[1]eBrain Lab, Division of Engineering, New York University Abu Dhabi, PO Box
129188, Abu Dhabi, UAE [2]Center for Quantum and Topological Systems (CQTS),
NYUAD Research Institute, New York University Abu Dhabi, PO Box 129188, Abu
Dhabi, UAE

# A Survey on Quantum Machine Learning: Current Trends, Challenges,
Opportunities, and the Road Ahead

Kamila Zaman1,2<EMAIL_ADDRESS>Alberto Marchisio1,2<EMAIL_ADDRESS>Muhammad Abdullah Hanif1,2<EMAIL_ADDRESS>Muhammad Shafique1,2
<EMAIL_ADDRESS>
###### Abstract

Quantum Computing (QC) claims to improve the efficiency of solving complex
problems, compared to classical computing. When QC is applied to Machine
Learning (ML) applications, it forms a Quantum Machine Learning (QML) system.
After discussing the basic concepts of QC and its advantages over classical
computing, this paper reviews the key aspects of QML in a comprehensive
manner. We discuss different QML algorithms and their domain applicability,
quantum datasets, hardware technologies, software tools, simulators, and
applications. In this survey, we provide valuable information and resources
for readers to jumpstart into the current state-of-the-art techniques in the
QML field.

##  1 Introduction

Machine Learning (ML) systems are established tools to find patterns in data
and successfully generalize complex/nonlinear problems. They can be applied to
several domains such as computer vision, health care, finance, and automotive
[1, 2, 3, 4]. However, ML practitioners have to be wary of the computing
resources required to run large ML models. Despite employing numerous
hardware-aware optimizations such as compression and approximations [5, 6, 7,
8, 9], the current computing infrastructures and technologies limit the
computation capability of the given ML system.

The high computational requirements demand advanced hardware development.
Hence, increasing the number of transistors on a chip to maximize the physical
capacity has been the most obvious solution used in the last few years.
Moore’s law states that the number of transistors doubles every two years on
an Integrated Circuit. This doubling pattern is now at full capacity with
high-end CPUs and GPUs [10]. Such physical saturation limits the computation
power and causes processing, developmental, and discovery delays in the ML
community. For instance, training Large Language Models with hundreds of
billions of parameters with trillions of tokens is extremely long and compute-
intensive [11]. It requires massive investments in time and hardware resources
that only a few high-end companies can afford. Hence, to overcome this
physical limit and support further discoveries, there is a dire need to look
for new technological avenues of hardware systems that can enhance the
computation efficiency for solving a given real-world problem by closely
simulating it and comprehending it.

Table 1: Qualitative comparison between our work and related QML surveys.

QML Survey | Year | # Pages | # Refs. | QC Details | QC Advantages | QML Algs. | Datasets/Encoding | HW Techs. | SW Tools | Applications
---|---|---|---|---|---|---|---|---|---|---
[12] | 2006 | 12 | 24 | ✓ | ✗ | ✓ | ✓ | ✗ | ✗ | ✗
[13] | 2014 | 19 | 71 | ✗ | ✓ | ✓ | ✗ | ✗ | ✗ | ✗
[14] | 2015 | 38 | 97 | ✗ | ✗ | ✓ | ✓ | ✗ | ✗ | ✓
[15] | 2016 | 17 | 100 | ✗ | ✓ | ✓ | ✗ | ✓ | ✗ | ✓
[16] | 2017 | 106 | 309 | ✓ | ✓ | ✓ | ✓ | ✗ | ✗ | ✓
[17] | 2018 | 33 | 187 | ✓ | ✓ | ✓ | ✗ | ✗ | ✗ | ✓
[18] | 2019 | 15 | 74 | ✓ | ✗ | ✓ | ✗ | ✗ | ✗ | ✓
[19] | 2019 | 29 | 309 | ✓ | ✓ | ✓ | ✗ | ✓ | ✗ | ✓
[20] | 2020 | 8 | 71 | ✓ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗
[21] | 2020 | 6 | 47 | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✓
[22] | 2020 | 17 | 165 | ✓ | ✗ | ✓ | ✗ | ✗ | ✗ | ✓
[23] | 2021 | 9 | 75 | ✗ | ✓ | ✓ | ✓ | ✗ | ✗ | ✗
[24] | 2021 | 4 | 41 | ✓ | ✗ | ✓ | ✗ | ✗ | ✗ | ✓
[25] | 2021 | 14 | 73 | ✗ | ✓ | ✓ | ✗ | ✗ | ✗ | ✗
[26] | 2022 | 6 | 21 | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗
[27] | 2022 | 17 | 154 | ✓ | ✗ | ✓ | ✗ | ✗ | ✗ | ✓
[28] | 2022 | 12 | 59 | ✗ | ✗ | ✓ | ✗ | ✗ | ✓ | ✗
[29] | 2023 | 19 | 116 | ✗ | ✗ | ✓ | ✓ | ✓ | ✓ | ✓
[30] | 2023 | 49 | 199 | ✓ | ✓ | ✓ | ✓ | ✗ | ✗ | ✓
[31] | 2023 | 337 | 851 | ✓ | ✓ | ✓ | ✓ | ✗ | ✗ | ✓
Our work | 2023 | 43 | 165 | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓

Thus, one of the most promising solutions to this bottleneck seems to be the
idea of employing Quantum Computing (QC) as an alternate technology to
distribute the demand with new chip designs and offload the server industry
into diverse directions. Initially presented by Feynman [32] and explained by
Preskill [33, 34], QC systems, by exploiting quantum mechanical phenomena,
show promising capabilities to significantly improve performance and
information processing as compared to classical systems. While natural
processes cannot be truly encompassed in classical systems, the deployment of
quantum mechanical principles on quantum computers should help to understand
their nature due to the similar principles they operate on. Moreover, QC holds
the potential to solve problems with cost efficiency using optimizations,
time-saving, and environmentally-friendly designs.

Merging QC with ML forms the Quantum Machine Learning (QML) paradigm. It
represents an excellent opportunity for researchers and industries to make
phenomenal discoveries and unravel efficient ways to solve complex real-world
problems with significant direction toward practicality and improved
performance compared to classical systems. QML opens new avenues for the
community to discover, build, and align their designs to different levels of
the quantum stack [35].

###  1.1 Scope and Contributions

In this work, we discuss how far this overlap of ML and QC has been explored
and what further potential lies ahead. This paper presents a systematic
discussion of the current status and a well-sorted outlook of the future
perspectives in the QML field. It enables the readers to grasp an adequate
understanding of QML with currently available algorithms, frameworks,
technologies, and tools. Compared to prior QML surveys [12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], we provide a
well-structured collection of information and concepts that summarize the
current state of development of QML. Table 1 illustrates the comparison
between this work and a list of related QML surveys, considering all the
important features and topics covered. From the table, it is clear that each
prior survey has at least one missing feature, while this work provides a
detailed review covering all the key aspects of QML. The contributions of this
paper are summarized below.

  * •

We present an overview of quantum computers, the current state of challenges,
existing techniques, and efforts leading toward finding viable methods to
solve them.

  * •

We review the current state-of-the-art QML algorithms, encoding techniques,
datasets, hardware technologies, software tools, and their applications.

  * •

We discuss the open research challenges and future directions.

Figure 1: Paper outline.

###  1.2 Outline

This paper is organized into different sections and sub-sections, as depicted
in Figure 1. Section 1 introduces the problem, the scope, and the
contributions of this paper. Section 2 presents foundational theoretical
concepts of general quantum computers, the challenges of the current
technologies, and the state-of-the-art methodologies to overcome these issues.
Section 3 provides an overview of the claimed and proven advantages of QC
compared to classical computing. This section discusses the motivations that
are driving researchers and industries to design QC and QML systems. Section 4
discusses the plethora of the most common QML algorithms that are present in
the literature and provides an overview of their applicability. Section 5
provides the necessary information to understand the existing methods for
manipulating quantum data in a way that it can be used in the QML workflow,
along with a curated list of dataset resources from the QML perspective.
Section 6 introduces the existing tools and technologies that are available
for QML practitioners to experiment with and investigate further. Section 7
presents the current and potential applications of QML that majorly highlight
their benefits over classical computing. Section 8 concludes the paper and
provides future outlooks in the QML field.

##  2 Quantum Computing Preliminaries

This section provides an overview of the fundamental concepts in the field of
quantum systems. It explains the core features and characteristics of QC
systems to build a strong conceptual model of quantum computers, with
necessary details for readers to understand the rest of the paper.

###  2.1 Understanding the Qubit

The bit forms the fundamental processing unit of a classical computing system.
As shown in Figure 2a, it can hold a value of $0$ or $1$ at one instance of
time. On the other hand, a quantum computer’s fundamental unit is a qubit. The
difference lies in what the qubit can represent at an instance of time against
a classical bit. As opposed to a bit, the qubit can represent a mixture of the
0 and 1 state values simultaneously (see Figure 2b). The final state is
dictated by the probabilistic occurrence of the collective system’s states of
qubits resulting from the measurement over the superposition of 0’s and 1’s
followed by the final probabilistic distribution over possible state outcomes
as solutions to the problem [36].

Figure 2: (a) Classical bit vs. (b) Quantum Bit with $\alpha$ and $\beta$ as
respective amplitude values of a state-vector, creating a superposition state
where $|\alpha|^{2}+|\beta|^{2}=1$ as per the Max-Born Rule. $|\alpha|^{2}$
and $|\beta|^{2}$ give the respective probability for each state.

The Bloch sphere is an intuitive visual representation of a two-level qubit
system for a single qubit. As shown in Figure 3, it represents the pure and
mixed states of a qubit. The pure states are distributed over the surface of
the sphere, and all the internal states within the sphere are referred to as
mixed-qubit states. It has three axes with their respective basis states on
opposite ends.

Figure 3: Bloch sphere for a single two-level qubit system.

###  2.2 Superposition

Before delving into the details of the superposition principle, we discuss two
fundamental concepts of quantum mechanics, the Wave–Particle duality and the
Max-Born rule.

The Wave-Particle duality theory states that waves can exhibit particle-like
properties while particles can exhibit wave-like properties. This definition
opposes classical mechanics or Newtonian Physics [37].

In its simplest form, the Max-Born rule states that the probability density of
finding a system in a given state is proportional to the square of the
amplitude of the system’s wave function measured at that state, usually
comprised within a state-vector representation. It is intertwined with the
measurement process of qubits [38].

The superposition concept states that a quantum system can exist in multiple
states simultaneously until it is measured. The space of the solutions is
defined as the Hilbert space. A common physical interpretation of the
superposition principle is represented by the spin of a particle, which can
assume two different spin directions (up or down) or any spin position in
between those two. Additionally, energy levels of free electrons of an atom
are also a common physical interpretation for a qubit capable of superposition
where one level corresponds to the ground state ($0$) and the other one to the
excited state ($1$) with additional possible states in between.

####  2.2.1 Single Qubit Superposition

As illustrated in Figure 4, the state of the qubit is composed of a mixed
presence of state $0$ and state $1$, i.e., there is a superposition. This
presence is dictated quantitatively by the amplitude value in context to the
wave function driven by the underlying quantum mechanics of the system. Using
the Max-Born rule, we sum the squared norm of all amplitudes of the super-
position states, which in our example are $\alpha$ and $\beta$. Converting the
states into probabilities makes the system more intuitive and interpretable.

Figure 4: A few examples for single qubit Superposition cases. (a) Fully
measured to be state $0$. (b) $70\%$ probability to be measured in state $0$
and $30\%$ for state $1$. (c) Equal probability to be measured in state $0$
and state $1$. (d) $30\%$ probability to be measured in state $0$ and $70\%$
for state $1$. (e) Fully measured to be in state $1$.

Quantum physics uses state-vectors to describe the state of the system. Figure
4 illustrates various superposition states possible for a single qubit system
resulting in probabilities. Currently, there are $5$ different arbitrarily
chosen superposition states shown for a single qubit, where each block is an
independent qubit system with its own corresponding state-vector containing
amplitude values $\alpha$ and $\beta$. We can observe that the amplitudes of
$\alpha$ and $\beta$ can hold any value from the complex number space defining
the quantum mechanical system. However, for practical use of holding and
processing information, some rules apply to the amplitude values in the state-
vector. According to the Max-Born rule, all amplitude values in the state-
vector are such that the sum of their squared norm is always equal to $1$. In
other words, we observe from the Max-Born rule that amplitudes and
probabilities are related to each other. To use the probabilities, we should
satisfy the property that their sum is equal to $1$. To meet these conditions,
the amplitude values of the state-vector must be normalized.

####  2.2.2 Two-Qubit Superposition

Let us now take one step forward and visualize the superposition samples for a
two-qubit system. In Figure 5 and Figure 6, all the principles involved in the
system are the same as those applied in a single qubit. Still, now the quantum
system can exist in $4$ (in general, $2^{n}$ for an n-qubit system)
superpositioned states with $2$ qubits at once in the system with respective
amplitudes for each state. Each figure shows one independent quantum system
with various examples of superpositions to elaborate the idea. Equation 1 also
illustrates that every state has a corresponding amplitude value ($\alpha$,
$\beta$, $\gamma$, $\eta$), whose squared value corresponds to the probability
of that state being the final state.

$system\ state=\alpha(00)+\beta(01)+\gamma(10)+\zeta(11)$ (1)

Figure 5: A few examples of two-qubit superposition cases. (a) State $00$ as
final state with $100\%$ probability. (b) State $01$ as final state with
$100\%$ probability. (c) State $10$ as final state with $100\%$ probability.
(d) State $11$ as final state with $100\%$ probability.  Figure 6: Other
examples of two-qubit superposition cases. (a) $25\%$ equal probability to be
measured in states $00$, $01$, $10$, and $11$. (b) $50\%$ probability to be
measured in state $00$ and $50\%$ for state $10$. (c) $50\%$ probability to be
measured in state $01$ and $50\%$ for state $11$. (d) $25\%$ probability to be
measured in state $00$, $50\%$ for state $01$, and $25\%$ for state $10$.

To understand the basic difference between classical and quantum systems, let
us consider an analogy that presents a problem to determine the position (A,
B, C, or D equivalent to states $00$, $01$, $10$, and $11$) of an object at a
given instance. Figure 7 shows an example of how the solution approaches
differ for classical and quantum systems. Suppose that every state corresponds
to the position of the object. In that case, the classical system determines
the object’s position by sequentially checking each of the possible states of
the solution and validating whether the object is present. It ends up with one
single state as the final solution, which in the example is position B (state
$01$), with $100\%$ probability that the object is at that position (state).

On the contrary, a quantum system for the same problem will result in the same
solution for identifying the position (state) of the object, but the approach
is different. Based on quantum mechanics principles, the quantum system
identifies the solution that results in a probabilistic distribution of the
possible solutions to the given problem, i.e., in this case, the probabilistic
positions of the object’s placement. The quantum system’s solution is that the
object is at position B (state $01$). However, it is only $60\%$ confident
that it is there and considers other possible outcomes. The mechanisms that
quantum systems follow to understand and find the solution are much closer to
the real world than classical systems.

Figure 7: Example of the probability of states at one point in time for (a) a
classical system and (b) a quantum system. This example helps to understand
the differences between the two systems. We assume that each state shows a
particular position where an object can be, i.e., positions A, B, C, or D.
Both classical and quantum computers give the same solution. The main
difference is that the classical system finds the solution after sequential
iterations on each possible state. On the other hand, the quantum system gives
a probability distribution of where the object is positioned. Both ultimately
convey that the position of the object is B.

###  2.3 Quantum Gates

Quantum logic gates are one of the essential parts of a quantum computer and
are the building blocks of all quantum algorithms. These gates are
mathematically described by unitary matrices, and their action is always
logically reversible. The gates correspond to operators used to manipulate the
quantum state of qubits. The only condition on a valid gate (and its
corresponding quantum operator) is that it should be unitary. By unitary we
mean that it meets the $U\textsuperscript{\textdagger}=U^{-1}$ condition,
where $U$ is the gate operation matrix and $U\textsuperscript{\textdagger}$ is
the adjoint of $U$. This property also ensures that it is a reversible gate
operation [39].

###  2.4 Quantum Circuits

A quantum circuit is a collection of quantum gates interconnected by quantum
wires. The actual structure of a quantum circuit, the number and the types of
gates, as well as the interconnection scheme, are dictated by the unitary
transformation, U, executed by the circuit [40]. A circuit is an independent
module composed of gates, wires, and qubits arranged in a certain way to
perform a task. Hence, its functionality is similar to what we refer to as an
algorithm. A collection of circuits are independent functional quantum units
using the output of a circuit as the input of another one to form a complete
algorithm for the solution of the given problem [41] (see Figure 8).

Figure 8: An example of a quantum circuit depicting input qubits, gates,
measurement, and output.

###  2.5 Quantum Correlations

Correlation is a term derived from statistics. Given a system that consists of
two (or more parts), the correlation quantifies how much we can predict about
the second part when knowing of the first part, compared to how much we can
predict about the second part without that knowledge. Therefore, the
correlation in classical mechanics considers statistical systems in a
deterministic way. Any observable has only one possible outcome, and the
fluctuations around that value may be due to errors or inaccuracies. However,
in quantum mechanics, the outcome of an observable does not always have to be
the same value [42]. In fact, there exists a set of possible values, playing
the role of the eigenvalues of the observable, which becomes a hermitian
matrix. Hence, in quantum mechanics, even with no perturbation on the system,
we may get varying results with respect to particular eigenvalues over time.
Such a system is described by eigenfunctions that belong to a Hilbert space.

####  2.5.1 Quantum Entanglement

Quantum entanglement is one of the central concepts in quantum information
theory. It is an unconventional and counter-intuitive phenomenon that explains
how two particles can be linked to each other even if they are separated by a
great distance. Despite their separation, a change induced in one particle
will affect the other [43]. Einstein called this phenomenon a “spooky action
at a distance” when referring to the non-local forces in a system [44]. The
generic mechanism of their interaction is illustrated in Figure 9. While the
quantum entanglement has a nondeterministic behavior, a deterministic control
over it can open avenues for new applications in several domains. Quantum
development tools (e.g., Qiskit [45], PennyLane [46]) give access to creating
and using these entanglement states for real-world problem-solving.

Figure 9: Overview of the mechanism of two-qubit entanglement. The bell states
are four specific maximally entangled (shared) quantum states of two qubits.
They are in a superposition of 0 and 1.

####  2.5.2 Quantum Decoherence

A quantum system is coherent if a definite phase relation exists between
different quantum states. This property informs us about how long a qubit can
retain its state without being affected by the external environment. It is a
desired feature for qubits and allows us to perform computations in a suitable
time frame while maintaining the system states over the required time for
processing without information loss.

Quantum decoherence is a phenomenon that describes the loss of quantum
coherence. It is evident when the persistence of a pure coherent state of the
quantum system is altered due to the system’s interaction with the external
environment. The decoherence can be interpreted as a loss of information from
the system to the environment or an interference that induces a wave function
collapse [47]. This phenomenon constitutes one of the biggest obstacles that
prevent the development of quantum computers with stable quantum states at a
large scale.

###  2.6 Quantum Noise

Leading from the idea of coherence and decoherence, quantum noise makes the
implementation of large-scale and reliable quantum computers with low error
rates extremely challenging. Therefore, it is imperative to properly
characterize the noise in quantum systems and devise mitigation techniques to
detect and correct the errors [48].

####  2.6.1 Uncertainty Principle

To some extent, quantum noise can be characterized using Heisenberg’s
uncertainty principle. According to Heisenberg’s uncertainty principle, it is
impossible to accurately know the values of specific related pairs of physical
quantities of a particle, such as velocity, position, and momentum [49]. Such
an uncertainty phenomenon generates quantum noise in the indeterminate quantum
state. Since qubits physically function on quantum mechanical principles, they
are susceptible to interference from the external environment and imperfect
fabrication.

####  2.6.2 Noisy Intermediate-Scale Quantum Era

The long-term goal for QC is to develop functional quantum algorithms that can
solve problems despite the noisy environmental systems they work in.
Completely eliminating the noise errors is extremely difficult. Hence, the
community has defined a set of achievable intermediate goals to evolve from
Noisy Intermediate-Scale Quantum (NISQ)111NISQ is a term coined by John
Preskill [33]. to Fault Tolerant Quantum Computers (FTQC); see Figure 10.

The current state of quantum computing is referred to as the NISQ era [33].
Current quantum processors only support $50$-$100$’s of qubits but are not
advanced enough to guarantee complete fault tolerance. Despite that, they
represent a valid infrastructure for experimenting and improving the designs
towards ideal systems. In the NISQ era, near-term hybrid quantum-classical
algorithms are designed and applied to various fields, such as quantum
chemistry, QML, and combinatorial optimization.

Figure 10: Quantum Information Processing Development Stages highlighting NISQ
and FTQC era phases.

####  2.6.3 Error Correction

Due to the need for algorithms that are capable of handling quantum noise in
the NISQ era, several techniques have been proposed for error correction and
mitigation. In classical systems, error correction mechanisms are based on
redundancy. If the copies do not retain the same value, a majority vote
determines the correct value. This process works well in systems with a
sufficiently low error probability, since it is most likely that only single
errors appear.

Similarly, error correction techniques for quantum systems do not correct
$100\%$ of the errors but help to reduce the effect of the noise. Unlike
classical bits that can only be affected by a flip between $0$ and $1$, qubits
can also experience phase errors, i.e., errors appearing when a qubit state
changes its phase. Moreover, quantum errors are continuous since the rotation
angle can assume any value.

Due to the no-cloning theorem [50], creating a perfect copy of a quantum state
is impossible. Hence, Quantum Error Correction (QEC) codes introduce
redundancy by spreading the information of a single qubit onto an entangled
state of multiple qubits. With this approach, it is possible to perform multi-
qubit (syndrome) measurements to extract the information about the error
without altering the quantum information. Typical QEC schemes are the three-
qubit bit flip code [51], the three-qubit phase flip code [52] and Shor’s nine
qubit code [53].

As shown in Figure 11, it is currently not possible to engineer systems with
noise rates lower than $10^{-2}$ or $10^{-3}$ per gate, but the quantum
community is confident that this threshold can be overcome in the next
$5$-$10$ years.

Figure 11: Current state of QEC development. The figure shows the physical
error rate threshold per gate that is currently not possible to overcome in
the NISQ era.

####  2.6.4 Error Mitigation

Despite sharing the same goal as QEC to reduce the impact of quantum noise,
Quantum Error Mitigation (QEM) techniques operate differently. While QEC aims
to restore the correct value after the error occurs, QEM aims to reduce or
suppress the errors that occur during computation. Moreover, as shown in
Figure 12, QEC completely removes the noise, while QEM keeps a small amount of
noise under control. A scenario where QEC is applied, called ‘hard regime’, is
theoretically ideal but difficult to realize. The other scenario where QEM is
employed, called ‘easy regime’, is experimentally easier to implement but at
the price of lower qubit protection. For QEM deployment, Mitiq [54] is an
open-source error mitigation package in Python that implements QEM techniques
like zero-noise extrapolation [55] and probabilistic error cancellation [56]
for quantum computers.

Figure 12: Error correction vs. error mitigation.

The NISQ era is characterized by relatively small-sized quantum circuits.
Moreover, quantum noise affects state preparation, gate operations, and
measurement. Due to the many qubits and large circuit depth required by QEC
codes, it is impossible to implement QEC on NISQ devices. On the other hand,
QEM techniques offer low-overhead solutions to implement quantum circuits in
an accurate and reliable way [48][52].

####  2.6.5 Fault-Tolerant Quantum Computing

Fault tolerance is a property that enables a system to continue operating not
only in normal conditions but also in the presence of hardware or software
failures. While there is not often a direct comparison between classical and
quantum information processing, the principles of fault tolerance in classical
systems are easily transferable.

In a quantum computer, the basic gates are much more vulnerable to noise than
classical transistors, since qubits’ implementations depend on manipulating
single electron spins, photon polarization, or similar fragile subatomic
particle systems. Currently, the main challenge for developing efficient and
reliable quantum hardware is the ability to maintain qubit states long enough
to perform useful computations without requiring redundancy efforts that cause
high compute inefficiency and overhead due to the duplication of states.
However, it might not be possible to engineer systems with lower error rates.
An important ongoing research direction dictates investigating alternative
technologies for qubit materials to build quantum systems that achieve
reasonable coherence times.

Additionally, the phenomenon of entanglement makes quantum systems inherently
fragile. The interactions between qubits during the execution of the quantum
circuit lead to errors. If several errors appear in an uncontrolled manner,
the QEC or QEM is overwhelmed, and the computation will fail. Therefore, the
main goal of FTQC is to control the propagation of faults between processes.

###  2.7 Quantum Stack

Quantum computers operate under a fundamentally different set of rules than
classical computers and can solve certain problems faster by exploiting
quantum effects. The development stack in Figure 13 shows an abstraction of
the quantum stages formulating a generalized quantum stack. Starting from the
Hardware or qubit level, the abstractions go up to the application level
composed of Quantum Algorithms. Since QC is at an early development stage,
industries primarily focus on the Quantum Hardware level, where the
reliability of measurements plays a crucial role in designing and choosing the
physical mediums. While simulated quantum hardware can deal with perfect
qubits, realistic quantum hardware can be implemented with superconducting
qubits, ion trapped qubits, quantum dots, and neutral atom qubits [57].

Figure 13: Quantum stack - from Hardware to Applications.

##  3 Benefits of Quantum Computing

This section presents an overview of the benefits of QC over classical
systems. It provides the necessary motivations to conduct further research and
development of quantum computers.

###  3.1 Quantum Dimensionality Reduction

A common way to reveal properties of an unknown quantum state, having many
copies of a system in that state, is to measure different observables and to
analyze the results statistically. The unknown quantum state can play an
active role in this analysis. Given multiple copies of a quantum system with
its respective density matrix, it is possible to perform the unitary
transformation. Its result generates quantum coherence among different copies
of the system. Popular techniques for dimensionality reduction are the
Principal Component Analysis (PCA) [58], where the eigenvectors corresponding
to the unknown state’s eigenvalues state are revealed in a very fast manner,
and the Quantum Linear Discriminant Analysis Dimensionality Reduction (QLDADR)
[59], where the vectors having a high-dimensional feature space are projected
onto a low-dimensional feature space.

###  3.2 Quantum Supremacy for Solving Complex Problems

The inherent problems’ difficulty, or hardness, is a fundamental concept in
computational complexity theory (see Appendix A for more details). The
hardness quantifies the resources required by different models of computation
to solve problems, such as the number of steps of a deterministic Turing
machine [60]. Various models and resources are often considered, including
deterministic, nondeterministic, and probabilistic models, time and space
constraints, and interactions across different models. “Quantum Supremacy” is
a concept that indicates that a programmable quantum device can solve a
problem that no classical computer can solve in a feasible amount of time. For
instance, Google’s Sycamore processor [61] performs in $200$ seconds a task
that would require $10,000$ years using a classical computer. Such an
acclaimed advantage of QC compared to classical computers demands a
modification of the complexity theory for problems that are solvable on a
quantum computer. Hence, the classical complexity classification diagram has
been redefined to incorporate the quantum perspective and evaluate the
complexity of the issues, considering their ability to be solvable on a
quantum computer. Figure 14 shows that a set of problems outside the P space
that can be solved in Bounded-error, Quantum, and Polynomial-time (BPQ) [62].
Identifying BPQ problems is crucial to prove the overwhelming ability of
quantum computers compared to classical computers.

Figure 14: Quantum Computing complexity theory. The classical complexity
theory is extended with the class of Bounded-error, Quantum, and Polynomial-
time (BPQ) problems.

###  3.3 Preserving Maximum Information

Quantum computers are machines that execute calculations using the
characteristics of quantum mechanics. What is occurring at the quantum scale
follows the different laws of physics from what is known empirically from
classical physics. Due to the superposition phenomenon, a qubit can represent
the dual state of $0$ and $1$ simultaneously, while a classic binary bit can
only take either $0$ or $1$ at a time. Hence, to express n-bit combinations
using a classical computer, we need $2^{n}$ combinations, which can be checked
sequentially before the classical computer can find the solution. On the other
hand, a quantum computer takes advantage of its wave nature so that the wave
interference increases the probability of the desired state and decreases the
other to reach a correct solution effectively all at once.

###  3.4 Large-Scale Infrastructures and Design Automation

Infrastructure is an asset to any economy. Its development and maintenance,
despite being difficult, have an enormous contribution to economic development
and prosperous future growth. Continuous needs for humanitarian survival and
improved quality of life are attracting necessary investments in
transportation, civil engineering, and health sectors. Developing systems that
provide smooth processing for these domains is challenging, given the burden
of data required to process and handle with the aspects of time criticality,
user safety, and reliability. The potential of QC to provide extensive grounds
to optimize these processes defines new out-of-the-box ways for
infrastructural development that are entirely new and creatively solve the
problems at hand in a way that was impossible with classical computers.

The realization of large-scale quantum systems will take several years.
However, it is the ultimate target for all significant quantum hardware
companies. Their roadmaps aim to have such large-scale systems available for
industrial and practical use around the end of the NISQ era. Figure 15 shows
the development roadmaps provided by IBM [63] and Honeywell [64] for their
current progress and plans towards achieving large-scale quantum
infrastructures. These milestones demonstrate the continuous focus on
developing quantum chips with more qubits every year, like the 432-qubit
Osprey machine by IBM [65]. Similarly, Honeywell’s H-series [66] has stages
planned from H1 to H5 as their progress from 10 qubit systems to larger scale
qubit quantum systems envisioned to be designed around the year 2030.

Figure 15: Roadmap to design large-scale quantum chips by Honeywell and IBM.

##  4 Quantum Machine Learning Algorithms

This section provides an overview of QML algorithms and their domain
applicability. ML is a class of advanced algorithms that perform a certain
task. Given a large number of inputs and desired outputs, an ML model can be
trained to make predictions on unseen data. If it is executed on quantum
computers, it becomes a quantum ML algorithm. An overview of the existing QML
algorithms is shown in Figure 16.

Figure 16: Overview of QML algorithms.

###  4.1 Categorization of QML Approaches

Before diving into the details of QML algorithms, it is important to
characterize different approaches based on the type of data and type of
processor used to solve the problem [12]. The four categories (see Figure 17)
are formed based on whether the data is classical (C) or quantum (Q) and
whether the algorithm runs on a classical (C) or quantum (Q) computer.

Figure 17: Categories of QML based on types of data and processor.

  * •

CC refers to processing Classical data using Classical computers, but using
algorithms inspired by quantum computing, such as the recommendation system
algorithm.

  * •

CQ refers to processing Classical data using Quantum machine learning
algorithms and will be the main focus of this chapter.

  * •

QC refers to processing Quantum data using Classical machine learning
algorithms. This is an active area of investigation, with classical machine
learning algorithms used in many quantum computing areas, such as qubit
characterization, control, and readout

  * •

QQ refers to processing Quantum data using Quantum machine learning
algorithms. It is also known as Fully Quantum Machine Learning (FQML). It can
be considered a future investigation area that can be developed during a more
mature stage of quantum computing.

###  4.2 Parametrized Quantum Circuits

Variational or Parametrized Quantum Circuits (PQCs) are specific types of
quantum algorithms that depend on free parameters. PQCs allow us to utilize
the existing quantum computers to their full extent. In the context of QML,
PQCs are used either to encode the data, where the parameters are determined
by the data being encoded, or as a quantum model, where the parameters are
determined by an optimization process. PQCs can be interpreted as ML models,
considering a variational quantum classifier that uses two variational
circuits (see Figure 18). The first circuit associates the gate parameters
with fixed data inputs, while the second circuit depends on free and trainable
parameters. This setup provides the basic building blocks to build QML
algorithms on NISQ devices [67].

Figure 18: Example of a parametrized quantum circuit.

A Variational Quantum Algorithm (VQA) combines a classical optimizer with a
PQC. VQAs represent a valid solution to enhance the quantum advantage over
classical systems, despite using NISQ devices with several constraints, such
as the limited number and connectivity of qubits and decoherence errors that
limit quantum circuit depth [68].

###  4.3 Quantum Optimization Problems

An optimization problem aims to find the best solution to a given challenge.
For instance, the goals of a business are to minimize the production cost or
maximize its revenue. The solution can either be discrete if the variable to
optimize is determined from a countable set or continuous if the optimized
value is found from a continuous function. It can also be either single-
objective or multi-objective. Applying QML to solve optimization problems
enables us to employ quantum classification and regression programs on QML
infrastructure (see Figure 19). The intrinsic parallelism of quantum computing
can speed up the optimization computation to compute the global minimum (or
maximum) faster than classical computing [69].

Figure 19: Solving a combinatorial Optimization Problem with hybrid quantum
models. Hamiltonians (H) are always square matrices with lengths equal to
$2^{n}$, where $n$ is the number of qubits in the Hamiltonian.

####  4.3.1 Quantum Approximate Optimization Algorithm

The Quantum Approximate Optimization Algorithm (QAOA) [70] is a technique that
finds approximate solutions to combinatorial optimization problems. It is
based on PQCs that approximate the adiabatic evolution from an initial
Hamiltonian. A set of parameters are tweaked to optimize a cost function out
of the quantum circuit output. The QAOA method is a promising candidate for
achieving quantum advantage over classical systems in the NISQ era.

####  4.3.2 Variational Quantum Eigensolver

A Variational Quantum Eigensolver (VQE) is a hybrid algorithm that uses both
classical computers and quantum computers to find the ground state of a given
physical system. The VQE algorithm is versatile and can be applied to a wide
range of tasks, making it a promising candidate for use in many applications
of quantum computers, including machine learning and control theory [71]. One
particularly useful application of the VQE algorithm is finding the ground
state energy (i.e., the minimum eigenvalue) of a Hamiltonian. This is
accomplished by minimizing a cost function based on the expectation value of
the Hamiltonian. An additional term in this algorithm accounts for the overlap
between the excited and ground states.

An instance of VQE requires the definition of two algorithmic sub-components,
which are a quantum trial state (also called Ansatz [72]) and a classical
optimizer. The optimizer varies the parameters of the Anzatz, such that it
works towards a state, as determined by its parameters, that results in the
minimum expectation value being measured by the input operator (Hamiltonian).
Additionally, the VQE can be used to model complex wave functions in
polynomial time, making it one of the most promising NISQ applications for
quantum computing.

####  4.3.3 Quantum Annealing

The Quantum Annealing (QA) method is an algorithm to solve combinatorial
optimization problems. Instead of using the temperature to explore the problem
space, the QA uses the laws of quantum mechanics to measure the energy state.
Starting from the qubits’ dual state, where all the possible combinations of
the values are equally likely to happen, the quantum mechanical effects are
gradually reduced through a process called quantum fluctuation [73] to reach
the most optimal point where the lowest energy state is achieved.

Figure 20: Example of quantum annealing.

QA techniques require an objective function to be defined in the Ising model
or Quadratic Unconstrained Binary Optimisation (QUBO). The Ising model is a
mathematical model in statistical mechanics [74]. The QUBO [75] is
mathematically equivalent to the Ising model and can be used to formulate a
problem in a more simple way.

In physics, the minimum energy principle [76] states that the internal energy
decreases and approaches the minimum values. If we can formulate our problems
as an energy minimization problem, the QA method can search for the best
possible solution by utilizing quantum methods over an energy landscape.

###  4.4 Quantum Neural Networks

Quantum Neural Networks (QNNs) are computational Artificial Neural Network
(ANN) models that are based on the principles of quantum mechanics [77]. As
shown in Figure 21, typically in QNNs, the input data is loaded with classical
data inputs. The quantum circuit contains a feature map module with input
parameters and an Ansatz module with trainable weights. Measurements are
conducted to obtain the outputs.

Figure 21: Example of a Quantum Neural Network.

What makes QNNs exciting compared to traditional ANNs is the differentiable
nature of the quantum circuits. Quantum computers can compute the changes in
the control parameters needed to make the QNN better at a given task.

During the NISQ era, the main focus is on Hybrid Quantum Neural Networks
(HQNNs). The generic structure of an HQNN is illustrated in Figure 22, while
specialized model types are discussed in the following sections.

Figure 22: Functionality of a Hybrid Quantum Deep Neural Network. The
classical input is converted into the quantum domain by encoding the data as a
quantum state. After measurements, a quantum to classical output mapping is
required to obtain the final results.

####  4.4.1 Quantum Convolutional Neural Networks

Convolutional Neural Networks(CNNs) are established architectures for image
classification tasks in the computer vision domain [78]. Quantum Convolutional
Neural Networks (QCNNs) are HQNN architectures whose structure is inspired by
classical CNNs (see Figure 23). The QCNN circuit model proposed by Cong et al.
[79] extends fundamental properties of CNNs to the quantum domain by using
only $O(log(N))$ variational parameters for input sizes of $N$ qubits.
Therefore, it enables efficient training and implementation of QCNNs on
realistic, near-term quantum devices.

Figure 23: Functionality of a Quantum Convolutional Neural Network.

The structure of a classical CNN consists of applying alternating
convolutional layers (with an activation function) and pooling layers,
typically followed by fully-connected layers before the output is generated.
In QCNNs, the convolution operations are performed as parameterized unitary
rotations based on PQCs, executed on neighboring pairs of qubits. These
convolutions are followed by pooling layers. The scope of pooling layers is to
down-sample the results of the previous layer by reducing the data dimensions
while inherently extracting the most relevant features. The dimensionality
reduction with pooling layers reduces the dimensions of the quantum circuits,
which in turn reduces the number of qubits in the circuit while retaining the
maximum information possible from predecessor layers. Consequently, it reduces
the computational cost of the complete circuit since the number of learnable
parameters of the QCNN is equal to the qubits in the circuit. However, in
quantum physics, it is not possible to directly remove qubits from the
circuit. Hence, the pooling layers in quantum circuits are deployed by
measuring a subset of the qubits and using these measurement results to
control subsequent operations. The fully-connected layers are implemented as a
multi-qubit operation on the remaining qubits before the final measurement.
All the parameters involved in these operations are learned during training.

While the classification task is the primary purpose for using QCNNs, they can
also devise a quantum error correction scheme optimized for a given error
model [79]. However, QCNNs with large circuit depths are challenging to
implement due to the high coherence required for qubits.

####  4.4.2 Quanvolutional Neural Networks

Motivated by the idea of a convolution layer where, instead of processing the
entire input data with a global function, a local convolution is applied,
Quanvolutional Neural Networks (QuanCNNs) [80] are based on a “quanvolutional”
kernel. Compared to QCNNs, where the complete architecture is implemented with
PQCs, QuanCNNs focus only on building efficient convolutional layers using
quantum circuits (see Figure 24). Considering that the underlying basic
principle of convolutional layers is to extract features from data
hierarchically, a quanvolutional layer leverages the ability of quantum
systems to contextualize greater information capacity. Therefore, QuanCNNs
introduce preliminary layers before the classical CNN to manipulate (i.e.,
pre-process through transformations) the inputs using a single or a set of
consecutive quantum layers. The input of the first classical layer is a
quantum output measured classically that preserves a large amount of
information for improving the contextual feature extraction in the classical
layers. In other words, it aims to encapsulate the input information using a
representation that results from the quantum space transformations. This
approach complements and facilitates the architecture in identifying
correlated features in the subsequent (classical) layers.

Figure 24: Functionality of a Quanvolutional Neural Network.

Considering the current limits of the quantum hardware, where the quantum
circuit implementation of many qubits is challenging, QuanCNNs represent a
valid solution for circumventing this obstacle since the deeper layers are
still implemented in classical computing. The advantage of quantum computing
is that it is used for implementing convolutional layers, where low-qubit
circuits are sufficient to perform the required task well.

####  4.4.3 Qauntum Recurrent Neural Networks

Recurrent Neural Networks (RNNs) are the foundational models for using
sequential data to solve temporal problems such as natural language
processing. Inspired by the structure of a VQE quantum circuit, a Quantum
Recurrent Neural Network (QRNN) replaces the RNN layers with PQCs. While VQE
circuits are very dense, QRNN cells are highly structured circuits with fewer
parameters that are reused. Each parameter has a higher-level logical unit
than the VQE components [81]. Applying multiple QRNN cells iteratively on the
input sequence generates a QRNN (see Figure 25) that behaves similarly to
classical RNNs. However, due to the decoherence issues, the QRNNs’ performance
on long sequential data is typically worse than classical RNNs. The coherence
constraints can be relaxed if the QRNN architecture is built by stacking the
recurrent blocks in a staggered way [82].

Figure 25: Functionality of a Quantum Recurrent Neural Network.

####  4.4.4 Quantum Generative Adversarial Networks

A Generative Adversarial Network (GAN) aims to generate data that resembles
the original samples used in the training dataset. This is typically
implemented using a generator that produces new (fake) data and a
discriminator that distinguishes fake from real data. Fully Quantum GANs
(QGANs) typically require too large resources that cannot be implemented in
near-term quantum devices. To overcome this limitation, it is possible to
design hybrid QGANs where either the generator or the discriminator is
implemented with classical computing. For example, the QGAN architecture
proposed by Huang et al. [83] has a quantum generator and a classical
discriminator. The generator is divided into multiple sub-generator blocks,
each of them responsible for generating a patch of the full image. As shown in
Figure 26, the generated image is formed by concatenating the patches. This
approach can be implemented on quantum computers with a limited number of
available qubits, where the same device can execute the sub-generator
sequentially. This architecture provides guidance for developing advanced QGAN
models on near-term quantum devices and opens up an avenue for exploring
quantum advantages in various GAN-related tasks.

Figure 26: Functionality of a Quantum Generative Adversarial Network, where
the quantum generator is split into multiple sub-generators that craft patches
of the full image.

####  4.4.5 Quantum Autoencoders

Autoencoders (AEs) are self-supervised ML models that reduce the size of the
input data by reconstructing it. It can compress and encode information from
the input using representation learning. Quantum counterparts of AEs, called
Quantum Autoencoders (QAEs) [84], aim to reduce the dimensionality of quantum
states. As shown in Figure 27, a QAE is composed of three layers, which are
input, bottleneck, and output. Its primary use is for digital compression,
where the information can be encoded into a smaller amount of qubits. QAEs
enable the transformation and mapping of large problems into smaller quantum
circuit equivalents. Moreover, QAEs are useful for denoising since they
extract relevant features from the initial quantum state into encoded data
while neglecting the additional noise [48].

Figure 27: Functionality of a Quantum Autoencoder, where the input state is
compressed through the encoder and then decompressed through the decoder.

###  4.5 Quantum Kernels

Kernel methods are a collection of pattern analysis algorithms that employ
kernel functions to operate in a high-dimensional feature space, such as
Support Vector Machines (SVMs) and clustering techniques. The main objective
of SVMs is to find decision boundaries to group a given set of data points
into classes. When these classes’ data spaces are not linearly separable, SVMs
can benefit from using kernels to find these boundaries.

Similarly to classical kernels, Quantum Kernels (QKs) help to deal with data
points in high dimensional feature space. A quantum kernel can be defined as
in Equation 2, where $x$ and $y$ are $n$-dimensional inputs and $Q(x,y)$ is
the kernel function. $f$ denotes a function that performs the mapping from an
$n$-dimensional space into an $m$-dimensional space, where $n<m$. $\langle
f(x),f(y)\rangle$ denotes the dot product between the functions. This function
serves as the foundation for performing clustering and regression algorithms
in the quantum domain.

$Q(x,y)=\langle f(x),f(y)\rangle$ (2)

Constructing QKs for SVMs has been under investigation in recent works. The
performance of the resulting quantum SVMs for classification problems with a
small number of training points exceeds that of optimized classical models
with conventional kernels [85][86]. Moreover, QKs allow us to interpret the
learning process with the help of quantum information tools [87].

Quantum clustering is a class of data clustering techniques that use the
mathematical and conceptual tools of quantum mechanics. It belongs to a family
of density-based clustering algorithms, where clusters are defined by areas of
higher-density data points. Its design involves a scale-space probability
function viewed at the lowest eigenstate of a Schrödinger equation, followed
by simple analytic operations to derive a potential function whose minima
determine cluster centers (see Figure 28). The design proposed by Horn et al.
[88] was applied to a two-dimensional system but is also scalable to higher
dimensions. As demonstrated by Aïmeur et al. [89], quantum clustering for
different algorithms (divisive clustering, k-medians, and construction of a
neighborhood graph) can obtain significant speedup compared to classical
clustering.

Figure 28: Overview of Quantum Clustering, where the cluster centers are
represented in the Hilbert space.

##  5 Datasets

This section provides an overview of the existing methods for generating and
manipulating quantum data to be usable for QML tasks. After discussing the
data pre-processing methods and the data encoding techniques, we present a
list of quantum datasets.

###  5.1 Quantum Data

In quantum mechanics, the fundamental principle is the evolution of energy
systems over time, i.e., the Hamiltonian of the systems. In the domain of
quantum computing, we deal with varying states of the system for computation
and processing. Hence, quantum computing systems must be capable of handling
data dictated by these transformations, mechanics, and quantum constraints.
Quantum data can be referred to as relevant information describing the states
and evolution of the quantum system over time. These quantum features
represent input and output arguments for various quantum operations and
functions that transform the system’s quantum state from one form to another.
The collection of such data is referred to as quantum datasets.

Some examples of quantum data can be:

  * •

Factors affecting the relevant characteristics of a quantum system such as the
Hamiltonian and other observables.

  * •

A particular stationary state of the quantum system (e.g., the ground state),
from where the process of evolution begins encapsulated in a quantum
computation process.

  * •

Unitary transformations, the possible quantum operations applicable to the
system.

  * •

Result of computation attained by measurement or additional projection
operators which allow extraction of resulting distributions and expectation
values of the system.

  * •

Noise-relevant data for the systems’ evolution and other phenomena collected
via the control parameters.

The data collected for solving real-life problems are currently in classical
form. To use quantum models for these solutions, we need to understand and
process data in its quantum equivalent form. The possible types of classical
data required to create quantum state representations for quantum processing
are the following:

  * •

Discrete data (binary or integer values).

  * •

Real continuous data (floating-point values).

  * •

Complex continuous data (complex values).

While fully quantum datasets will drive the FTQC era, the goal in the NISQ era
is to understand how to encapsulate the current classical information into a
quantum state. In this regard, after understanding the quantum pre-processing,
we discuss encoding and embedding techniques for classical data into quantum
data. Then, we provide a curated list of quantum datasets along with their
properties and application domains.

###  5.2 Motivation for QML Datasets

A valuable lesson from the success of classical ML is that easily accessible
and high-quality datasets catalyze the development of new algorithms and the
improvement of existing ones. Learning from quantum data is more efficient
when executed on quantum computers. Hence, deploying ML algorithms on quantum
computers promises efficiency, accuracy, and rich data encapsulation into
computing information about real-world systems.

Readily available data drives the efficiency of research quality. High-quality
data curated for research and development catalyzes practical and
collaborative research across disciplines and fosters insightful growth.
Similarly, to experience the full potential of quantum computers to extract
information and deeply understand a system through accurate simulations of its
real-world counterpart, it is important to have quantum data available to
diverse users.

Figure 29: Data pre-processing pipelines. (a) Generic pipeline. (b) Quantum
pipeline.

###  5.3 Data Pre-Processing

Individual independent variables operating as inputs to the ML model are
referred to as features. They can be thought of as representations or
attributes that describe the data. An ML model makes accurate and precise
predictions if the algorithm can easily interpret the data’s features.

However, applying ML algorithms to noisy data would not give quality results
as the system would fail to identify features effectively. Noisy data would
introduce factors and patterns in the learned model that are different from
the actual distribution of the given problem. Therefore, data pre-processing
is important to improve the overall data quality to feed the ML models. The
generic data pre-processing pipeline in an ML system includes the steps needed
to transform or encode data so that it can be easily parsed by the machine, as
shown in Figure 29a. The same principle applies in the quantum domain, but the
quantum machines should also interpret the classical data. In this regard, a
coherent and structured pre-processing mechanism for QML in the NISQ era is an
inevitable need. Despite the generic pre-processing pipeline applicable in any
case, Figure 29b illustrates a hybrid pre-processing pipeline for quantum
model development. It ensures that the information extraction is maximized
using quantum machines and allows quantum computation to apply to any kind of
classical data.

Since QML has not yet extensively demonstrated its advantages compared to
classical ML, it is more susceptible to adoption by the industry. The current
quantum computers have few qubits to test and are noisy, making it difficult
to demonstrate the current and potential quantum advantage of QML methods.
Utilizing quantum methods for ML in the pre-processing phase, it is possible
to achieve better classical encoding and performance of quantum classifiers
[90]. Since many proposed QML applications rely on using well-known datasets,
especially in the context of quantum data with constraints generated due to
the underlying quantum mechanics for representing it as quantum states, it is
important to formulate and standardize its collection and encoding processes
over time to maintain quality [91].

###  5.4 Data Encoding

Even though the current quantum resources and methodologies provide close
estimations and approximations of real-world systems, they do not always
constitute adequate datasets for quantum classification models. The most
common approach in the NISQ era is transforming the well-known classical
datasets into the corresponding quantum representation and vice versa [91].
The fundamental step in the quantum processing pipeline is referred to as
state preparation. It allows us to process data on quantum circuits before
applying any specific quantum computation. Figure 28 shows an example of how
quantum encoding is used to project data from the classical space to the
quantum (Hilbert) space.

For tackling the data constraints in QML implementations, data encoding is a
fundamental step for representing classical data as quantum states. Encoding
layers largely influence the QML model expressivity since the data encoding
strategy defines and drives the relevant QML parameters, e.g., the features
the QML model can represent [29]. Knowing the various encoding techniques is
necessary to choose the most suitable one for solving the particular QML task.
Each encoding method has constraints due to the unstable quantum mechanical
properties that hinder the complete data encapsulation in quantum
representation. However, despite that, they successfully encode the
information into quantum states and have large margins of improvement over
time [92, 93, 94]. Figure 30 shows a generic overview of the quantum state
preparation stage. The process of embedding/encoding generates a quantum
representation of the classical data governed by the selected encoding
technique within the Hilbert space dimension.

Figure 30: Quantum state preparation model in the data processing pipeline.

Data encoding patterns describe a particular encoding as a tradeoff between
three major objectives:

  * •

The number of qubits needed for the encoding should be minimal because current
quantum devices only support a limited number of qubits.

  * •

The number of parallel operations needed to realize the encoding should be
minimal to minimize the width of the quantum circuit. Ideally, the loading
routine should have constant or logarithmic complexity.

  * •

The data must be represented appropriately for further calculations, e.g.,
arithmetic operations.

Figure 31: Overview of quantum data encoding techniques. (a) Basis Embedding.
(b) Amplitude Embedding. (c) Angle Embedding. (d) Quantum Random Access Memory
(QRAM) Encoding. (e) Qsample Encoding. (f) Instantaneous Quantum Polynomial-
time (IPQ) Embedding. (g) Quantum Approximate Optimization Algorithm (QAOA)
Embedding. (h) Hamiltonian Embedding. (i) Displacement Embedding.

The following list contains the encoding techniques that are used in the
literature. Note that the terms encoding and embedding can be used
interchangeably as they refer to the same process. Figure 31 shows an overview
of the data encoding techniques and their key characteristics.

  1. (a)

The Basis Embedding encodes the binary feature vector into a basis state. It
is primarily used when real numbers are mathematically manipulated in quantum
algorithms. Such an encoding represents real numbers as binary numbers and
transforms them into quantum states. However, since the binary features are
not differentiable, the basis encoding does not allow gradient computations
[95].

  2. (b)

In the Amplitude Embedding technique, classical data is encoded into the
amplitudes of a quantum state. Besides defining the measurement probabilities,
each quantum system’s wavefunction can be used to represent its amplitude as a
data value. The main advantage of the amplitude embedding is that it can
encode $n$ real values (or $n$ fixed-point precision approximation of real
numbers) using $\mathcal{O}(\log{n})$ qubits, i.e., it has a logarithmic
runtime dependency on the dataset size. Due to its efficiency, it is employed
in many QML algorithms [95].

  3. (c)

The Angle Embedding is performed by applying rotations on the x-axis or y-axis
using specialized quantum gates, called Pauli rotation gates, along with the
values that must be encoded. If we apply angle embedding on a dataset, the
number of rotations is the same as the number of features in the dataset [95].

  4. (d)

The Quantum Random Access Memory (QRAM) Encoding [96] is a mechanism that
allows accessing classically stored information in superposition by querying
an index register. In other words, it enables direct access to the quantum
data given the corresponding classical data value. By introducing a
conditional rotation and branch selection procedure, we can feed the QRAM with
the classical address of the required quantum data and get the qubit state at
the output.

  5. (e)

The Qsample Encoding associates a real amplitude vector with a classical
discrete probability distribution. It can be considered a hybrid case of Basis
and Amplitude Encoding because amplitudes represent the encoded information,
but the features are encoded in the qubits [95].

  6. (f)

The Instantaneous Quantum Polynomial-time (IQP) Embedding [97] employs the so-
called IQP circuit, which is a quantum circuit of a block of Hadamard gates,
followed by a set of gates that are diagonal in the computational basis. Each
diagonal gate is composed of a single-qubit RZ rotation gate, encoding the $n$
features, followed by a two-qubit ZZ entangler. The entangler, whose pattern
can be customized, encodes the product between features.

  7. (g)

The Quantum Approximate Optimization Algorithm (QAOA) Embedding [70] encodes
$n$ features into $m$ qubits, where $m>n$. It uses a layered trainable quantum
circuit inspired by the QAOA Ansatz. The feature-encoding circuit associates
features with the angles of RX rotations. The main advantage is that it
supports gradient computations with respect to both the features and the
weights.

  8. (h)

The Hamiltonian Embedding scheme encodes features into the values of the
Hamiltonian operator matrix [95]. The Hamiltonian operator of a system is a
quantum mechanical operator that corresponds to the total energy of the
system. Unlike previous methods that encode the data in the qubits’ states,
the Hamiltonian embedding method encodes the data into the operator. Since it
is highly correlated with physics, various physical QML algorithms, including
VQE and QAOA, are developed using the Hamiltonian encoding method.

  9. (i)

The Displacement Embedding encodes $n$ features into the displacement of
amplitudes or phases of $m$ modes, where $n\leq m$. It is used in continuous-
variable quantum computing models, where classical information is encoded in
the displacement operator parameters [29].

###  5.5 Quantum Datasets Collection

Table 2 shows the collection of quantum datasets with key features. The
following list provides a brief description of each dataset.

Table 2: Quantum datasets with key features.

Dataset | Ref. | Type | Volume/Count | Release Date
---|---|---|---|---
QDataSet | [98] |  One-qubit and two-qubit systems evolving |  $14$ TB (compressed), $10,000$ samples each | Aug.2021
QM7, QM8, QM9 |  [99], [100] [101], [102] | Organic molecules |  Between $7,165$ and $133,885$ molecules, with $7$-$9$ heavy atoms | 2012-2015
PennyLane Datasets | [46] | Organic molecules and spin |  $42$ geometries of $34$ molecules, $4$ spin models with $100$ configurations | Apr.2022
TensorFlow Quantum Data | [103] | Image classification | One-qubit per pixel | Mar.2020
Mendley Datasets for Quantum Circuit Mapping | [104] |  Circuit mapping problem as classification task |  $188,434$ random quantum circuits | Sep.2021
QFlow Lite | [105] | Charge state recognition |  $1,001$ \- $1,599$ measurements, $100\times 100$ \- $250\times 250$ pixel maps | Sep.2018
NTangled Datasets | [106] |  Quantum states with multipartite entanglement |  $12,040$ training samples, $4,060$ test samples | Nov.2021

  * •

The QDataSet [98], released in 2022, is composed of $52$ datasets based on
simulations of one-qubit and two-qubit systems evolving in the presence or
absence of noise. It provides a large-scale set of datasets for QML
practitioners to train, benchmark, and develop classical and quantum
algorithms for common tasks in quantum sciences, such as quantum control,
quantum tomography, and noise spectroscopy. It has been generated using
customized code running on base-level Python packages to facilitate
interoperability and portability across common QML platforms. Each dataset
consists of $10,000$ samples and includes a range of information (stored in
list, matrix, or tensor format) regarding quantum systems and their evolution,
such as quantum state vectors, drift and control Hamiltonians, unitaries,
Pauli measurement distributions, time series data, pulse sequence data for
square and Gaussian pulses, noise and distortion data. The total compressed
size of the QDataSet (compressed with Pickle and zip) is around $14$ TB, while
its uncompressed size is around $100$ TB.

  * •

The QM7, QM8, and QM9 datasets are subsets of the GDB-13 [107] and GDB-17
[108] databases, which contain billions of organic molecules. The QM7 [99] is
composed of $7,165$ molecules of up to $23$ atoms (including $7$ heavy atoms
such as C, N, O, and S), represented using the Coulomb matrix. An extension of
the QM7 dataset for multitask learning, called QM7B [100], has $13$ additional
properties (e.g., polarizability, HOMO and LUMO eigenvalues, excitation
energies) and contains a total of $7,211$ molecules. The QMB8 [101] has a
training set of $10,000$ molecules with up to $8$ heavy atoms. The QM9 [102]
contains $133,885$ molecules with up to $9$ heavy atoms. These databases serve
as benchmarks for hybrid QML models and systematic identification of
structure-property relationships.

  * •

The PennyLane Datasets [46] are composed of the Quantum Chemistry (QChem)
Datasets for common molecular systems and the Quantum Many-Body Physics (QMBP)
Datasets. The QChem datasets contain the electronic structure data for $42$
different geometries of molecules such as linear hydrogen chains, metallic and
non-metallic hydrides, and charged species. For each geometry of the molecule,
it is possible to extract molecular data (i.e., information regarding the
molecule), Hamiltonian data (i.e., Hamiltonian for the molecular system),
tapering data (i.e., features based on Z2 symmetries of the molecular
Hamiltonian for performing tapering), tapered observables data, and
variational data. The QMBP datasets contain spin models displaying quantum
correlations. For each spin system, datasets are generated for $1$-D lattices
(linear chain) and $2$-D lattices (rectangular grid). Each dataset contains
values for $100$ different configurations of tunable parameters such as the
external magnetic field and coupling constants.

  * •

The TensorFlow Quantum Data builds upon classical datasets such as MNIST [109]
and Fashion-MNIST [110]. It implements the method proposed by Farhi et al.
[103] to use binary encoding for representing each pixel with a qubit, with
the state depending on the pixel’s value. According to the experimental
results of Huang et al. [111], this method achieves quantum speedup in the
fault-tolerant regime compared to the classical system.

  * •

The Mendley Datasets for Quantum Circuits Mapping [104] helps to solve the
circuit mapping problem as a classification task. Each dataset contains
features related to the calibration data of the physical device and others
related to the generated quantum circuit for specific IBM quantum machines,
such as IBMQ Santiago, IBMQ Athens, and IBMQ 16 Melbourne.

  * •

The QFlow Lite dataset [105] contains a Python-based software suite to train
neural networks to recognize the state of a device and differentiate between
states in experimental data. It consists of $1,001$ idealized simulated
measurements with gate configurations sampling (stored as $100\times 100$
pixel maps) over different realizations of the same type of device. The QFlow
Lite represents a reference dataset for researchers to use in their
experiments for developing QML approaches and concepts. The expanded dataset,
denominated QFlow 2.0, consists of $1,599$ idealized simulated measurements
stored as $250\times 250$ pixel maps. In addition, the QFlow 2.0 dataset
includes two sets of noisy simulated measurements, one with a noise level
varied around $1.5$ times the optimized noise level, and the other with a
noise level ranging from $0$ to $7$ times the optimized noise level.

  * •

The NTangled Datasets [106] consist of trained weights for three hardware-
efficient Ansatzes), strongly entangling, and convolutional, varying the
number of qubits, depths, and types of multipartite entanglement. The states
are generated by sending product state inputs into PQCs. The NTangled Datasets
are used to benchmark QML models for supervised learning classification tasks.

##  6 Tools and Technologies

This section introduces the tools, simulators, and hardware models that are
available for QML practitioners. Since the infrastructure provided by QML
industries is rapidly evolving, it is important to keep updated about the new
technologies and software.

###  6.1 Quantum Computing Models

The first concept of a quantum computer was introduced in 1982 by Richard
Feynman [32]. Since then, a significant amount of research efforts have been
conducted that resulted in the development of real quantum computers at our
disposal for experiments [112]. Due to their different use cases and
architectures, quantum computers can be classified into quantum gate models
(or quantum circuits) and quantum annealers.

####  6.1.1 Quantum Gate Model

The gate model involves operations in quantum computation using matrix
operators, which are applied to the input qubits. These operators process the
information and generate the final output distributions. They are designed
with unitary constraints to qualify as valid quantum operators, making them
reversible. The reversibility property is necessary for introducing some form
of determinism in the quantum computation. A single operator is referred to as
a gate in the quantum circuit and can be applied independently or grouped
together with other similar quantum operators to perform computation on
qubits. A gate can be combined with other gates in varying orientations and
series such that they solve certain problems. The collection of gates arranged
in configurations respective to the problem being solved along with qubits and
measurements is referred to as a circuit (see Section 2.4). Complex problems
in the quantum domain can be solved by configuring gates in meaningful ways.
The computation done with this quantum model is referred to as the gate model.
Since it is easily interpretative and intuitive, it is nowadays the most used
quantum computation model.

####  6.1.2 Quantum Annealer

Quantum Annealing (QA) is an adiabatic quantum computation method that finds
optimal solutions to problems. It uses quantum tunneling, entanglement, and
superposition to find the solution in terms of the system’s energy. It usually
corresponds to finding the global energy minimum in the quantum energy state
plane. Instead of using temperature to explore the problem space, QA uses the
laws of quantum mechanics to measure the energy state. A quantum annealer is
relatively more noise-tolerant compared to the gate model. Quantum annealers
shine when they are used to search for the best from many possible
combinations. Such challenges are called combinatorial optimization problems
(see Section 4.3.3).

###  6.2 Quantum Hardware

Single qubits can be implemented using different materials and principles. As
shown in Figure 32, companies do not use the same technology for implementing
their quantum computers. For example, IBM Quantum uses niobium and aluminum on
a silicon base to achieve superconductivity under near absolute zero
temperature [65]. IonQ uses trapped ion quantum made of Ytterbium, a rare
metal [113]. Another example is Xanadu, which applies photonics (light
particles) in the quantum processing unit [114]. Using ion and light has
advantages over the superconductivity materials as it works under room
temperature. The superconducting qubits are more scalable and are implemented
by other companies like Google [61] and D-Wave [115]. It is exciting to see
future innovations in this area. In this section, after discussing the set of
conditions necessary for designing qubits (denoted as DiVincenzo’s criteria),
we review the qubit technologies to understand their advantages and
disadvantages and the reasons for their adoption in the industry. A detailed
view of the quantum hardware technologies is also illustrated in Table 3.

Figure 32: Qubit technologies used by companies for their quantum computers.
Table 3: Quantum hardware technologies with key features.

Brand & Technology | Processor | Ref. | Connectivity | # Qubits | Single-Qubit Native Gates | Multi-Qubit Native Gates | Release Date
---|---|---|---|---|---|---|---
IBM Superconducting Qubit | Osprey | [65] | All-to-All | $433$ | ID, RZ , SX, X |  Echoed cross resonance gate (ECR) | Nov.2022
Eagle | [116] | $127$ | Nov.2021
Egret | $33$ | CZ | Apr.2022
Hummingbird | $65$ | CX | Nov.2020
Falcon | $27$ | Nov.2019
Canary |  $5$-$16$ | Jan.2017
IonQ Trapped Ions | Forte | [117] | All-to-All | $32$ | Gpi, GPi2, Virtual Z |  Fully Entangling MS, Partially Entangling MS | 2022
Aria | [113] | $21$ | 2022
Harmony | [113] | $9$ | 2020
Xanadu Photonic Qubits | Borealis | [114] | 3D connectivity | $216$ | NA | NA | Jun.2022
X-Series | [118] | NA | NA | Mar.2021
Google Superconducting Qubits |  Sycamore (Weber QC) | [61] |  Square-Lattice Grid | $54$ |  Phased XZ, Virtual Z, Physical Z, Pauli Gates XYZ, PhasedXPowGate |  Sycamore Gate, Square Root of iSWAP, CZ, FSim Gateset, Wait Gate, Parameterized Gates | Oct.2019
Bristlecone | [119] |  $9$-qubit linear array | $72$ | NA | NA | Mar.2018
Foxtail | [120] | NA | 2016
Rigetti Superconducting Qubits | Aspen-M-3 | [121] |  Octagonal with $3$-fold connectivity ($2$-fold for edges) | $80$ |  RX (angle, qubit) RZ (angle, qubit) |  CPHASE, MEASURE (qubit, classical reg), CZ | Dec.2022
Aspen-M-2 | $80$ | Aug.2022
Aspen-M-1 | $80$ | Feb.2022
Aspen-11 | $40$ | Dec.2021
Aspen-10 | $32$ | Nov.2021
Aspen-9 | $32$ | Feb.2021
Aspen-8 | $31$ | May.2020
Aspen-7 | $28$ | Nov.2019
Aspen-4 | $13$ | Mar.2019
Aspen-1 | $16$ | Nov.2018
Acorn | $19$ | Dec.2017
Agave | $8$ | Jun.2017
Intel Spin Qubits |  Intel Quantum Hardware | [122] | NA | $12$ | NA | NA | 2023
Quantinuum (Honeywell) Trapped Ions | H2-1 | [123] |  fully connected qubits | $32$ | Rotation Gates | ZZ | Aug.2023
H1-3 | $20$ | Jun.2022
H1-2 | $12$ | Jul.2021
H-1 | $10$ | Sep.2020
Atom Computing Neutral Atoms | Phoenix | [124] |  $8$:$1$ connectivity | $100$ | NA | NA | Jul.2021
QuEra Neutral Atoms | Aquila | [125] | NA | $256$ |  Rabi Oscillations Time-dependent protocols Dynamic Decoupling Protocols |  Adiabatic State Preparation and Rydberg Blockade Rabi Frequency Enhancement Levin-Pichler gate analogues Non-Equlibrium dynamics of $2$ qubits | Jul.2021
D-Wave Systems Superconducting Qubits | Advantage | [115] |  Degree-$15$ Lattice ($15\times 15\times 12$) |  $5,000$+ | superconducting quantum annealing | superconducting quantum annealing | Sep.2022
2000Q | [126] |  Degree-$6$ Lattice ($8\times 8\times 8$) |  $2,000$+ | Quantum Annealing | Quantum Annealing | Mar.2017

####  6.2.1 DiVincenzo’s Criteria

Introduced by theoretical physicist David P. DiVincenzo in the year 2000,
DiVincenzo’s criteria [127] defines a set of conditions necessary for
constructing a quantum computer. The community widely accepts it, and, in the
current state of the field, it represents a useful tool for designing qubits.
The following list contains the fundamental characteristics for designing new
qubit modalities in suitable experimental setup.

  * •

Scalability with well-characterized qubits

  * •

Ability to initialize qubits to a simple fiducial state

  * •

Long relevant decoherence times

  * •

A “universal” set of quantum gates

  * •

A qubit-specific measurement capability

While the above criteria are necessary for designing qubits, the following two
criteria are the minimum fundamental characteristics required for quantum
communication.

  * •

Ability to interconvert stationary and flying qubits

  * •

Ability to faithfully transmit flying qubits between specified locations

####  6.2.2 Superconducting Qubits

Superconducting circuits are macroscopic in size but detain generic quantum
properties such as entanglement, quantized energy levels, and superposition of
states, all of which are more commonly associated with atoms. Their quantum
state is manipulated using electromagnetic pulses to control the electric
charge, the magnetic flux, or the phase difference across a Josephson junction
[128] (a device with nonlinear inductance and no energy dissipation). As such,
superconducting qubits represent the primitive building blocks of quantum
computers [129]. There exist multiple types of superconducting qubits, such as
flux qubits, phase qubits, charge qubits, and Transmon qubits. The most
popular type is the Transmon qubit [130] given its design capability to limit
noise effects, supporting error mitigation and correction.

####  6.2.3 Semiconducting Quantum Dots & Spin Qubits

Semiconductor charge and spin qubits based on gate-controlled semiconductor
quantum dots, shallow dopants (i.e., traces of an impurity element introduced
into a chemical material to alter its original electrical or optical
properties), and color centers in wide-bandgap materials. Semiconductor
Quantum Dots (QDs) [131] are nanoscale material clusters composed of
$102$–$105$ atoms. The size of the QDs is orders of magnitude larger than a
typical atomic radius, yet small enough to provide quantum confinement of
electrons and holes in all three spatial dimensions. Depending on their
configuration, they can be controlled by both magnetic and electric fields.
Therefore, they can be dephased by electric and magnetic field noise, with
different types of spin qubits having various control mechanisms and noise
susceptibilities.

In semiconductor quantum dots, spin qubits represent a prominent class of
solid-state qubits in the effort to build a quantum computer. The simplest
spin qubit is a single electron spin located in a quantum dot. However, many
additional varieties have been designed, some containing multiple spins in
multiple quantum dots, each with different benefits and drawbacks. Spin qubits
based on solid-state defects have emerged as promising candidates because
these qubits can be initialized, selectively controlled, and read with high
fidelity at ambient temperatures.

####  6.2.4 Trapped Ions

As the name suggests, the qubits are ions trapped by electric fields and
manipulated with lasers [52]. Trapped ions have relatively long coherence
times, which implies that the qubits are long-lived. Moreover, they can easily
interact with their neighbors.

####  6.2.5 Neutral Atoms

In the neutral atom quantum computing model, light beams manipulate arrays of
single neutral atoms to encode and measure quantum states. In these quantum
processors, a qubit is defined by one of two electronic states of an atom, and
these single neutral atoms are arranged in configurable arrays. Neutral atom
hyperfine qubits provide inherent scalability owing to their identical
characteristics, long coherence times, and the ability to be trapped in dense,
multidimensional arrays [132]. Combined with the strong entangling
interactions provided by Rydberg states [133], all the necessary
characteristics for quantum computation are met.

####  6.2.6 Photonic Atoms

One way to get scalable structures is to use photons (i.e., light particles)
to represent qubits [134]. It consists of a ring that is used for photon
storage, along with the scattering unit. Unlike other physical systems,
photonics allows us access to an infinite number of states.

####  6.2.7 Transition Metal Dichalcogenides

Among the possibilities in the solid state, a defect in Transition Metal
Dichalcogenides (TMDs), known as the nitrogen-vacancy (NV-1) center, stands
out for its robustness. Its quantum state may be initialized, manipulated, and
measured with high fidelity at room temperature. These controls can be used in
conjunction with electronic structure theory to smartly sort through candidate
defect systems for probable qubit representation [135]. However, this
technology has not really been popular in the industry until now.

####  6.2.8 Topological Nanowire Qubits (Majorana Qubits)

Majorana particles are a unique class of topological nanowire particles that
are their own antiparticles. The quasi-particles emerge from the topological
nature of systems and are bound at zero energy. They are predicted to obey
non-abelian braiding statistics such that they can “remember” their histories.
A pair of Majorana fermions represents a single qubit, and its realization
could lead the way to fault-tolerant quantum computing [136]. Due to their
stability at close to normal temperatures, Majorana Qubits work fine and are
the technology chosen by Microsoft [137].

###  6.3 Quantum Circuit Mapping

The process of circuit mapping consists of compiling a quantum circuit to the
given device. The circuit is transformed to comply with the architecture’s
limited qubit connectivity.

Combining one-qubit rotation gates and two-qubit Controlled-NOT (CNOT) gates
can realize any quantum circuit. However, CNOT gates cannot be applied to all
pairs of qubits in the currently available NISQ quantum devices. Therefore,
the circuit must be transformed into an equivalent circuit that does not
violate the limitation of the device. The transformation often requires
additional gates that limit the usability of the device [138]. The quality of
the circuit mapping can be assessed via different metrics, such as the
resulting circuit’s (two-qubit) gate count, depth, or the expected fidelity
[139].

Circuit mapping is a fundamental step in the context of QML. The factors
affecting the development of QML algorithms directly depend on the design and
optimization of circuit mapping. For instance, considering a circuit mapping
capability of coupling up to 4 qubits significantly limits the depth and size
of QNN layers.

###  6.4 Quantum Simulators and Primitives

Quantum Simulators are cloud-based classical systems emulating quantum
systems. These are useful in the NISQ era as designers fully control the
hardware characteristics that enable experiments of fault-tolerant quantum
systems. Ideal quantum systems can be simulated for long-term design and
development with no noise in the system. Moreover, noise models can be
introduced and analyzed to comprehend the intuitive and counterintuitive
behaviors of the quantum systems under noise.

Quantum Primitives are foundational building blocks for designing and
optimizing quantum workloads. They provide options to customize the iteration
and execution of programs to maximize the solution quality. A primitive is a
set of key language elements that serve as the foundation for a programming
language. Every language supports a core set of primitives that grant the
basic building blocks for instructing a processor on executing specific
operations.

To be broadly applicable, quantum computers must accurately control and
process the information stored in thousands of quantum bits. However, as the
number of qubits increases, noise and cross-talk tend to increase the error
rate rapidly. In this thrust, it is important to study how the error rates
scale with the increased qubit number and system connectivity. To mitigate
this, quantum computers can be built in a modular fashion, with adequate
infrastructure needed to control them efficiently, and the software
compilation of quantum algorithms should be tailored to specific system
architectures and error characteristics.

Several software simulators have been released by major companies involved in
QML research. While Table 4 summarizes the key features of the simulators, the
following list provides key details of these tools.

Table 4: Quantum simulators with key features.

Brand Simulator Name Type # Qubits Noise Model IBM `simulator_statevector`
Schrödinger wavefunction $32$ Yes `simulator_stabilizer` Clifford 5000 Yes
(Clifford only) `simulator_extended_stabilizer` Extended Clifford $63$ No
`simulator_mps` Matrix Product State $100$ No `ibmq_qasm_simulator` General,
Context-Aware $32$ Yes `sampler` Qiskit Runtime Primitive (quasi-probability
dist. generator) NA (inherited from backend) Yes (via options parameter)
`estimator` Qiskit Runtime Primitive (expectation values calculator) NA Yes
(via options parameter) Xanadu `default.qubit` Qubit Simulator up to $26$ Yes,
non-native `default.mixed` Qubit Simulator up to $16$ Yes (Native support)
`default.gaussian` Qubit Simulator NA Yes, non-native `lightning.qubit`
Quantum Photonic Simulator up to $26$ Yes, non-native `lightning.gpu` GPU-
Accelerated Simulator up to $29$ (+$3$ with MPI optimization) Yes, non-native
`lightning.kokkos` Plugin for Kokkos Library up to $29$ (+$3$ with MPI
optimization) Yes, non-native Jet tensor network contractions (acceleration)
NA Yes, non-native Rigetti Quantum Virtual Machine (QVM) Emulated Execution NA
Yes (built-in and custom) Intel Intel Quantum Simulator (IQS) Generic Qubit
Simulator $32$, $40$ NA Quantinuum (Honeywell) H2 Emulator Functional
Simulator $32$ Yes (via Quantinuum API) H1 Emulator Functional Simulator $20$
Yes (via Quantinuum API) Google Qsim Full Wave Function Simulator $40$ Yes
(via channel in Criq)

  * •

IBM Qiskit [45] is an open-source software development kit for quantum systems
at the level of circuits, pulses, and algorithms. With the help of domain-
specific APIs, Qiskit builds a software stack that makes it easy for users to
use quantum computers, as it allows practitioners to easily design experiments
and applications and run them on classical simulators or real quantum
computers. An overview of the main characteristics of Qiskit is illustrated in
Figure 33. It comprises the following components:

    * –

Qiskit Runtime: It contains computational primitives to perform foundational
quantum computing tasks and supports error correction and mitigation
techniques.

    * –

Qiskit Composer: A graphical quantum programming tool that lets users describe
the operations to build quantum circuits and run them on real quantum hardware
or simulators.

    * –

Qiskit Nature: It supports different applications and provides the necessary
components to convert classical codes into representations required by quantum
computers.

    * –

Qiskit Finance: It provides a set of illustrative applications and tools,
including data providers for real or random data, Ising translators for
portfolio optimization, and implementations for pricing different financial
options or for credit risk analysis.

    * –

Qiskit Machine Learning: It provides fundamental QK and QNN building blocks
for QML algorithms that apply them to solve different tasks such as regression
and classification. Moreover, it connects to PyTorch to enhance classical ML
workflows with quantum components.

    * –

Qiskit Optimization: It provides a range of optimizations, including high-
level modeling of optimization problems, automatic conversion of problems to
different required representations, and a suite of easy-to-use quantum
optimization algorithms.

    * –

Qiskit Experiments: It runs characterization, calibration, and verification
steps for the experiments.

    * –

OpenQASM: It is a simple text-format quantum assembly language for describing
acyclic quantum circuits composed of single-qubit, controlled single-qubit,
multiple-qubit, and controlled multiple-qubit gates. It is used to implement
experiments with low-depth quantum circuits. OpenQASM represents universal
physical circuits with straight-line code that includes measurement, reset,
fast feedback, and gate subroutines. The simple text language can be written
by hand or by higher-level tools and may be executed on the IBM Q Experience.

Figure 33: Qiskit features and modules.

  * •

Xanadu PennyLane [46] is the leading tool for programming quantum computers.
It is a cross-platform Python library that enables quantum differentiable
programming and integration with ML tools to train a quantum computer like you
would train a DNN. PennyLane also supports a comprehensive set of features,
hardware, simulators, and community-led resources that enable users of all
levels to build, optimize, and deploy quantum-classical applications easily.
An overview of its main features is depicted in Figure 34.

Figure 34: PennyLane features and modules.

  * •

Xanadu Strawberry Fields [140] is a full-stack Python library for designing
quantum algorithms and executing them directly on photonic quantum computers
such as Xanadu’s next-generation quantum hardware.

  * •

Rigetti PyQuil [141, 142] is a Python library for writing and running quantum
programs using Quantum instruction language (Quil) programs. PyQuil allows
developers to easily generate Quil programs from quantum gates and classical
operations, compile and simulate these programs using the Quil compiler
(Quilc) and the Quantum Virtual Machine (QVM), or execute them on a Rigetti
quantum processor. Quilc is an optimizing compiler for gate-based quantum
programs, capable of performing circuit transformations to produce optimal
circuit implementations for a specific Rigetti processor. Such optimizations
allow developers to write programs faster while preserving (or improving)
their execution fidelity on a given hardware system. Quil is built and
executed using the Forest SDK, a software tool that allows users to write
quantum programs in Quil.

  * •

Quantinuum (Honeywell) TKET [143] is an advanced software development kit,
accessible through the PyTKET Python package, for the creation and execution
of programs for gate-based quantum computers. It is platform-inclusive, and
its state-of-the-art circuit optimization routines allow users to extract as
much power as possible from any of today’s NISQ devices.

  * •

Google Cirq [144] is an open-source framework for programming quantum
computers, whose software library can be used for writing, manipulating,
optimizing, and running quantum circuits on quantum computers and quantum
simulators. Cirq provides valuable abstractions for dealing with today’s NISQ
computers, where hardware details are vital to achieving state-of-the-art
results.

  * •

Google OpenFermion [145] is an open-source library for analyzing and compiling
quantum algorithms to simulate fermionic systems, including quantum chemistry.
The package provides efficient data structures for representing fermionic
operators and fermionic circuit primitives for their execution on quantum
devices. Plugins to OpenFermion provide users with efficient and low-overhead
means of translating electronic structure calculations into quantum circuit
calculations.

  * •

Google TensorFlow Quantum (TFQ) [146] is a QML library for rapid prototyping
hybrid quantum-classical ML models. Research in quantum algorithms and
applications can leverage Google’s TFQ, all from within TensorFlow. It
integrates quantum computing algorithms and logic designed in Cirq and
provides quantum computing primitives compatible with existing TensorFlow
APIs, along with high-performance quantum circuit simulators.

  * •

D-Wave Leap [147] is a programming model and execution framework to execute
quantum and hybrid workloads effectively.

###  6.5 Supporting Infrastructure Tools and Hardware

To help users access quantum resources, several cloud-based platforms and
development support infrastructures have been built. The following list
describes the most popular resources used in the community.

  * •

qBraid [148] is a cloud-based quantum computing platform designed to
accelerate the pace of progress in the field. It offers access to the
ecosystem of quantum software, hardware, and an entire suite of application-
specific quantum algorithms, helping software engineers run programs on
quantum computers available from AWS, IBM, Rigetti, IonQ, and QuEra within
minutes. qBraid Lab is a web-based IDE interface (through JupiterLab) that
provides software tools for developers and access to quantum hardware. It also
includes CLI, a Command Line Interface for interacting with all parts of the
qBraid platform.

  * •

IonQ Quantum Cloud [149] provides access compatible with most Quantum SDKs
with their continuous effort to extend the support to newer tools. The cloud
provides access to all IonQ’s quantum processors and supports noisy and ideal
simulators.

  * •

Amazon Braket by AWS [150] is a fully managed quantum computing service
designed to speed up scientific research and software development for quantum
computing. Its support is useful for experimenting with quantum computing
algorithms, testing different quantum hardware, and building and developing
quantum software. It provides access to different quantum computers and
simulators using a consistent collection of development tools. Users can build
quantum projects on a trusted cloud with affordable pricing and management
controls for both quantum and classical workloads. It allows faster execution
of hybrid quantum-classical algorithms with priority access to quantum
computers and no classical infrastructure to manage.

  * •

Microsoft Azure Quantum [151] provides quantum cloud access equipped with
open-source development toolkits and environments for most leading quantum
technologies with runtime metrics. Microsoft is adopting topological qubits as
their qubit modality for developing their quantum computers. Their aim behind
using these qubits directly corresponds with the long-term scalability of
quantum computers. Since the topological qubits can survive at regular room
temperatures, their usage improves the systems’ stability compared to other
qubit technologies. The resource estimator tool provided by Azure monitors the
quantum system’s runtime statistics. It estimates the resources allocated by
the quantum code and refines the solution to a reasonable extent. This tool
calculates the number of logical and physical qubits and runtime requirements
to execute quantum applications on future-scaled quantum computers. Additional
features, like determining the number of physical qubits required by circuits
and evaluating the differences across qubit technologies, facilitate the
design of quantum solutions with varying technologies and refining them to
scaled versions for FTQC.

  * •

Zapata Orquestra [152] is a platform to develop and deploy generative AI
applications. It allows rapid prototyping, benchmarking, and executing
workflows across quantum and classical computing resources. Their objectives
focus on prioritizing use cases that can deliver a near-term financial impact
and build customer first-mover capabilities, ultimately leading to solving
computationally complex problems.

  * •

Xanadu Cloud [153] is an interface that allows the simulation and execution of
programs on quantum photonic hardware. Users can develop quantum applications
and program quantum computers with PennyLane’s library of tools,
demonstrations, tutorials, and community support forums.

##  7 Applications

QC has been pacing itself from theory to practicality over the past decades,
with immense efforts and investment attracting its attention now more than
ever. Although there are significant hurdles in achieving high computing
efficiency, a path is still paved in the direction of robust, reliable, and
useful quantum computers. There is great potential in solving QML problems in
many application areas that are discussed in this section.

It is not necessarily claimed that these applications are beneficial only when
they are fully quantum. On the contrary, investigating it under the quantum
physics lens improves some aspects of the problem compared to its basic
(classical) version. In the following domains, the introduction of quantum
technology introduces improvements in speed, compatibility, accuracy, or
memory that make them suitable to be solved using quantum technologies.

###  7.1 Molecular Simulation: Drug Design

Identifying and developing small molecules and macromolecules that may help
cure illnesses and diseases is the primary activity of pharmaceutical
companies. Given its focus on molecular formations, the pharma industry is a
natural candidate for deploying quantum computing. The molecules (also
including those that might be used for drugs) are actually quantum systems,
i.e., systems based on quantum physics. Quantum systems can predict and
simulate these molecules’ structure, properties, and behavior (or reactivity)
more effectively than conventional computing. Exact methods are
computationally intractable by standard computers, and often approximate
methods are not sufficiently accurate when there are critical interactions on
the atomic level, as is the case for many compounds. On the contrary, quantum
computers can efficiently simulate the complete system, including interactions
on the atomic level. As quantum computers become more powerful, tremendous
value will be at stake.

QC’s primary value for pharmaceuticals lies in the R&D phase of drug design.
Quantum computers can help drug developers build predictive models to identify
which molecules will likely be effective drug candidates. These models can
filter out less promising candidates early in the drug development process,
saving resources and time [154].

###  7.2 Quantum Sensing

Quantum Sensing is an advanced sensor technology that detects changes in
motion and electromagnetic fields by collecting data at the atomic level
[155].

GPS, radar, lidar, and other electromagnetic technologies use quantum physics
to provide increasingly common tools on city streets. In aircraft and even on
basic cell phones, quantum sensing is in the process of shifting from being a
highly-prized capability few can afford to being in daily use everywhere. Once
it achieves widespread adoption, quantum sensing is expected to improve
capabilities for the following domains dramatically:

  * •

Aircraft, Automobile and Electronics Manufacturers

  * •

Geology and Civil Engineering

  * •

Environmental Management

  * •

Weather Forecasting

  * •

Cosmic wave detection and simulations (gravitational wave sensing)

As witnessed by the increasing number of startups in this field, quantum
sensors are finding their way from laboratories to the real world. The atomic
length scale of quantum sensors and their coherence properties enable
unprecedented spatial resolution and sensitivity. Biomedical applications
ranging from brain imaging to single-cell spectroscopy [156] could benefit
from these quantum technologies, but evaluating the potential impact of the
techniques is not trivial.

Several projects in the field of civil engineering have launched the adoption
of quantum sensing technologies. Since quantum sensors allow more precise and
accurate measurements, they relieve and facilitate in making the right
decisions for critical tasks (e.g., underground detection of sinkholes, which
currently are manually done by experts using their personal knowledge with
minimum technological support. This causes a massive amount of resources and
time wasted due to the lack of accuracy in correctly identifying the areas
where to dig holes.

Metrology is another applicable domain for quantum sensing [157]. The current
interests in this direction demand compact and susceptible measurement
systems. Integrating electronics with quantum sensing technology should be one
of the viable areas of interest for NISQ devices.

###  7.3 Quantum Cryptography

This application domain entirely focuses on creating new cryptography schemes
that ensure security against attacks and threats by quantum breach technology.
As per our current understanding of the capability of Quantum Computers, it is
evident that they will soon reach a point where they perform reliable
computations. Keeping that in mind, currently used encryption methods for many
critical communication and transaction processes use the Rivest–Shamir–Adleman
(RSA) scheme [158]. It is fundamentally based on the keys generated from prime
factors of very large numbers. Although any classical computer can find prime
numbers of a given small number given the understanding of complexity theory,
it has been proved that for large numbers, the complexity of finding its prime
numbers becomes difficult to intractable, i.e., solvable only in indefinite
time. This particular property of the problem with classical computers made it
an obvious choice to use the RSA as an encryption method. Something that would
require thousands of years to solve with the available technology would raise
no harm. However, the works of Peter Shor [159, 160] showed the capability of
quantum computers to obtain solutions to problems that were presumably be
indefinitely solved in the classical domain. These claims opened new avenues
driven by the need to develop new cryptography methods that make the
communication Quantum Safe [161, 162]. Instead of completely changing a
classical algorithm into quantum, the focus is on adding quantum modules to
existing classical cryptography techniques. The advantage of this strategy is
that while the existing classical techniques remain intact, applying the
quantum methods improves their efficiency and speed [163]. This proved to be a
very useful domain of potential applications for QC that builds over the
existing schemes and makes them robust and efficient rather than re-inventing
the whole process.

###  7.4 Financial Modeling

By enabling more accurate and complex financial data modeling, improving risk
management, and optimizing investment portfolios, QC could lead to better
decision-making and higher returns for investors. It can also enable financial
services organizations to re-engineer operational processes [164]. QC’s
specific use cases for financial services can be classified into three main
categories: targeted predictions, trading optimizations, and risk profiling.
Quantum technology computes more efficiently than classical computing when
generating probability distributions, mapping data, testing samples, and
iterating. These properties align well with the financial modeling since they
can help enhance investment gains, reduce capital requirements, open new
investment opportunities, and improve the identification and management of
risk and compliance.

###  7.5 Logistics Optimization

Supply chain and logistics professionals have been overloaded over the past
several years. An increasing amount of uncertainties, extreme weather
conditions, and pandemic-fueled supply and demand changes have exponentially
increased logistics complexity. While businesses need supply chain
optimizations to account for the complexity of the entire ecosystem, existing
optimizations no longer provide the cure to supply chain and logistics
challenges. In this regard, QC could provide solutions for persistent supply
chain and logistics problems, such as air traffic routing, trade shipment
routing, telecom network, quantum internet, and healthcare. Organizations in
the transportation and logistics area could be among those to gain the most
advantage from quantum computing’s groundbreaking capabilities. An
organization’s path from the initial investigation of QC’s impact to actual
readiness for implementation can span a few years.

###  7.6 Environmental, Social, and Governance (ESG)

With the current environmental situation, we see the global need to establish
and ensure sustainability. Despite that, technological developments are no
further uncorrelated to the environment. Hence, while the quantum technology
itself should be designed to be environmentally sustainable, it can be a
catalyst in enabling sustainable energy solutions at a large infrastructure
scale. Quantum technology, apart from reducing the efficiency and execution
time, can also introduce and develop methods that complement Environmental,
Social, and Governance (ESG) goals. It can drastically reduce energy wastage
and fuel costs with its potential in route optimizations and sensing
technologies. Moreover, it provides solutions that empower people to leave
aside tedious and repetitive tasks and has a huge social impact by fulfilling
problems that drastically improve the quality of life. Therefore, there is
great potential to evolve the quantum field from the ESG sustainability
perspective.

##  8 Conclusion and Road Ahead

The development of the QC field, particularly QML, opens new avenues for
demonstrating the true potential of technological advancements in the above-
discussed subfields. Besides its potential to efficiently solve complex tasks,
we have to face the realistic perspective of what has been implemented and
proven so far and what is the impact of QML. Its development is rapid and
substantial, but the limitations due to the size and noise constraints
challenge the scalability and reliability of these systems. The path towards
FTQC needs to be followed promptly, aligning with the steps envisioned by the
roadmap. To enable groundbreaking advancements in this relatively new and
exciting field, it is recommended to think out of the box rather than
exclusively adopting the canonical R&D steps. For QML implementations, it is
not always necessary to find analogies between classical ML and QML, but
rather investigate alternative points of view to redefine metrics and
standards from the quantum perspective. The open research directions can be
summarized below.

  * •

It is crucial to define QML benchmark suites to provide a standard set of
metrics and workloads that enable the systematic design, evaluation, and
comparison of different QML algorithms and infrastructures.

  * •

To overcome the scalability challenge, it is important to go beyond the limit
of single quantum chips and develop multi-chip devices with efficient
interconnection and communication systems.

  * •

For noise suppression, or, in general, to enable the computations in noisy
environments, it is crucial to define protocols that characterize the noise
correlations between the different qubits in a cost-effective and scalable
manner.

  * •

Due to the current resource limitations, it is imperative to design tailored
optimizations that enable the execution of QML workloads on resource-
constraint quantum devices.

This survey provides a comprehensive overview of the current state-of-the-art
algorithms, datasets, tools, and applications of QML. It represents a
reference resource to instruct and motivate passionate readers to learn and
join forces in researching and designing innovative methodologies, tools,
algorithms, and technologies that facilitate advancements in this emerging
field.

## Acknowledgment

This work was supported in part by the NYUAD Center for Quantum and
Topological Systems (CQTS), funded by Tamkeen under the NYUAD Research
Institute grant CG008.

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