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On the Eﬀects of Pseudo and Quantum Random Number

Generators in Soft Computing

Jordan J. Bird ·Anik´o Ek´art ·Diego R. Faria

Received: date / Accepted: date

Abstract In this work, we argue that the implications

of Pseudo and Quantum Random Number Generators

(PRNG and QRNG) inexplicably aﬀect the performances

and behaviours of various machine learning models that

require a random input. These implications are yet to

be explored in Soft Computing until this work. We use a

CPU and a QPU to generate random numbers for mul-

tiple Machine Learning techniques. Random numbers

are employed in the random initial weight distributions

of Dense and Convolutional Neural Networks, in which

results show a profound diﬀerence in learning patterns

for the two. In 50 Dense Neural Networks (25 PRNG/25

QRNG), QRNG increases over PRNG for accent clas-

siﬁcation at +0.1%, and QRNG exceeded PRNG for

mental state EEG classiﬁcation by +2.82%. In 50 Con-

volutional Neural Networks (25 PRNG/25 QRNG), the

MNIST and CIFAR-10 problems are benchmarked, in

MNIST the QRNG experiences a higher starting ac-

curacy than the PRNG but ultimately only exceeds it

by 0.02%. In CIFAR-10, the QRNG outperforms PRNG

by +0.92%. The n-random split of a Random Tree is en-

hanced towards and new Quantum Random Tree (QRT)

model, which has diﬀering classiﬁcation abilities to its

classical counterpart, 200 trees are trained and com-

Jordan J. Bird

School of Engineering and Applied Science

Aston University

E-mail: birdj1@aston.ac.uk

Anik´o Ek´art

School of Engineering and Applied Science

Aston University

E-mail: d.faria@aston.ac.uk

Diego R. Faria

School of Engineering and Applied Science

Aston University

E-mail: d.faria@aston.ac.uk

pared (100 PRNG/100 QRNG). Using the accent and

EEG classiﬁcation datasets, a QRT seemed inferior to

a RT as it performed on average worse by -0.12%. This

pattern is also seen in the EEG classiﬁcation problem,

where a QRT performs worse than a RT by -0.28%.

Finally, the QRT is ensembled into a Quantum Ran-

dom Forest (QRF), which also has a noticeable eﬀect

when compared to the standard Random Forest (RF).

10 to 100 ensembles of Trees are benchmarked for the

accent and EEG classiﬁcation problems. In accent clas-

siﬁcation, the best RF (100 RT) outperforms the best

QRF (100 QRF) by 0.14% accuracy. In EEG classiﬁca-

tion, the best RF (100 RT) outperforms the best QRF

(100 QRT) by 0.08% but is extremely more complex,

requiring twice the amount of trees in committee. All

diﬀerences are observed to be situationally positive or

negative and thus are likely data dependent in their

observed functional behaviour.

Keywords Quantum Computing ·Soft Computing ·

Machine Learning ·Neural Networks ·Classiﬁcation

1 Introduction

Quantum and Classical hypotheses of our reality are

individually deﬁnitive and yet are independently para-

doxical, in that they are both scientiﬁcally veriﬁed though

contradictory to one another. These concurrently anti-

thetical, nevertheless infallible natures of the two mod-

els have enﬂamed debate between researchers since the

days of Albert Einstein and Erwin Schr¨odinger during

the early 20th century. Though the lack of a Standard

Model of the Universe continues to provide a problem

for physicists, the ﬁeld of Computer Science thrives by

making use of both in Classical and Quantum comput-

ing paradigms since they are independently observable

2 Jordan J. Bird et al.

in nature.

Though the vast majority of computers available are

classical, Quantum Computing has been emerging since

the late 20th Century, and is becoming more and more

available for use by researchers and private institutions.

Cloud platforms developed by industry leaders such as

Google, IBM, Microsoft and Rigetti are quickly growing

in resources and operational size. This rapidly expand-

ing availability of quantum computational resources al-

lows for researchers to perform computational experi-

ments, such as heuristic searches or machine learning,

but allow for the use of the laws of quantum mechanics

in their processes. For example, for ncomputational

bits in a state of entanglement, only one needs to be

measured for all nbits to be measured, since they all

exist in parallel or anti-parallel relationships. Through

this process, computational complexity has been re-

duced by a factor of n. Bounded-error Quantum Polyno-

mial time (BQP) problems are a set of computational

problems which cannot be solved by a classical com-

puter in polynomial time, whereas a quantum processor

has the ability to with its diﬀerent laws of physics.

Optimisation is a large multi-ﬁeld conglomeration of

research, which is rapidly accelerating due to the grow-

ing availability of powerful computing hardware such

has CUDA. Examples include Ant Colony Optimisa-

tion inspired by the pheromone-dictated behaviour of

ants Deng et al. (2019), orthoganal translations to de-

rive a Principle Component Analysis Zhao et al. (2019),

velocity-based searches of particle swarms Deng et al.

(2017), as well as entropy-based methods of data anal-

ysis and classiﬁcation Zhao et al. (2018).

There are several main contributions presented by

this research:

1. A comparison of the abilities of Dense Neural Net-

works with their initial random weight distributions

derived by Pseudorandom and Quantum Random

methods.

2. An exploration of Random Tree models compared to

Quantum Random Tree models, which utilise Pseu-

dorandom and Quantum Random Number Genera-

tors in their generation respectively.

3. A benchmark of the number of Random Trees in

a Random Forest model compared to the number

of Quantum Random Trees in a Quantum Random

Forest model.

4. A comparison of the eﬀects of Pseudo and True ran-

domness in initial random weight distributions in

Computer Vision, applied to Deep Neural Networks

and Convolutional Neural Networks.

Although Quantum, Quantum-inspired, and Hybrid Clas-

sical/Quantum algorithms are explored, as well as the

likewise methods for computing, the use of a Quantum

Random Number Generator is rarely explored within a

classical machine learning approach in which an RNG

is required Kretzschmar et al. (2000).

This research aims to compare approaches for ran-

dom number generation in Soft Computing for two laws

of physics which directly defy one another; the Classi-

cal true randomness is impossible and the Quantum

true randomness is possible Calude and Svozil (2008).

Through the application of both Classical and Quan-

tum Computing, simulated and true random number

generation are tested and compared via the use of a

Central Processing Unit (CPU) and an electron spin-

based Quantum Processing Unit (QPU) via placing the

subatomic particle into a state of quantum superposi-

tion. Logic would conjecture that the results between

the two ought to be indistinguishable from one another,

but experimentation within this study suggests other-

wise. The rest of this article is structured as follows:

Section 2 gives an overview of the background to

this project and important related theories and works.

Speciﬁcally, Quantum Computing, the diﬀering ideas

of randomness in both Classical and Quantum com-

puting, applications of quantum theory in computing

and ﬁnally a short subsection on the machine learning

theories used in this study. Section 3 describes the con-

ﬁguration of the models as well as the methods used

speciﬁcally to realise the scientiﬁc studies in this arti-

cle, before being presented and analysed in Section 4.

The experimental results are divided into four individ-

ual experiments:

–Experiment 1 - On random weight distribution in

Dense Neural Networks: Pseudorandom and Quan-

tum Random Number Generators are used to ini-

tialise the weights in Neural Network models.

–Experiment 2 - On Random Tree splits: The nRan-

dom Splits for a Random Tree classiﬁer are formed

by Pseudo and Quantum Random numbers.

–Experiment 3 - On Random Tree splits in Random

Forests: The Quantum Tree model derived from Ex-

periment 2 is used in a Quantum Random Forest

ensemble classiﬁer.

–Experiment 4 - On Computer Vision: A Deep Neu-

ral Network and Convolutional Neural Network are

trained on two image recognition datasets with pseudo

and true random weight distributions for the appli-

cation of Computer Vision.

Experiments are separated in order to focus upon the

eﬀects of diﬀering random number generators on a spe-

ciﬁc model. Explored in these are the eﬀects of Pseudo-

random and Quantum Random number generation in

On the Eﬀects of Pseudo and Quantum Random Number Generators in Soft Computing 3

their processes, and a discussion of similarities and dif-

ferences between the two in terms of statistics as well as

their wider eﬀect on the classiﬁcation process. Section

5 outlines possible extensions to this study for future

works, and ﬁnally, a conclusion is presented in Section

6.

2 Background and Related Works

2.1 Quantum Computing

Pioneered by Paul Benioﬀ’s 1980 work Benioﬀ (1980),

Quantum Computing is a system of computation that

makes computational use of phenomena outside of clas-

sical physics such as the entanglement and superposi-

tion of subatomic particles Gershenfeld and Chuang

(1998). Whereas classical computing is concerned with

electronic bits that have values of 0 or 1 and logic gates

to process them, quantum computing uses both classi-

cal bits and gates as well as new possible states; such as

a bit being in a state of superposition (0 and 1) or en-

tangled with other bits. Entanglement means that the

value of the bit, even before measurement, can be as-

sumed to be parallel or anti-parallel to another bit of

which it is entangled to Bell (1964). These extended

laws allow for the solving of problems far more eﬃ-

ciently than computers. For example, a 64-bit system

(263 −1) has approximately 9.22 quintillion values with

its individual bits at values 1 or 0, whereas unlike a

three-state ternary system which QPUs are often mis-

taken for, the laws superposition and the degrees of

state would allow a small array of qubits to represent

all of these values at once - theoretically allowing quan-

tum computers to solve problems that classical comput-

ers will never be able to possibly solve. Since the sta-

bility of entanglement decreases with the more compu-

tational qubits used, only very small-scale experiments

have been performed as of today. Quantum Processing

Units (QPUs) made available for use by Rigetti, Google

and IBM have up to 16 available qubits for computing

via their cloud platforms.

2.2 Randomness in Classical and Quantum Computing

In classical computing, randomness is not random, rather,

it is simulated by a pseudo-random process. Processor

architectures and Operating Systems have individual

methods of generating pseudo-random numbers which

must conform to cybersecurity standards such as NIST

Barker and Kelsey (2007). Major issues arise with the

possibility of backdoors, notably for example Intel’s pseudo

random generator which, after hijacking, allowed for

complete control of a computer system for malicious in-

tent Degabriele et al. (2016); Schneier (2007). The Intel

issue was far from a lone incident, the RANDU system

was cracked by the NSA for unprecedented access to the

RSA BSAFE cryptographic library, as well as in 2006

when Debian OpenSSL’s random number generator was

also cracked, leading to Debian being compromised for

two years Markowsky (2014). Though there are many

methods of deriving a pseudo-random number, all clas-

sical methods, due to the laws of classical physics pro-

viding limitation, are sourced through arbitrary yet de-

terministic events Gallego et al. (2013); such as a com-

bination of, time since nlast key press, hardware tem-

perature, system clock, lunar calendar etc. This arbi-

tration could possibly hamper or improve algorithms

that rely on random numbers, since the state of the ex-

ecuting platform could indeed directly inﬂuence their

behaviour.

According to Bell’s famous theorem, ”No physical

theory of local hidden variables can ever reproduce all

of the predictions of quantum mechanics” Bell (1964).

This directly argued against the position put forward by

Einstein et. al in which it is claimed that the Quantum

Mechanical ’paradox’ is simply due to incomplete the-

ory Einstein et al. (1935). Using Bell’s theorem, demon-

strably random numbers can be generated through the

fact that observing a particle’s state while in superpo-

sition gives a true 50/50 outcome (qubit value 0, 1)

Pironio et al. (2010). This concretely random output

for the value of the single bit can be used to build in-

tegers comprised of larger numbers of bits which, since

they are all individually random, their product is too.

This process is known as a Quantum Random Number

Generator (QRNG).

Behaviours in Quantum Mechanics such as, but not

limited to, branching path superposition Jennewein

et al. (2000), time of arrival Wayne et al. (2009), parti-

cle emission count Ren et al. (2011), attenuated pulse

Wei and Guo (2009), and vacuum ﬂuctuations Gabriel

et al. (2010) are all entirely random - and have been

used to create true QRNGs. In 2000, it was observed

that a true random number generator could be formed

through the observation of photons Stefanov et al.

(2000). Firstly, a beam of light is split into two streams

of entangled photons, noise is reduced after which the

photons of both streams are observed. The two detec-

tors correlate to 0 and 1 values, and a detection will

amend a bit to the result. The detection of a photon

is non-deterministic between the two, and therefore a

completely random series of values are the result of this

4 Jordan J. Bird et al.

Fig. 1 The Famous Schr¨odinger’s Cat Thought Experiment.

When unobserved, the cat arguably exists in two opposite

states (alive and dead), which itself constitutes a third super-

state Schr¨odinger (1935).

experiment.

This study makes use of the branching path super-

position method for the base QRNG, in that the ob-

served state of a particle cat time t, the state of cis

non-deterministic until only after observation t. In the

classical model, the law of superposition simply states

that for properties Aand Bwith outcomes Xand Y,

both properties can lead to state XY. For example, the

translation and rotation of a wheel can lead to a rolling

state Cullerne (2000), a third superstate of the two

possible states. This translates into quantum physics,

where quantum states can be superposed into an addi-

tional valid state Dirac (1981).

This is best exempliﬁed with Erwin Schr¨odinger’s

famous thought experiment, known as Schr¨odinger’s

Cat Schr¨odinger (1935). As seen in Fig. 1, a cat sits

in a box along with a Geiger Counter and a source

of radiation. If alpha radiation is detected, which is a

completely random event, the counter releases a poi-

son into the box, killing the cat. The thought exper-

iment explains superposition in such a way, that al-

though the cat has two states (Alive or Dead), when

unobserved, the cat is both simultaneously alive and

dead. In terms of computing, this means that the two

classical behaviours of a single bit, 1 or 0, can be super-

posed into an additional state, 1 and 0.Just as the cat

only becomes alive or dead when observed, a superposed

qubit only becomes 1 or 0 when measured.

A Bloch Sphere is a graphical representation of a

qubit in superposition Bloch (1946) and can be seen in

Fig. 2. In this diagram, the basis states are interpreted

by each pole, denoted as |0iand |1i. Other behaviours,

the rotations of spin about points ψ,φ, and θare used

to superpose the two states to a degree. Thus depend-

ing on the method of interpretation, many values can

Fig. 2 A Bloch Sphere Represents the Two Basis States of a

Qubit (0, 1) as well as the States of Superposition In-between.

be encoded within only a single bit of memory.

The Hadamard Gate within a QPU is a logical gate

which coerces a qubit into a state of superposition based

on a basis (input) state. 0 is mapped as follows:

|0i 7→ |0i+|1i

√2(1)

The other possible basis state, 1, is mapped as:

|0i 7→ |0i−|1i

√2(2)

This single qubit quantum Fourier transform is thus

represented through the following matrix:

H=1

√21 1

1−1(3)

Just as in the thought experiment described in which

Schr¨odinger’s cat is both alive and dead, the qubit now

exists in a state of quantum superposition; it is both

1 and 0. That is, until it is measured, in which there

will be an equal probability that the observed state is 1

or 0, giving a completely randomly generated bit value.

This is the logical basis of all QRNGs.

2.3 Quantum Theory in Related State-of-the-art

Computing Application

The ﬁeld of Quantum Computing is young, and thus

there are many frontiers of research of which none have

been mastered. Quantum theory, though, has been shown

in some cases to improve current ideas in Computer

Science as well as endow a system with abilities that

On the Eﬀects of Pseudo and Quantum Random Number Generators in Soft Computing 5

would be impossible on a classical computer. This sec-

tion outlines some of the state of the art applications

of quantum theory in computing.

Quantum Perceptrons are a theoretical approach to

deriving a quantum equivalent of a perceptron unit

(neuron) within an Artiﬁcial Neural Network Schuld

et al. (2014). Current lines of research focus around

the possibilities of associative memory through quan-

tum entanglement of internal states within the neurons

of the network. The approach is heavily inspired by

the notion that the biological brain may operate within

both classical and quantum physical space Hagan et al.

(2002). Preliminary works have found Quantum Neural

Networks have a slight statistical advantage over clas-

sical techniques within larger and more complex do-

mains Narayanan and Menneer (2000). A very limited

extent of research suggest quantum eﬀects in a network

to be the possible source of consciousness Hameroﬀ and

Penrose (1996), providing an exciting avenue for Arti-

ﬁcial Intelligence research in the ﬁeld of Artiﬁcial Con-

sciousness. Inspiration from quantum mechanics has

led to the implementation of a Neural Networks based

on fuzzy logic systems Purushothaman and Karayian-

nis (1997), research showed that QNNs are capable of

structure recognition, which sigmoid-activated hidden

units within a network cannot.

There are many statistical processes that are either

more eﬃcient or even simply possible through the use of

Quantum Processors. Simon’s Problem provides initial

proof that there are problems that can be solved expo-

nentially faster when executed in quantum space Arora

and Barak (2009). Based on Simon’s Problem, Shor’s

Algorithm uses quantum computing to derive the prime

factors of an integer in polynomial time Shor (1999),

something which a classical computer is not able to do.

Some of the most prominent lines of research in

quantum algorithms for Soft Computing are the ex-

ploration of Computational Intelligence techniques in

quantum space such as meta-heuristic optimisation, heuris-

tic search, and probabilistic optimisation etc. Pheromone

trails in Ant Colony Optimisation searches generated

and measured in the form of qubits with operations of

entanglement and superposition for measurement and

state scored highly on the Tennessee Eastman Process

benchmark problem, due to the optimal operations in-

volved Wang et al. (2007). This work was applied by

researchers, who in turn found that combining Support

Vector Machines with Quantum Ant Colony Optimi-

sation search provided a highly optimised strategy for

solving fault diagnosis problems Wang et al. (2008),

greatly improving the base SVM. Parallel Ant Colony

Optimisation has also been observed to greatly improve

in performance when operating similar techniques You

et al. (2010). Similar techniques have also been used

in the genetic search of problem spaces, with quantum

logic gates performing genetic operations and proba-

bilistic representations of solution sets in superposi-

tion/entanglement, the technique is observed to be su-

perior over its classical counterpart when benchmarked

on the combinatorial optimisation problem Han et al.

(2001).

Statistical and Deep Learning techniques are often

useful in other scientiﬁc ﬁelds such as engineering Nader-

pour et al. (2019); Naderpour and Mirrashid (2019),

medicine Khan et al. (2001); Penny and Frost (1996),

chemistry Sch¨utt et al. (2019); Gastegger et al. (2019),

and astrophysics Krastev (2019); Kimmy Wu et al.

(2019) among a great many others Carlini and Wagner

(2017). As of yet, the applications of quantum solutions

have not been applied within these ﬁelds towards the

possible improvement of soft computing technique.

3 Experimental Setup and Design

For the generation of true random bit values, an electron-

based superposition state is observed using a QPU. The

Quantum Assembly Language code for this is given in

Appendix A; an electron is transformed using a Hadamard

Gate and thus now exists in a state of superposition.

When the bit is observed, it takes on a state of either

0 or 1, which is a non-deterministic 50/50 outcome ie.

perfect randomness. A VM example of how these op-

erations are formed into a random integer are given in

Appendix B; the superposition state particle is sequen-

tially observed and each derived bit is amended to a

result until 32 bits have been generated. These 32 bits

are then treated as a single binary number. The result

of this process is a truly random unsigned 32-bit integer.

For the generation of bounded random numbers,

the result is normalised with the upper bound being

the highest possible value of the intended number. For

those that also have lower bounds below zero, a simple

subtraction is performed on a higher bound of normali-

sation to give a range. For example, if a random weight

distribution for neural network initialisation is to be

generated between -0.5 and 0.5, the random 32-bit in-

teger is normalised between 0-1 and 0.5 is subtracted

from the result, giving the desired range. This process

is used for the generation of both PRN and QRN since

they are therefore then directly comparable with one

6 Jordan J. Bird et al.

another and thus also directly relative in their eﬀects

upon a machine learning process.

For the ﬁrst dataset in each experiment, a publicly

available Accent Classiﬁcation dataset is retrieved1. This

dataset was gathered from subjects from the United

Kingdom and Mexico, all speaking the same seven pho-

netic sounds ten times each. A ﬂat dataset is produced

via 27 logs of their Mel-frequency Cepstral Coeﬃcients

every 200ms to produce a mathematical description of

the audio data. A four-class problem arises in the pre-

diction of the locale of the speaker (West Midlands,

London, Mexico City, Chihuahua). The second dataset

in each experiment is an EEG brainwave dataset sourced

from a previous study Bird et al. (2018)2. The wave

data has been extracted from the TP9, AF7, AF8 and

TP10 electrodes, and has been processed in a similar

way to the speech in the ﬁrst dataset, except is done so

through a much larger set of mathematical descriptors.

For the four-subject EEG dataset, a three-class problem

arises; the concentrative state of the subject (concen-

trating, neutral, relaxed). The feature generation pro-

cess from this dataset was observed to be eﬀective for

mental state classiﬁcation in the aforementioned study,

as well as for emotional classiﬁcation from the same

EEG electrodes Bird et al. (2019a).

For the ﬁnal experiment, two image classiﬁcation

datasets are used. Firstly, the MNIST image dataset is

retrieved3LeCun and Cortes (2010) for the MLP. This

dataset is comprised of 60,000 32x32 handwritten single

digits 0-9, a 10-class problem with each class being that

of the digit written. Secondly, the CIFA-10 dataset is

retrieved4Krizhevsky et al. (2009) for a CNN. This, as

with the MNIST dataset, is comprised of 60,000 32x32

10-class images of entities (eg. bird, cat, deer).

For the generation of pseudorandom numbers, an

AMD FX8320 processor is used with given bounds for

experiment 1a and 1b. The Java Virtual Machine gen-

erates pseudorandom numbers for experiments 2 and 3.

All of the pseudorandom number generators had their

seed set to the order of execution, ie. the ﬁrst model

has a seed of 1 and the nth model has a seed of n.

Due to the high resource usage of training a large vol-

ume of neural networks, the CUDA cores of an Nvidia

GTX980Ti were utilised and they were trained on a

1https://www.kaggle.com/birdy654/speech-recognition-

dataset-england-and-mexico

2https://www.kaggle.com/birdy654/eeg-brainwave-

dataset-mental-state

3http://yann.lecun.com/exdb/mnist/

4https://www.cs.toronto.edu/kriz/cifar.html

70/30 train/test split of the datasets. For the Machine

Learning Models explored in Experiments 2 and 3, 10-

fold cross validation was used due to the availability of

computational resources to do so.

3.1 Experimental Process

In this subsection, a step-by-step process is given de-

scribing how each model is trained towards comparison

between PRNG and QRNG methods. MLP and CNN

RNG methods are operated through the same technique

and as such are described together, following this, the

Random Tree (RT) and Quantum Random Tree (QRT)

are described. Finally the ensembles of the two types of

trees are then ﬁnally described as Random Forest (RF)

and Quantum Random Forest (QRF). Each set of mod-

els is tested and compared for two diﬀerent datasets, as

previously described. For replicability of these exper-

iments, the code for Random Bit Generation is given

in Appendix A (for construction of an n-bit integer).

Construction of the n-bit integer through electron ob-

servation loop is given in Appendix B.

For the Random Neural Networks, all use the ADAM

Stochastic Optimiser for weight tuning Kingma and

Ba (2014), and the activation function of all hidden

layers is ReLU Agarap (2018). For Random Trees, K

randomly chosen attributes is deﬁned below (acquired

via either PRNG or QRNG) and the minimum possi-

ble value for kis 1, no pruning is performed. Minimum

class variance is set to −inf since the datasets are well-

balanced, the maximum depth of the tree is not limited,

and classiﬁcation must always be performed even if con-

fusion occurs. The chosen Random Tree attributes are

also used for all trees within Forests, where the random

number generator for selection of data subsets is also

decided by a PRNG or QRNG. The algorithmic com-

plexity for a Random Tree is given as O(v×nlog(n))

where nis the number of data objects in the dataset

and vis the number of attributes belonging to a data

object in the set. Algorithmic complexity of the neural

networks are dependent on chosen topologies for each

problem, and the complexity is presented as an O(n2)

problem.

Given nnumber of networks to be benchmarked for

xepochs, generally, the MLP and CNN experiments are

automated as follows:

1. Initialise n/2 neural networks with initial random

weights generated by an AMD CPU (pseudoran-

dom).

On the Eﬀects of Pseudo and Quantum Random Number Generators in Soft Computing 7

2. Initialise n/2 neural networks with initial random

weights generated by a Rigetti QPU (true random).

3. Train all nneural networks.

4. Consider classiﬁcation accuracy at each epoch5for

comparison as well as statistical analysis of all n/2

networks.

Given nnumber of trees with a decision variable Kx

(Krandomly chosen attributes at node x), the process

of training Random Trees (RT) and Quantum Random

Trees (QRT) are given as follows:

1. Train n/2 Random Trees, in which the RNG for

deciding set Kfor every xis executed by an AMD

CPU (pseudorandom)

2. Train n/2 Quantum Random Trees, in which the

RNG for deciding set Kfor every xis executed by

a Rigetti QPU (true random).

3. Considering the best and worst models, as well as

the mean result, compare the two sets of n/2 models

in terms of statistical diﬀerence6

Finally, the Random Tree and Quantum Random

Tree are benchmarked as an ensemble, through Ran-

dom Forests and Quantum Random Forests. This is

performed mainly due to the unpruned Random Tree

likely overﬁtting to training data Hastie et al. (2005).

The process is as follows7:

1. For the Random Forests, benchmark 10 forests con-

taining {10, 20, 30 ... 100}Random Tree Models (as

generated in the Random Tree Experimental Process

list above).

2. For the Quantum Random Forests, benchmark 10

forests containing {10, 20, 30 ... 100}Quantum Ran-

dom Tree Models (as generated in the Random Tree

Experimental Process list above).

3. Compare abilities of all 20 models, in terms of clas-

siﬁcation ability as well as the statistical diﬀerences,

if any, between diﬀerent numbers of trees in the for-

est.

4 Results and Discussion

In this section, results are presented and discussed for

multiple Machine Learning models when their random

number generator is either Pseudo-randomly, or True

(Quantum) Randomly generated. Please note that in

neural network training, lines do not correlate on a one-

to-one basis. Each line is the accuracy of a neural net-

work throughout the training process, and line colour

5Accuracy/epoch graphs are given in Section 4

6Box and whisker comparisons given in Section 4.

7For further detail on the Random Decision Forest classi-

ﬁer selected for this study, please refer to Breiman (2001)

Fig. 3 The Main Learning Curve Experienced for 50 Dense

Neural Networks, 25 with PRNG and 25 with QRNG Initially

Distributed Weights in Accent Classiﬁcation

deﬁnes how that network had its weights initialised ie.

whether or not it has pseudo or quantum random num-

bers as its initial weights.

4.1 MLP: Random Initialisation of Dense Neural

Network Weights

For Experiment 1, a total of ﬁfty dense neural networks

were trained for each dataset. All networks were iden-

tical except for their initial weight distributions. Initial

random weights within bounds of -0.5 and 0.5 were set,

25 of the networks derived theirs from a PRNG, and

the other 25 from a QRNG.

4.1.1 Accent Classiﬁcation

For Experiment 1a, the accent classiﬁcation dataset

was used. In this experiment, we observed initial sparse

learning processes before stabilisation occurs at approx-

imately epoch 30 and the two converge upon a similar

result. Fig. 3 shows this convergence of the learning

processes the initial learning curve experienced during

the ﬁrst half of the process, in this graph it can be

observed that the behaviour of pseudorandom weight

distribution is far less erratic than that of the quan-

tum random number generator. This shows that the

two methods of random number generators do have an

observable eﬀect on the learning processes of a neural

network.

For PRNG, the standard deviation between all 25

ﬁnal results was 0.00098 suggesting that a classiﬁcation

maxima was being converged upon. The standard devi-

ation for QRNG was considerably larger, but statisti-

8 Jordan J. Bird et al.

Fig. 4 The Full Learning Process of 50 Dense Neural Net-

works, 25 with PRNG and 25 with QRNG Initially Dis-

tributed Weights in Mental State EEG Classiﬁcation

Fig. 5 The Final Epochs of Learning for 50 Dense Neural

Networks, 25 with PRNG and 25 with QRNG Initially Dis-

tributed Weights in Mental State EEG Classiﬁcation

cally minimal at 0.0017. Mean ﬁnal results were 98.73%

for PRNG distributions and 98.8% for QRNG distribu-

tions. The maximum classiﬁcation accuracy achieved

by the PRNG initial distribution was 98.8% whereas

QRNG achieved a slightly higher result of 98.9% at

epoch 49. For this problem, the diﬀerences between the

initial distribution of PRNG and QRNG are minimal,

QRNG distribution results are somewhat more entropic

than PRNG but otherwise the two sets of results are in-

distinguishable from one another, and most likely sim-

ply due to random noise.

4.1.2 Mental State Classiﬁcation

For Experiment 1b, the Mental State EEG classiﬁca-

tion dataset was used Bird et al. (2018). Fig. 4 shows

the full learning process of the networks from initial

epoch 0 up until backpropagation epoch 100, though

this graph is erratic and crowded, the emergence of

a pattern becomes obvious within epochs 20-30 where

the learning processes split into two distinct groups. In

this ﬁgure, a more uniform behaviour of QRNG meth-

ods are noted, unlike the previous experiment. The be-

haviours of PRNG distributed models are extremely er-

ratic and in some cases, very slow in terms of improve-

ments made. Fig. 5 show a higher resolution view of the

data in terms of the end of the learning process when

terminated at epoch 100, a clear distinction of results

can be seen and a concrete separation can be drawn

between the two groups of models except for two inter-

secting processes. It should be noted that by this point,

the learning process has not settled towards a true best

ﬁtness, but a vast and clear separation has occurred.

For PRNG, the standard deviation between all 25

results was 0.98. The standard deviation for QRNG

was somewhat smaller at 0.74. The mean of all results

was 63.84% for PRNG distributions and 66.45% for

QRNG distribution, a slightly superior result. The max-

imum classiﬁcation accuracy achieved by the PRNG ini-

tial distribution was 65.35% whereas QRNG achieved a

somewhat higher best result of 68.17%. The worst-best

result for PRNG distribution networks was 62.28%, and

was 65.31% for QRNG distribution networks. For this

problem, the diﬀerences between the initial distribution

of PRNG and QRNG weights are noticeable, QRNG

distribution results are consistently better than PRNG

approaches to initial weight distribution.

4.2 Random Tree and Quantum Random Tree

Classiﬁers

Experiments 2a and 2b make use of the same datasets as

in 1a and 1b respectively. In this experiment, 200 Ran-

dom Tree classiﬁers are trained for each dataset. These

are, again, comprised of two sets; ﬁrstly 100 Random

Tree (RT) classiﬁers which use Pseudorandom numbers,

and secondly, 100 Quantum Random Tree (QRT) clas-

siﬁers, which source their random numbers from the

QRNG. Random Numbers are used to select the n-

random attribute subsets at each split.

4.2.1 Accent Classiﬁcation

200 Experiments are graphically represented as a box-

and-whisker in Fig. 6. The most superior classiﬁer was

On the Eﬀects of Pseudo and Quantum Random Number Generators in Soft Computing 9

Fig. 6 A Comparison of results from 200 Random Tree Clas-

siﬁers, 100 using PRNG and 100 using QRNG on the Accent

Classiﬁcation Dataset

the RT with a best result of 86.64% and worst of 85.68%,

on the other hand, the QRT achieved a best accuracy

of 86.52% and worst of 85.62%. Best and worst results

of the two models are extremely similar. The standard

deviation of results of the RT was 0.19 and the QRT

similarly had a standard deviation of 0.17. The range

of the RT results was 0.96 and QRT results had a sim-

ilar range of 0.9. Interestingly, a similar pattern is not

only found in results, but also with the high outlier too

when considered relative to the model’s median point.

Though an overall slight superiority is seen in pseudo-

random number generation, the two models are consid-

erably similar in their abilities.

4.2.2 Mental State Classiﬁcation

Fig. 7 shows the distribution for the 200 Random Tree

classiﬁers trained on the Mental State dataset. The

standard deviation of results from the RT was 0.81

whereas it was slightly lower for QRT at 0.73. The

best result achieved by the RT was 79.68% classiﬁca-

tion accuracy whereas the best result from the QRT

was 79.4%. The range of results for RT and QRT were

a similar 3.31 and 3.47 respectively. Overall, very little

diﬀerence between the two models occurs. The distri-

bution of results can be seen to be extremely similar to

the ﬁrst RT/QRT experiment when compared to Fig.

6.

4.3 Random Forest and Quantum Random Forest

Classiﬁers

In this third experiment, the datasets are classiﬁed us-

ing two models. Random Forests (RF) which use a com-

Fig. 7 A Comparison of results from 200 Random Tree Clas-

siﬁers, 100 using PRNG and 100 using QRNG on the Mental

State EEG Dataset

10 20 30 40 50 60 70 80 90 100

90

90.5

91

91.5

92

92.5

Number of Trees in the Forest

Classiﬁcation Accuracy

Accent Classiﬁcation Experiment

Pseudorandom

True Random

Fig. 8 Classiﬁcation Accuracies of 10 Random Forest and 10

Quantum Forest Models on the Accent Classiﬁcation Dataset

mittee of Random Trees to vote on a Class, and Quan-

tum Random Forests (QRF) which use a committee of

Quantum Trees to vote on a class. For each dataset, 10

of these models are trained, with a committee of 10 to

100 Trees respectively.

4.3.1 Accent Classiﬁcation

The results from the Accent Classiﬁcation dataset for

the RF and QRF methods can be observed in Fig. 8.

The most superior models both used a committee of 100

of their respective trees, scoring two similar results of

91.86% with Pseudo-randomness and 91.78% for Quan-

tum randomness. Standard deviation of RF results are

0.5% whereas QRF has a slightly lower deviation of

10 Jordan J. Bird et al.

10 20 30 40 50 60 70 80 90 100

0.84

0.85

0.86

0.87

0.88

Number of Trees in the Forest

Classiﬁcation Accuracy

Mental State Classiﬁcation Experiment

Pseudorandom

True Random

Fig. 9 Classiﬁcation Accuracies of 10 Random Forest and

10 Quantum Forest Models on the EEG Mental State Clas-

siﬁcation Dataset

0.43. The worst result by RF was 90.31% classiﬁcation

accuracy at 10 Random Trees, the worst result by the

QRF was similarly 10 Quantum Trees at 90.36% classi-

ﬁcation accuracy (+0.05). The range of RF results was

1.55, compared to the QRF results with a range of 1.43.

4.3.2 Mental State Classiﬁcation

The results from the Mental State EEG Classiﬁcation

dataset for the RF and QRF methods can be observed

in Fig. 9. The most superior model for the RF was

86.91% with a committee of 100 trees whereas the best

result for QRF was 86.83% achieved by committees of

both 100 and 60 trees. The range of QRF results were

slightly lower than that of the RF, measured at 2.34

and 2.42 respectively. Although initially considered neg-

ligible, this same pattern was observed in the previous

experiment in Fig. 8. Additionally, the standard devia-

tion of RF was higher at 0.69 compared to 0.65 in QRF.

Though very similar results were produced, the ﬁrst

QRF best result required approximately 60% of the

computational resources to achieve compared to the

best RF result. Unlike the ﬁrst Forest experiment, the

patterns of the two diﬀerent models are vastly diﬀerent

and often alternate erratically. This suggests somewhat

that the two models should both be benchmarked in

order to increase the chances of discovering a more su-

perior model, considering the level of data dependency

on the classiﬁcation accuracies of the models.

Fig. 10 The Full Learning Process of 50 Deep Neural Net-

works, 25 with PRNG and 25 with QRNG Initially Dis-

tributed Weights in MNIST Image Dataset Classiﬁcation

4.4 CNN: Initial Random Weight Initialisation for

Computer Vision

Experiment 4a and 4b make use of the MNIST and

CIFAR-10 image datasets respectively. In 4a, an ANN is

initialised following the same PRNG and QRNG meth-

ods utilised in Experiment 1 and trained to classify

the MNIST handwritten digits dataset. In 4b, the ﬁ-

nal dense layer of the CNN are initiliased through the

same methods.

4.4.1 MNIST Image Classiﬁcation

For the purpose of scientiﬁc recreation, the architec-

ture for MNIST classiﬁcation is derived from the of-

ﬁcial Keras example8. This is given as two sets of two

identical layers, a hidden layer of 512 densely connected

neurons followed by a dropout layer of 0.2 to prevent

over-ﬁtting. All hidden neurons, as with other experi-

ments in this study, are initialised randomly within the

standard -0.5 to 0.5 range. 25 of these are generated

by a PRNG and the other 25 by a QRNG, producing

observable results of 50 models in total.

Due to the concise nature and close results observed

in the full process showed in Fig. 10, two additional

graphs are presented; ﬁrstly, the graph in Fig. 11 shows

the classiﬁcation abilities of the models before any train-

ing occurs. Within this, a clear distinction can be made,

the starting weights generated by QRNG are almost ex-

clusively superior to those generated by PRNG, provid-

ing the QRNG models with a superior starting point for

learning. The distinction continues to occur through-

8https://github.com/keras-team/keras/tree/master/examples

On the Eﬀects of Pseudo and Quantum Random Number Generators in Soft Computing 11

Fig. 11 Initial (pre-training) Classiﬁcation Abilities of 50

Deep Neural Networks, 25 with PRNG and 25 with QRNG

Initially Distributed Weights in MNIST Image Dataset Clas-

siﬁcation

Fig. 12 The Initial Learning Curve Experienced for 50 Deep

Neural Networks, 25 with PRNG and 25 with QRNG Initially

Distributed Weights in MNIST Image Dataset Classiﬁcation

out the initial learning curve, observed in Fig. 12, not

too dissimilar to the results in the previous experi-

ment. At the pre-training abilities of the two methods

of weight initialisation, dense areas can be observed at

approx 77.5% Finally, at around epochs 10-14, the re-

sultant models begin to converge and the separation

becomes less prominent. This is shown through both

sets of models having identical best classiﬁcation accu-

racies of 98.64%m suggesting a true best ﬁtness may

possibly have been achieved. Worst-best accuracies are

also indistinguishably close, 98.27% for QRNG models

and 98.25% for PRNG models, population ﬁtnesses are

extremely dense and little entropy exists throughout

the whole set of ﬁnal results.

Fig. 13 The Full Learning Process of 50 Convolutional Neu-

ral Networks, 25 with PRNG and 25 with QRNG Initially

Distributed Weights for the Final Hidden Dense Layer in

CIFAR-10 Image Dataset Classiﬁcation

4.4.2 CIFAR-10 Image Classiﬁcation

In the CNN experiment, the CIFAR-10 image dataset

is used to train a Convolutional Neural Network. The

two number generators are applied for the initial ran-

dom weight distribution of the ﬁnal hidden dense layer,

after feature extraction has been performed by the CNN

operations. The network architecture is constructed as

is the oﬃcial Keras Development Team example for Sci-

entiﬁc purposes in ease of recreation of the experiment.

In this architecture, one hidden dense layer of 512 units

precedes the ﬁnal classiﬁcation output, and weights are

generated within the bounds of -0.5 and 0.5 as is a

standard in neural network generation. 50 CNNS are

trained, all of which are structurally identical except

for that 25 have their dense layer weights initialised by

PRNG and the other 25 have their dense layer weights

initialised by QRNG.

Fig. 13 shows the full learning process of the two

diﬀerent methods of initial weight distribution. It can

be observed that there are roughly three partitions of

results between the two methods, the pattern is visually

similar to the ANN learning curve in the MNIST Com-

puter Vision experiment. Fig 14 shows the pre-training

classiﬁcation abilities of the initial weights, distribution

is relatively equal and unremarkable unless compared to

the ﬁnal results of the training process in Fig. 15; the

four best initial distributions of network weights, all are

of that which have been generated by the QRNG, con-

tinue to be the four superior overall models. It must be

noted although, that the rest of the models regardless

of RNG method, are extremely similar and no other di-

12 Jordan J. Bird et al.

Fig. 14 Initial (pre-training) Classiﬁcation Abilities of 50

Convolutional Neural Networks, 25 with PRNG and 25 with

QRNG Initially Distributed Weights for the Final Hidden

Dense Layer in CIFAR-10 Image Dataset Classiﬁcation

Fig. 15 The Learning within the Final Epochs for 50 Convo-

lutional Neural Networks, 25 with PRNG and 25 with QRNG

Initially Distributed Weights for the Final Hidden Layer in

CIFAR-10 Image Dataset Classiﬁcation

vide is seen by the end of the process.

The six overall most superior models were all ini-

tialised by QRNG, the best result being a classiﬁca-

tion accuracy of 75.35% at epoch 50. The seventh best

model was the highest scoring model that had dense

layer weights initialised by PRNG, scoring a classiﬁca-

tion accuracy of 74.43%. The worst model produced by

the QRNG was that which had a classiﬁcation accuracy

of 71.91%, slightly behind this was the overall worst

model from all experiments, a model initialised by the

PRNG with an overall classiﬁcation ability of 71.82%.

The QRNG initialisation therefore outperformed PRNG

by 0.92 in the best case, and outperformed PRNG by

0.09 in the worst case. The average result from both

methods of distribution. The average result between the

two models was equal, at 73.3% accuracy.

It must be noted that by epoch 50 the training pro-

cess was still producing increasingly better results, but

computational resources available limited the 50 net-

works to be trained for this amount of time.

5 Future Work

It was observed in those experiments that did stabilise,

results as expected reached closer similarities. With re-

sources, future work should concern the further training

of models to observe this pattern with a greater reach of

examples. Extensive computational resources would be

required to train such an extensive amount of networks.

Furthermore, the patterns in Fig. 9, Quantum vs

Random Forest for Mental State Classiﬁcation, suggest

that the two forests have greatly diﬀerent situational

classiﬁcation abilities and may produce a stronger over-

all model if both used in an ensemble. This conjecture is

strengthened through a preliminary experiment; a vote

of maximum probability between the two best models in

this experiment (QF(60) and RF(100)) produces a re-

sult of 86.96% which is a slight, and yet superior classiﬁ-

cation ability. The forests ensembled with other forests

of their on type on the other hand do not improve.

With this discovery, a future study should consider en-

semble methods between the two for both deriving a

stronger overall classiﬁcation process, as well as to ex-

plore the patterns in the ensemble of QRNG and PRNG

based learning techniques. This, at the very minimum,

would require the time and computational resources to

train 100 models to explore the two sets of ten models

produced in the related experiment, though exploring

beyond this, or even a full bruteforce of each model in-

creasing their population of forests by 1 rather than 10

would produce a clearer view of the patterns within.

Of the most noticeable eﬀects of QRNG and PRNG

in machine learning, many of the neural network ex-

periments show greatly diﬀering patterns in learning

patterns and their overall results when using PRNG

and QRNG methods to generate the initial weights for

each neuron within hidden layers. Following this, fur-

ther types of neural network approaches should be ex-

plored to observe the similarities and diﬀerences that

occur. In addition to this, the architectures of networks

are by no means at an optimum, the heuristic nature

of the network should also be explored, by techniques

On the Eﬀects of Pseudo and Quantum Random Number Generators in Soft Computing 13

such as a genetic search, for it too requires the idea of

random inﬂuence Bird et al. (2019b,c).

6 Conclusion

To conclude, this study performed 8 individual exper-

iments to observe the eﬀects of Quantum and Pseudo-

random Number Generators when applied to multiple

machine learning techniques. Some of the results were

somewhat unremarkable as expected, but some eﬀects

presented profound diﬀerences between the two, many

of which are as of yet greatly unexplored. Based on

these eﬀects, possibilities of future work has been laid

out in order to properly explore them.

Though observing superposition provides perfectly

true randomness, this also provides a scientiﬁc issue in

the replication of experiments since results cannot be

coerced in the same nature a PRNG can through a seed.

In terms of cybersecurity, this nature is ideal Yang et al.

(2014); Stipcevic (2012), but provides frustration in a

research environment since only generalised patterns at

time tcan be analysed Svore and Troyer (2016). This

is overcome to an extent by the nature of repetition

in the given experiments, many countless classiﬁers are

trained to provide a more average overview of the sys-

tems.

The results for all of these experiments suggest that

data dependency leads to no concrete positive or nega-

tive eﬀect conclusion for the use of QRNG and PRNG

since there is no clear superior method. Although this

is true, pseudo-randomness on modern processors are

argued to be indistinguishable from true randomness,

but clear patterns have emerged between the two. The

two methods do inexplicably produce diﬀerent results

to one another when employed in machine learning, an

unprecedented, and as of yet, relatively unexplored line

of scientiﬁc research. In some cases, this was observed

to be a relatively unremarkable, small, and possibly co-

incidental diﬀerence; but in others, a clear division sep-

arated the two.

The results in this study are indicative of a pro-

found eﬀect on patterns observed in machine learning

techniques when random numbers are generated either

by the rules of classical or quantum physics. Their ef-

fects being positive or negative are seemingly dependent

on the data at hand, but regardless, the fact that two

methods of randomness ostensibly cause such disparate

eﬀects juxtapose to the current scientiﬁc processes of

their usage should not be underestimated. Rather, it

should be explored.

Appendices

1 Quantum Assembly Language for Random Num-

ber Generation

Note: Code comments (#) are not Quantum Assembly

Language and are simply for explanatory purposes. The

following code will place a quanta into superposition via

the Hadamard gate and then subsequently measure the

state and store the observed value. The state is equally

likely to be observed at either 1 or 0.

#Electron zero to Hadamard Gate

H 0

#Declare memory space ’ro’ of one bit

DECLARE ro BIT[1]

#Measure the qubit at 0th index of ’ro’

MEASURE 0 ro[0]

2 Python Code for Generating a String of Ran-

dom Bits

The following code generates a random 32-bit integer by

observing an electron in superposition which produces

a true random result of either 1 or 0. The result is

amended at each individual observation until 32 bits

have been generated. Decimal conversion takes place

and two ﬁles are generated; a raw text ﬁle containing

the decimal results and a CSV containing a column of

binary integers and their decimal equivalents.

from pyquil.quil import Program

from pyquil.gates import H

# Select the lattice of Qubits

lattice = "Aspen-1-5Q-B"

# Initialise QPU

qpu = get_qc(lattice)

#Place electron 0 into superposition

numbers = Program(H(0))

#Observe the superposition

getNum = numbers.measure_all()

#Print the Quantum Assembly Language

print(getNum)

compiled_program = qpu.compile(numbers)

#Length of integer to generate

numbers = 32

#How many integers to generate

toGenerate = 1

print("\n Random number of " +str(numbers) + "

bits:")

for yin range(0, 10000):

output = ""

14 Jordan J. Bird et al.

for xin range(0, toGenerate):

#Run the code on a Quantum Processing Unit

result = qpu.run(compiled_program)

#Observe the superposition

result = result[0][0]

output += str(result)

print("\n\n Random no." +str(y) + " is: " +

output)

decimal = int(output, 2)

with open("numbers.txt","a") as myfile:

myfile.write("\n" +str(decimal))

with open("random.csv","a") as myfile:

myfile.write("\n" +str(output) + "," +

str(decimal))

Acknowledgments

The authors would like to thank Rigetti Computing for

granting access to their Quantum Computing Platform.

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