Article

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

Abstract and Figures

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 multiple 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 classification at +0.1%, and QRNG exceeded PRNG for mental state EEG classification by +2.82%. In 50 Convolutional Neural Networks (25 PRNG/25 QRNG), the MNIST and CIFAR-10 problems are benchmarked, in MNIST the QRNG experiences a higher starting accuracy 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 enhanced towards and new Quantum Random Tree (QRT) model, which has differing classification abilities to its classical counterpart, 200 trees are trained and compared (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 Random 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 classification, the best RF (100 RT) outperforms the best QRF (100 QRF) by 0.14% accuracy. In EEG classification, 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.
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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
11(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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