-suited for numerical simulations in
quantum computing, particularly those pertinent to SD or BM. Finally, it’s
important to note that our chosen default parameters are not intended to
represent optimal configurations but rather to serve as a reference framework
for evaluating Gaussian Random Variables (GRVs). A more detailed account of
our research methodology will follow in subsequent sections of this paper.

In light of these developments, it becomes pertinent to explore how quantum
computing can further enrich the field of generative models, particularly
diffusion models. The following sections delve into this subject, examining
the foundational principles of diffusion models and their potential for
quantum augmentation.

Figure 1: An illustrative depiction of our proposed quantum pipeline. 1
Utilizing $N$ qubits, an initial non-parametric (3.2) quantum circuit, founded
on Hadamard transformations, engenders a uniform distribution (3.3) across
$2^{N}$ discrete values through sampled measurement outcomes, rather than
expectation values (further elucidated in 3). These outcomes are subsequently
transmuted into equidistant angles, which are then channeled into a static
quantum circuit that neither adapts nor learns, with the purpose of inducing
stochastic rotations. 2 This static assembly, encompassing $M>N$ qubits,
functions as follows: a Hadamard gate is applied to each qubit, thus creating
a superposition of states. Upon activation of the useRot flag, the circuit
administers a rotation (3.5) to every qubit, utilizing the previously
generated angles as the parameters for these rotations. The output 3 from this
circuit is then employed in the 4 Marsaglia polar method to transmute the
uniform distribution into a zero-mean Gaussian 5. This transformation is
achieved through the equation $Z=\sqrt{-2\log U}\cos(2\pi V)$, where $U$ and
$V$ are two uniform pseudo-random numbers (PRN’s), and by substituting our
quantumly-generated uniform random bits for $U$ and $V$, we synthesize a
source of quantum (LABEL:sec:guss) Gaussian random variables. 6 The Marsaglia
polar method serves as a paradigmatic example of how quantum random bits can
seamlessly supplant classical PRN’s.

###  1.2 A general overview of our pipeline

In the hybrid quantum-classical framework we describe, as illustrated in
Figure 1, we make use of two individual quantum circuits in a sequential
manner. The first circuit, which utilises $N$ qubits, aims to produce a
uniform distribution over $2^{N}$ unique states. These states are subsequently
converted into equiangular values. These equiangular values are then fed into
the second, pre-determined quantum circuit. This latter circuit, containing
$M>N$ qubits, applies a Hadamard gate to each qubit and carries out rotations
as determined by the angles generated in the first circuit. The output from
this series of quantum operations is then used in the classical Marsaglia
polar method to transform the uniform distribution into a Gaussian
distribution centred at zero. This serves as an illustrative example of how
QRNGs can replace classical PRNGs in models like SD or BM, particularly during
the forward diffusion stage. The numerical trials are conducted on idealised
quantum simulators running on classical hardware, thereby approximating the
essential elements of diffusion methodologies.

As we transition from the quantum-classical hybrid framework to a more focused
discussion on diffusion models, it is crucial to remember that our ultimate
aim is to explore the quantum-classical interface. The subsequent section on
diffusion models should be viewed as an extension of this aim, offering a
detailed background against which our quantum contributions can be more fully
understood.

###  1.3 Diffusion Models

Our work is fundamentally anchored in the generation of pseudo-random
sequences, a critical aspect of diffusion techniques that often takes up a
significant portion of the computational resources. While this is a focal
point of our study, it’s worth noting that the applications of SD models are
manifold, including, but not limited to, the realms of speech, image, and
audio generation, as well as denoising techniques [41, 17, 51, 2, 31]. Given
the extensive scope of these applications, a complete discussion in this
article would be impractical.

Figure 2: An illustrative schema elucidating the employment of quantum-sourced
uniform random variables for the generation of Gaussian noise, specifically
for the perturbation of images in a Stable Diffusion context. 1 Presents a
histogram of the quantum-derived uniform random variables. 2 Demonstrates the
Gaussian distribution fitted to the uniform variables, facilitated by the
Marsaglia polar method. 3 Exhibits an image in two dimensions that has been
disrupted by quantum-originated Gaussian noise.

Figure 2 captures a singular moment from an animated sequence that visualizes
the operational stages of the QRNG. The complete animation can be viewed at
https://boltzmannentropy.github.io/qonfusion.github.io/. This visualization
serves as a practical illustration of our central research theme: the
integration of QRNGs into classical diffusion models. While the field of
diffusion models is vast and rich in applications, our focus remains on
identifying areas where quantum computing, particularly QRNGs, can make a
meaningful impact.

####  1.3.1 Stable Diffusion

Building on the theme of QRNG integration, the domain of diffusion models is
rich in seminal contributions. Works by Sohl-Dickstein et al. [43] and Song et
al. [44, 46, 42] stand as milestones in establishing the theoretical
underpinnings of this field. Their groundbreaking research has set the stage
for the work by Ho et al. [23], which offers a unified, practical framework
that has become the go-to standard for multiple implementations. For readers
seeking a deeper theoretical dive, the studies by Song et al. [46, 42] are
invaluable. On the mathematical front, Luo et al. [29] have compiled essential
formulations for anyone interested in a comprehensive understanding of
diffusion models. Furthermore, for an overview of the diversity of
applications, one cannot overlook the work by Yang et al. [52].

####  1.3.2 Brownian Motion

Having established the foundational principles and key contributions in the
domain of SD models, it is instructive to delve into a specific, yet seminal,
stochastic process that often serves as the underpinning of a different group
of diffusion models. This leads us to the discussion of Brownian Motion, a
process that not only has historical significance but also offers a robust
mathematical framework for understanding diffusion dynamics. BM is a
cornerstone of stochastic processes and is unique for being the first to
incorporate continuous time and state variables, thus leaving an indelible
mark on subsequent research, notably in Gaussian processes, martingales, and
Markov processes. Initially brought to light by Einstein [15], BM provides a
robust framework for the analytical representation of the random motion of
particles in fluids, a subject further explored by Perrin [37], Chandrasekhar
[13], and Langevin [27]. The process is governed by stochastic differential
equations and, distinctively, employs Gaussian-distributed stochastic
variables for its diffusion dynamics. This stands in contrast to other
stochastic processes, which often resort to Poisson or Bernoulli distributions
[19, 48].

The reason we are testing our QRNG in BM is that BM is highly sensitive to any
deviations in the mean of the Gaussian distribution, and this serves as a
benchmark for our quantum generator. Our findings concerning the application
of our QRNG to Brownian Motion simulations are detailed in Section 4.3,
entitled Quantum-Augmented Gaussian Fluctuations in Simulations of Brownian
Motion.

##  2 The Fundamentals of Classical Pseudo-Random Number Generation

As a prelude to our exploration of quantum-generated random numbers, it is
instructive to delineate the foundational methods employed in classical
computing to produce uniform and Gaussian distributions [1]. Classical Pseudo-
Random Numbers (PRNs) are generated using deterministic algorithms that offer
repeatability while producing a sequence of numbers that seemingly fall within
the random range of $(0,1)$ [9, 7]. The effectiveness of such generators is
gauged through two primary criteria:

  1. 1.

The Length of the Period: This measure signifies the duration before the
generated sequence repeats. Period lengths below $2^{32}$ are generally
considered insufficient, and a minimum period length of $2^{40}$ is usually
recommended for robust applications. For further insights into the nuances of
such sequences, refer to the works by [6, 7].

  2. 2.

Robustness in Statistical Terms: Any generator worth its salt will have
undergone meticulous statistical evaluation to ascertain its quality of
’randomness’. Software that passes these stringent tests is considered
reliable.

Moving forward, we delve into the techniques used to generate numbers
following a non-uniform, specifically Gaussian, distribution. The mathematical
expression for the probability density function (PDF) of such a Gaussian
distribution can be articulated as:

$P(x)=\exp\left(-\frac{(x-x_{0})^{2}}{2\sigma^{2}}\right)$ (1)

The Central Limit Theorem comes to the fore here. The theorem posits that the
summation of a large number of independent random variables tends to a
Gaussian distribution, with its width inversely related to $\sqrt{N}$.
Therefore, it’s not a stretch to infer that by summing $n$ uniform random
numbers, one can approximate a Gaussian distribution. The closer $n$ is to a
large number, the more faithfully the resulting distribution will mirror a
Gaussian form. To customise this distribution with a particular mean $\bar{x}$
and width $\sigma$, the following transformation of the summation $S$ of $n$
uniform random variables can be employed:

$x=\bar{x}-2\sigma\left(\frac{S}{n}-\frac{1}{2}\right)\sqrt{3}$ (2)

It’s worth mentioning that the traditional approach of approximating a
Gaussian distribution through the summation of $n$ uniform random variables is
not computationally efficient. This is because generating a single Gaussian
Random Variable (GRV) [9, 3, 7] requires the prior generation of $n$ uniform
random variables within the range $(0,1)$. While classical methods for
generating these uniform random variables abound, our attention is rather
riveted on how to transform quantum-generated uniform random numbers into GRVs
via classical computational techniques. Here, we elucidate four key methods
for this purpose:

  1. 1.

The Box-Muller Technique : This method stands out for its computational
efficiency and is well-adopted in practice. It takes two independent uniform
random variables as inputs and produces two independent standard GRVs as
outputs [33].

  2. 2.

Marsaglia’s Polar Approach : This is essentially a variant of the Box-Muller
method but avoids the use of trigonometric functions, offering a quicker
alternative in certain scenarios [24].

  3. 3.

Marsaglia’s Ziggurat Method : Known for its efficiency in generating normal
and exponential random variables, this method employs precomputed tables,
making it especially useful for large-scale simulations [35].

  4. 4.

Inverse Cumulative Distribution Function (CDF) Transformation : Though
computationally more demanding, this method has the flexibility to handle any
distribution with a known inverse CDF. It transforms a uniform random variable
into a GRV through the inverse of the CDF [50].

###  2.1 The Marsaglia Polar Method Explained

GRVs play a pivotal role in the forward phase of SD models, serving as the
source of Gaussian noise introduced into images. These variables find
extensive utility not only in machine learning but also in the physical
sciences. A standard GRV, denoted $Z$, has a mean $\mu$ of zero and a variance
$\sigma^{2}$ of one. Its Probability Density Function (PDF) and Cumulative
Distribution Function (CDF) are described as follows:

$\displaystyle\phi(z)$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{1}{2}z^{2}\right),$ (3)
$\displaystyle\Phi(z)$
$\displaystyle=\mathbb{P}[Z<z]=\int_{-\infty}^{z}\phi(s)\,\mathrm{d}s.$ (4)

The Marsaglia method, which can be viewed in Algorithm 2.1, commences by
generating two uniform random variables, $u1$ and $u2$. These are subsequently
transformed to lie within the interval $[-1,1]$, creating $u$ and $v$. Here,
$u$ and $v$ act as Cartesian coordinates on a two-dimensional plane. The
algorithm’s crux is to ascertain if the point $(u,v)$ resides within a unit
circle. This is determined by calculating $s=u^{2}+v^{2}$ and ensuring that
$s<1$ and $s\neq 0$. Should the point fall outside the unit circle, a new set
of $u$ and $v$ is generated. Once a valid point is identified, a scaling
factor is computed using $s$, and this factor is applied to $u$ and $v$ to
produce $z1$ and $z2$, two normally distributed variables. The algorithm then
outputs $z2$. Herein, we offer an exhaustive account of the Marsaglia method’s
inner workings.

The Marsaglia Polar Method for Producing Normally Distributed Random Variables

1:Initialise $u1,u2,u,v,s,z1,z2$

2:while True do

3: $u1,u2\sim U(0,1)$ $\triangleright$ Quantum uniform random number generator

4: $u=2u1-1$

5: $v=2u2-1$

6: $s=u^{2}+v^{2}$

7: if $s\geq 1$ or $s=0$ then

8: Continue $\triangleright$ Revert to step one

9: else

10: $z1=u\sqrt{-2\ln s/s}$

11: $z2=v\sqrt{-2\ln s/s}$

12: break

13: end if

14:end while

15:return $z2$

##  3 Quantum Random Number Generators (QRNGs)

While the prevailing literature often emphasises Parametric Quantum Circuits
(PQCs) [32, 16] and variational quantum eigensolvers (VQEs) [38, 25], our
approach diverges by adopting a non-parametric methodology, obviating the need
for hybrid optimisation strategies. For completeness, we first expound upon
the parametric methods.

###  3.1 Parametric Quantum Circuits (PQCs)

The quantum circuitry employed for generating GRVs has garnered significant
attention, particularly within the context of Quantum Circuit Born Machines
(QCBMs) [12]. A recurring theme in current methodologies is the integration of
quantum and classical systems. Romero and Aspuru-Guzik [39], for instance,
employed a variational quantum generator within a Generative Adversarial
Network (GAN), with the quantum circuit functioning as the generator and a
classical neural network acting as the discriminator. Similarly, works by Gili
et al. [20] and Liu and Wang [30] have further enriched the landscape of
quantum-classical hybrid systems in the context of probability distribution
learning and optimisation.

###  3.2 Non-Parametric Quantum Circuits

Our methodology, depicted in Figure 1, centres around non-parametric quantum
circuits [10, 36, 5, 22]. It specifically aims at transforming uniform
distributions to Gaussian noise, with the details elaborated upon
subsequently. The quantum-enhanced implementation of our algorithm, delineated
in Code snippet 1, is modelled on Algorithm 2.1.

⬇

1 while True:

2 u1 = (self.q_uniform.Q_sample())

3 u2 = (self.q_uniform.Q_sample())

4 u = 2 * u1 - 1

5 v = 2 * u2 - 1

6 s = u ** 2 + v ** 2

7 if s < 1 and s != 0:

8 break

9 factor = (-2 * np.log(s) / s) ** 0.5

10 z1 = u * factor

11 z2 = v * factor

12 return torch.tensor(z1, dtype=float), torch.tensor(z2, dtype=float)

Listing 1: Quantum-Enhanced Marsaglia Algorithm for Gaussian Noise Generation

###  3.3 Generation of Uniform Random Numbers

Figure 3 portrays a quantum circuit designed for generating random rotations.
These rotations, functioning as non-trainable parameters, are subsequently
utilised in a separate ansatz to generate uniform distributions. The quantum
noise emanating from these circuits is employed in image corruption tasks and
the simulation of diffusion processes akin to Brownian motion.

(a)

⬇

1 def Q_PQC_ROT_ANZATS(n_qubits):

2 for i in range(n_qubits):

3 qml.Hadamard(wires=i)

4 _expectations = [qml.sample(qml.PauliZ(i)) for i in range(n_qubits)]

5 return _expectations

(b)

Figure 3: A Simplified Circuit Ansatz for Generating Random Rotations: (a)
Circuit Ansatz visualisation, (b) Corresponding Python code snippet.

###  3.4 Sampling vs. Measuring Expectation Values

In quantum computing, sampling and calculating expectation values are distinct
yet interconnected concepts. Much like expectation values, sampling plays a
crucial role in quantum algorithms and simulations. While qml.expval()
computes expectation values without collapsing the wavefunction, qml.sample()
performs a measurement that results in wavefunction collapse, yielding a
specific eigenvalue of the measured operator [4].

######  Definition 1.

(Expectation Values) The expectation value of an operator $\hat{O}$ for a
state $|\Psi\rangle$ is the average value of the observable corresponding to
$\hat{O}$, given by

$\langle\hat{O}\rangle=\langle\Psi|\hat{O}|\Psi\rangle.$ (5)

######  Example 3.1.

For a qubit in the state
$|\Psi\rangle=\alpha|\uparrow\rangle+\beta|\downarrow\rangle$, the expectation
value of the Pauli-Z operator can be calculated as follows:

$\displaystyle\langle\hat{\sigma}_{z}\rangle$
$\displaystyle=\langle\Psi|\hat{\sigma}_{z}|\Psi\rangle,$ (6)
$\displaystyle=\left(\alpha^{*}\langle\uparrow|+\beta^{*}\langle\downarrow|\right)\hat{\sigma}_{z}\left(\alpha|\uparrow\rangle+\beta|\downarrow\rangle\right),$
(7)
$\displaystyle=\alpha^{*}\alpha\langle\uparrow|\hat{\sigma}_{z}|\uparrow\rangle+\beta^{*}\beta\langle\downarrow|\hat{\sigma}_{z}|\downarrow\rangle,$
(8) $\displaystyle=|\alpha|^{2}\cdot 1+|\beta|^{2}\cdot(-1),$ (9)
$\displaystyle=|\alpha|^{2}-|\beta|^{2}.$ (10)

######  Example 3.2.

Consider the state

$|\psi\rangle=\frac{3}{7}|00\rangle+\frac{6}{7}|01\rangle+\frac{2}{7}|10\rangle,$
(11)

and the operator $I\otimes\sigma_{z}$, representing the $\sigma_{z}$ operator
acting on qubit B. The expectation value of $\sigma_{z}$ for qubit B can be
calculated as:

$\displaystyle\langle\sigma_{z}\rangle_{B}$
$\displaystyle=\langle\psi|\left(I\otimes\sigma_{z}\right)|\psi\rangle$ (12)
$\displaystyle=\left(\frac{3}{7}\right)^{2}\cdot
1+\left(\frac{6}{7}\right)^{2}\cdot(-1)+\left(\frac{2}{7}\right)^{2}\cdot 1$
(13) $\displaystyle=\frac{9}{49}-\frac{36}{49}+\frac{4}{49}$ (14)
$\displaystyle=\frac{-23}{49}$ (15)

######  Definition 2.

(Sampling) Sampling in the context of a quantum state $|\Psi\rangle$ refers to
the process of performing a measurement on the state in a specific basis, such
as the Pauli-Z basis. The outcome is probabilistic, and the state collapses to
one of the eigenstates of the measured operator. Mathematically, if
$|\phi_{i}\rangle$ are the eigenstates of an operator $\hat{O}$ with
eigenvalues $o_{i}$, then sampling $|\Psi\rangle$ with respect to $\hat{O}$
yields $o_{i}$ with probability $|\langle\phi_{i}|\Psi\rangle|^{2}$.

######  Example 3.3.

Consider a qubit in the state
$|\Psi\rangle=\frac{1}{\sqrt{2}}(|\uparrow\rangle+|\downarrow\rangle)$. The
probabilities for each outcome when sampling this state in the Pauli-Z basis
can be calculated as follows:

$\displaystyle P(+1)$
$\displaystyle=\left|\langle\uparrow|\Psi\rangle\right|^{2}$ (16)
$\displaystyle=\left|\frac{1}{\sqrt{2}}\right|^{2}$ (17)
$\displaystyle=\frac{1}{2},$ (18) $\displaystyle P(-1)$
$\displaystyle=\left|\langle\downarrow|\Psi\rangle\right|^{2}$ (19)
$\displaystyle=\left|\frac{1}{\sqrt{2}}\right|^{2}$ (20)
$\displaystyle=\frac{1}{2}.$ (21)

Therefore, sampling would yield $+1$ with a probability of $0.5$ and $-1$ with
a probability of $0.5$.

######  Example 3.4.

For a two-qubit state

$|\psi\rangle=\frac{1}{\sqrt{3}}(|00\rangle+|01\rangle+|10\rangle),$ (22)

sampling the first qubit in the Pauli-Z basis would yield the following
probabilities:

$\displaystyle P(+1)$
$\displaystyle=\left|\frac{1}{\sqrt{3}}\right|^{2}+\left|\frac{1}{\sqrt{3}}\right|^{2}$
(23) $\displaystyle=\frac{1}{3}+\frac{1}{3}$ (24) $\displaystyle=\frac{2}{3},$
(25) $\displaystyle P(-1)$ $\displaystyle=\left|\frac{1}{\sqrt{3}}\right|^{2}$
(26) $\displaystyle=\frac{1}{3}.$ (27)

Hence, $+1$ would be obtained with a probability of $\frac{2}{3}$ and $-1$
with a probability of $\frac{1}{3}$.

We are concerned with sampling here. The primary distinction between the
expressions [qml.sample(qml.PauliZ(i)) ] and [qml.expval(qml.PauliZ(i))]
resides in the nature of the result they yield [4] . The former returns a
sampled measurement outcome, while the latter procures the expectation value.
qml.sample() executes a measurement on the qubit in the Pauli-Z basis, causing
the collapse of the wavefunction and randomly returning either 0 or 1 based on
the probabilities inherent to the qubit state. On the other hand, qml.expval()
computes the expectation value $\langle Z\rangle$ of the Pauli-Z operator on
the qubit, yielding the average value we would anticipate measuring, without
inducing the collapse of the wavefunction. In essence, qml.sample() provides a
random measurement sample (0 or 1) while qml.expval() offers the expected
average value of a measurement (a value between 0 and 1). Sampling induces the
collapse of the state, while expectation values permit further quantum
processing.

Figure 4: Plots of random rotation angles generated by a quantum circuit. The
circuit applies a Hadamard gate to each qubit, creating a superposition of
states. Each state is then measured in the Pauli-Z basis, resulting in a
series of 0s and 1s (binary outcomes). These binary outcomes are converted to
a binary index value, which is then normalised to a value between 0 and 1 by
dividing by the total number of possible states (the space span). Each plot
shows the distribution of these generated values for different numbers of
qubits. The limited range of qubit string bits is due to the finite number of
binary outcomes that can be generated by a given number of qubits.

The Python method toBinaryIndex performs the task of converting the sampling
outcomes to a binary index value. It takes a list of qubit measurements as
input and returns an integer value representing the binary index.

value $\displaystyle=\sum_{i=0}^{n-1}2^{i}\times\text{sample}_{i}$ (28)

The binary index is then normalised to a value between 0 and 1 by dividing it
by the total number of possible states, effectively spanning the state space.

Normalised Value $\displaystyle=\frac{\text{value}}{2^{n}-1}$ (29)

The distribution of these normalised values is plotted in Figure 4 for varying
numbers of qubits. The limited range of qubit string bits is a consequence of
the finite number of binary outcomes that can be generated by a given number
of qubits.

###  3.5 Generating Uniformly Distributed Rotations

As previously highlighted, a central premise of our manuscript is the
cascading of two quantum circuits, wherein the first circuit supplies the
second with random rotations, thereby introducing an additional layer of
stochasticity. Quantum rotations serve as the cornerstone for myriad quantum
manipulations and are essential for altering quantum states. Within this
article, we briefly examine the mathematical formulations of quantum
rotations, specifically concentrating on rotations about the $x,y$, and $z$
axes, symbolised as $R_{x},R_{y}$, and $R_{z}$ respectively.

Figure 5: Plots of random rotation angles generated by a quantum circuit. The
circuit employs a Hadamard gate on each qubit to create a superposition of
quantum states. Following state preparation, each qubit state is measured in
the Pauli-Z basis, yielding a series of binary outcomes, represented as 0s and
1s. These outcomes are then converted into a single binary index value, which
is normalized to a value between 0 and 1 by dividing by the total number of
possible states. This normalized value is subsequently scaled to a rotation
angle between 0 and $2\pi$ radians. Each plot illustrates the distribution of
these generated angles for varying numbers of qubits, depicted in polar
coordinates with the angle represented by the position on the circle. The
limited range of angles is a consequence of the finite number of binary
outcomes that can be generated by a given number of qubits.

######  Definition 3.

(Quantum Rotation Operators) The quantum rotation operators $R_{x}$, $R_{y}$,
and $R_{z}$ are defined as:

$\displaystyle R_{x}(\theta)=e^{-i\theta
X/2}=\cos\left(\frac{\theta}{2}\right)I-i\sin\left(\frac{\theta}{2}\right)X,$
(30) $\displaystyle R_{y}(\theta)=e^{-i\theta
Y/2}=\cos\left(\frac{\theta}{2}\right)I-i\sin\left(\frac{\theta}{2}\right)Y,$
(31) $\displaystyle R_{z}(\theta)=e^{-i\theta
Z/2}=\cos\left(\frac{\theta}{2}\right)I-i\sin\left(\frac{\theta}{2}\right)Z,$
(32)

where $X$, $Y$, and $Z$ are the Pauli matrices, $I$ is the identity matrix,
and $\theta$ is the angle of rotation.

######  Example 3.5.

The matrix representations of the quantum rotation operators $R_{x}$, $R_{y}$,
and $R_{z}$ are given by:

$\displaystyle R_{x}(\theta)$
$\displaystyle=\left[\begin{array}[]{cc}\cos\frac{\theta}{2}&-i\sin\frac{\theta}{2}\\\
-i\sin\frac{\theta}{2}&\cos\frac{\theta}{2}\end{array}\right],$ (35)
$\displaystyle R_{y}(\theta)$
$\displaystyle=\left[\begin{array}[]{cc}\cos\frac{\theta}{2}&-\sin\frac{\theta}{2}\\\
\sin\frac{\theta}{2}&\cos\frac{\theta}{2}\end{array}\right],$ (38)
$\displaystyle R_{z}(\theta)$
$\displaystyle=\left[\begin{array}[]{cc}e^{-i\frac{\theta}{2}}&0\\\
0&e^{i\frac{\theta}{2}}\end{array}\right].$ (41)

######  Example 3.6.

We consider a three-qubit quantum circuit where each qubit undergoes a
Hadamard transformation followed by a general rotation characterized by three
parameters: $\theta,\phi,\lambda$.

  1. 1.

Initial State: The initial state of the three qubits is prepared by another
quantum circuit consisting solely of Hadamard gates. Mathematically, this is
represented as:

$|\psi_{\text{initial}}\rangle=(H\otimes H\otimes H)|0\rangle^{\otimes 3}.$
(42)

  2. 2.

Total Operation: The total unitary operation acting on the initial state
consists of Hadamard gates followed by general rotations on each qubit. This
is given by:

$U_{\text{total}}=(U(\theta_{1},\phi_{1},\lambda_{1})\otimes
U(\theta_{2},\phi_{2},\lambda_{2})\otimes
U(\theta_{3},\phi_{3},\lambda_{3}))\cdot(H\otimes H\otimes H)\cdot(H\otimes
H\otimes H).$ (43)

  3. 3.

Final State: The final state of the system is then:

$|\psi_{\text{final}}\rangle=U_{\text{total}}|\psi_{\text{initial}}\rangle.$
(44)

Let’s consider a numerical example for the system. We’ll use the following
rotation angles for each qubit:  $\theta_{1}=\frac{\pi}{4}$
$\phi_{1}=\frac{\pi}{2}$ $\lambda_{1}=\pi$ $\theta_{2}=\frac{\pi}{3}$
$\phi_{2}=\frac{\pi}{4}$ $\lambda_{2}=\frac{\pi}{2}$
$\theta_{3}=\frac{\pi}{6}$ $\phi_{3}=\frac{\pi}{3}$
$\lambda_{3}=\frac{\pi}{4}$ The Hadamard gate $H$ is represented as:

$H=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix}$ (45)

And a general rotation $U(\theta,\phi,\lambda)$ is given by:

$U(\theta,\phi,\lambda)=\begin{pmatrix}\cos(\frac{\theta}{2})&-e^{i\lambda}\sin(\frac{\theta}{2})\\\
e^{i\phi}\sin(\frac{\theta}{2})&e^{i(\phi+\lambda)}\cos(\frac{\theta}{2})\end{pmatrix}$
(46)

First, we’ll find the initial state $|\psi_{\text{initial}}\rangle$ after
applying the Hadamard gates:

$|\psi_{\text{initial}}\rangle=\frac{1}{\sqrt{8}}(|000\rangle+|001\rangle+|010\rangle+|011\rangle+|100\rangle+|101\rangle+|110\rangle+|111\rangle)$
(47)

Next, we’ll find the unitary operation $U_{\text{total}}$ for the Hadamard and
rotation gates:

$U_{\text{total}}=(U(\theta_{1},\phi_{1},\lambda_{1})\otimes
U(\theta_{2},\phi_{2},\lambda_{2})\otimes
U(\theta_{3},\phi_{3},\lambda_{3}))\times(H\otimes H\otimes H)$ (48)

Finally, we’ll find the final state $|\psi_{\text{final}}\rangle$:

$|\psi_{\text{final}}\rangle=U_{\text{total}}|\psi_{\text{initial}}\rangle$
(49)

The evaluation of $|\psi_{\text{final}}\rangle$ is executed solely via
numerical software, notably the PennyLane quantum simulation library.

###  3.6 Generating GRV’s

In the present investigation, we solely employ the Marsaglia [47] polar
algorithm (2.1) as a mechanism for generating normally distributed random
numbers. While the extant literature presents a variety of enhancements or
alternatives to the Marsaglia polar method, such as the one mentioned above
(1), we have chosen not to explore these avenues, given the empirical efficacy
of the method under consideration. It is crucial to underscore that while
various studies propose potential refinements to the method, the selection of
an optimal approach is intrinsically tied to the specific requirements of the
application in focus. The prospect of harnessing the uncertainty of quantum
measurements for generative modelling presents an intriguing research
direction [39, 30]. The premise, posits that it is feasible to iteratively
superimpose tiny amounts of quantum noise Figure 6, denoted as $\epsilon$,
onto any image $X$ over $t$ timesteps, thereby transmuting $X$ into a pure
Gaussian noise sample $T$. In a reciprocal manner, given that $t$ is
adequately large the noise-laden sample $T$ can be reverted to the original,
noise-free image $X$ through the systematic elimination of the superimposed
noise.

Figure 6: This figure depicts the probability density function (PDF) of a
standard normal (Gaussian) distribution, generated from 2000 samples of a
quantum circuit. The red dashed lines mark positions of one, two, and three
standard deviations away from the mean, in both the positive and negative
directions [8]. The labels ’-1$\sigma$’, ’+1$\sigma$’, ’-2$\sigma$’,
’+2$\sigma$’, ’-3$\sigma$’, ’+3$\sigma$’ represent points one, two, and three
standard deviations away from the mean, respectively. This visualisation
assists in understanding the empirical rule for a standard normal
distribution, also known as the 68-95-99.7 rule.

In our software stack QonFusion, the QuantumRandomGenerator class is designed
to generate a uniform random distribution utilizing the quantum circuits
discussed above. The initialization of the class sets up a quantum device,
defines the number of qubits, the number of layers, and the quantum circuit.
An option to use rotations is also provided, which can be turned on or off
with the useRot flag. If the useRot flag is set, the generator employs the
QuantumRandomRotationGenerator to produce rotation angles that are
subsequently fed into the second quantum circuit. This circuit applies
Hadamard gates to each qubit, creating a superposition of states, and then
applies the rotations (if the useRot flag is set). Each qubit is then measured
in the Pauli-Z basis. The measurements produce a binary string, which is
converted to an index using the toBinaryIndex method. This method iterates
over the binary outcomes, treating each binary value as a digit in a binary
number and summing the corresponding powers of 2 to produce a decimal number.
The Q_uniform_rnd method runs the process, generating a random number
according to a uniform distribution. If the isIndex flag is set, it returns
the binary index directly. Otherwise, it normalizes the index to a value
between 0 and 1 by dividing by the total number of possible states (the space
span), effectively producing a continuous uniform random number. This process
exemplifies the potential of quantum computing in generating truly random
numbers and highlights the flexibility of quantum circuits in constructing
complex probability distributions.

##  4 Results

To test our quantum generative modeling techniques, we ran experiments both on
a simulator using classical computing resources and on actual quantum hardware
from IBM with 20 qubits. Running on the simulator allows us to validate the
theoretical performance of the quantum circuits before real-world execution.
We then benchmarked the same experiments on IBM’s quantum computer to compare
the practical results and quantify the error rates. While the simulator
produces idealized noise-free outputs, the real hardware introduces errors
from effects like decoherence and gate imperfections. By conducting
experiments under both simulated and real conditions, we can verify the
validity of our quantum approach and characterize the degree of deviation on
current noisy devices. Analyzing the discrepancies between these results is
crucial for determining the readiness of near-term quantum computers for
practical generative modeling. Our evaluation provides an empirical
demonstration of running quantum generation algorithms and contrasts the
accuracy achieved on simulators versus real quantum chips.

###  4.1 Statistical Evaluation of Quantum-Generated Distributions

Our investigation confirms that quantum circuits are capable of generating
both continuous uniform and Gaussian distributions. When employed as an
alternative to classical random number generators, quantum noise effectively
corrupts images for diffusion processes. The integrity of the quantum-
generated Gaussian distribution is corroborated through a series of
statistical tests [26, 28, 49], including a statistical permutation test, as
summarized in Table (1).

Test | Value
---|---
KS Statistic | 0.052
KS P-Value | 0.124
MMD | 0.001
KL Divergence | 0.030
Permutation test | 0.718
Table 1: The quantum random Gaussian generator underwent a battery of
statistical tests, juxtaposing the quantum-generated samples with classical
Gaussian samples. The Kolmogorov-Smirnov (KS) test yielded a statistic of
$S=0.052$ and a $p$-value of $p=0.124$. Given that the $p$-value exceeds the
conventional threshold of $0.05$, the null hypothesis, which posits that the
quantum and classical samples are drawn from the same continuous distribution,
could not be rejected. This suggests that the quantum samples do not
significantly deviate from their classical Gaussian counterparts. The Maximum
Mean Discrepancy (MMD), a measure that quantifies the difference between the
mean embeddings of two distributions in a Reproducing Kernel Hilbert Space
(RKHS), recorded a value of $MMD=0.001$. This minimal MMD value indicates an
inconsequential difference between the mean embeddings of the quantum and
classical Gaussian distributions in the RKHS. The Kullback-Leibler (KL)
divergence, which measures how much the quantum distribution diverges from the
classical Gaussian distribution, was $D_{KL}=0.030$. This modest KL divergence
value corroborates that the quantum distribution closely approximates the
classical Gaussian distribution. Finally, a statistical permutation test was
conducted, with high values suggesting that the quantum and classical samples
are indistinguishable. These results are consistent with the findings from the
KS test and MMD, further reinforcing the congruence between the quantum and
classical Gaussian distributions.

###  4.2 Quantum-Augmented Gaussian Noise in Forward Diffusion Processes

In this study, we conducted experiments with a variable number of qubits (4,
5, 6), different PennyLane devices (e.g., ‘qml.device(’default.qubit’)‘), and
varying numbers of shots. In PennyLane, as is the case with many quantum
simulators, the number of shots dictates the number of times the circuit is
executed.

Figure 7: This figure illuminates the role of Gaussian noise in forward
diffusion algorithms. These algorithms convert images into pure Gaussian noise
through the sequential addition of small noise perturbations. While the
reverse operation, which utilizes a U-Net neural architecture to remove the
noise, is not examined in this study, it plays a crucial role in reverting the
image to its original, noise-free state. The efficacy of stochastic diffusion
(SD) models is largely dependent on the successful integration of Gaussian
noise, an aspect that could be further optimized by employing quantum random
number generation (QRNG).

###  4.3 Quantum-Enhanced Gaussian Noise in Brownian Motion Simulations

This section provides an illustration of the utility of our Quantum Gaussian
generator in simulating Brownian motion, serving as a quality metric for our
QRNG. In conventional Brownian motion, any deviation of the Gaussian
distribution’s mean from zero disrupts the motion’s fidelity to true Brownian
behaviour (Theorem (1)); we prove this in Proof (4.3) . Our QRNG, however,
accurately encapsulates the fundamental characteristics of Brownian motion.

Figure 8: This figure showcases the efficacy of our Quantum Gaussian generator
in simulating Brownian motion. The plot serves as a quality metric for our
Quantum Random Number Generator (QRNG). Unlike conventional Brownian motion,
where any deviation from a zero mean in the Gaussian distribution disrupts the
fidelity to true Brownian behavior, our QRNG maintains this fidelity. The plot
illustrates that the generated Brownian motion closely adheres to the
theoretical expectations, thereby validating the utility of quantum-enhanced
Gaussian noise in stochastic simulations.

######  Definition 4.

Standard Brownian motion $W(t)$ is a stochastic process that satisfies the
following properties:  1. $W(0)=0$ 2. $W(t)$ has independent increments. 3.
$W(t)-W(s)\sim\mathcal{N}(0,t-s)$ for $0\leq s<t$. 4. $W(t)$ is continuous in
$t$.

######  Theorem 1.

Any deviation of the Gaussian distribution’s mean from zero disrupts the
motion’s fidelity to true Brownian behavior.

###### Proof.

Let’s consider a Gaussian distribution with a mean $\mu\neq 0$ and variance
$\sigma^{2}$. If this distribution is used for the increments in Brownian
motion, then the increment $W(t)-W(s)$ would be distributed as
$\mathcal{N}(\mu,t-s)$. We examine each of the properties of standard Brownian
motion to see if they still hold:  1. $W(0)=0$ still holds. 2. $W(t)$ would
still have independent increments. 3. $W(t)-W(s)$ would now be
$\mathcal{N}(\mu,t-s)$, which violates the standard definition requiring a
zero mean. 4. $W(t)$ would still be continuous in $t$. The violation occurs at
the third property, thus proving that any deviation from a zero mean disrupts
the fidelity to true Brownian motion. ∎

##  5 Discussion

This study presents an innovative yet straightforward strategy for generating
Gaussian random variables through the utilisation of non-parametric quantum
circuits. Empirical analyses substantiate a close concordance between quantum
and classical Gaussian distributions, thereby affirming the efficacy of the
quantum approach. Further experiments underscore the applicability of quantum-
generated Gaussian noise in diffusion-centric methods such as SD and Brownian
Motion.

Although the approach still necessitates classical post-processing, the
replacement of pseudo-random numbers with quantum-generated random bits marks
a noteworthy advancement towards more robust generative models. This hybrid
methodology dispenses with the laborious task of parametric optimisation
within quantum circuits. Nevertheless, the full realisation of quantum
generative modelling continues to pose considerable challenges, particularly
in the scaling of quantum convolutional layers. Despite these hurdles, the
present study lays the groundwork for an auspicious avenue of exploration.

##  6 Acknowledgements.

###### List of Figures

  1. 1 An illustrative Depiction of our Proposed Quantum Pipeline.
  2. 2 Quantum-Generated Gaussian Noise in Stable Diffusion Processes
  3. 3 A Simplified Circuit Ansatz for Generating Random Rotations
  4. 4 Plots of Random Rotation Angles Generated by a Quantum Circuit
  5. 5 Plots of Quantum Sampling Measurement Results on a Unit Circle
  6. 6 Plots of 2000 Gaussian samples from a quantum circuit
  7. 7 Quantum-Enhanced Gaussian Noise in Forward Diffusion Models
  8. 8 Quantum-Enhanced Gaussian Noise in Brownian Motion Simulations for 9 Qubits

###### List of Tables

  1. 1 Statistical tests

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