A Structured Survey of Quantum Computing for the Financial Industry
Franco D. Albareti111BME Inntech, Plaza de la Lealtad 1, 28014 Madrid, Phone:
+34 91 709 56 59, Email<EMAIL_ADDRESS>Thomas
Ankenbrand222Hochschule Luzern, Institut für Finanzdienstleistungen Zug IFZ,
Suurstoffi 1, 6343 Rotkreuz, Phone: +41 41 757 67 23, Email:
<EMAIL_ADDRESS>Denis Bieri333Hochschule Luzern, Institut für
Finanzdienstleistungen Zug IFZ, Suurstoffi 1, 6343 Rotkreuz, Phone: +41 41 757
67 49, Email<EMAIL_ADDRESS>Esther Hänggi444Hochschule Luzern,
Lucerne School of Computer Science and Information Technology, Suurstoffi 1,
6343 Rotkreuz, Phone: +41 41 757 68 97, Email<EMAIL_ADDRESS>Damian Lötscher555Hochschule Luzern, Institut für Finanzdienstleistungen Zug
IFZ, Suurstoffi 1, 6343 Rotkreuz, Phone: +41 41 757 67 94, Email:
<EMAIL_ADDRESS>Stefan Stettler666Inventx AG, Haldenstrasse 23
8306 Brüttisellen, Phone: +41 81 287 16 72, Email:
<EMAIL_ADDRESS>and Marcel Schöngens777SIX Financial Information
AG, Pfingstweidstrasse 110, 8005 Zürich, Phone: +41 58 399 58 28, Email:
<EMAIL_ADDRESS>

August 27, 2024

###### Abstract

Quantum computers can solve specific problems that are not feasible on
"classical" hardware. Harvesting the speed-up provided by quantum computers
therefore has the potential to change any industry which uses computation,
including finance. First quantum applications for the financial industry
involving optimization, simulation, and machine learning problems have already
been proposed and applied to use cases such as portfolio management, risk
management, and pricing derivatives. This survey reviews platforms,
algorithms, methodologies, and use cases of quantum computing for various
applications in finance in a structured way. It is aimed at people working in
the financial industry and serves to gain an overview of the current
development and capabilities and understand the potential of quantum computing
in the financial industry.

##  I Introduction

Moore’s Law states that the number of transistors in a microprocessor doubles
approximately every two years. For decades, it has correctly predicted the
increasing speed of computers and the constant reduction in the size of the
chips that drives them. Despite this continuous increase in computing power,
there are certain problems which are not solvable in practice today. Quantum
computers — computers which manipulate quantum information instead of
“traditional” bits — enable new approaches to certain complex problems. This
could reduce the computing time required to solve previously untrackable
problems to a manageable level.

A variety of applications of quantum computing for the financial industry have
already been proposed (see e.g. [1, 2] for overviews). The propositions differ
in the expected quantum speed-up, the necessary quantum resources, and the
type of quantum computer required which makes it difficult to compare them
directly. To this end, we propose a framework to analyze the use cases of
quantum computing for the financial industry. It proceeds along layers, with
each layer focusing on a different aspect.

This structured approach allows one to illustrate for which applications
quantum computing is proposed in the financial industry and to what extent it
can offer an advantage. Furthermore, it highlights how these applications are
implemented and whether there are already results of corresponding experiments
on quantum computing hardware. Quantum-inspired applications, such as tensor
networks [3], are out-of-scope in this paper. These are interesting
approaches, but they do not require quantum hardware.

The paper proceeds as follows: In Section II we provide an overview of the
current state of quantum computing. We also discuss the areas where quantum
computing could potentially be applied in the financial industry and the
implications of it. In Section III, we propose a framework for a systematic
analysis of proposals for the use of quantum computing in finance. We apply
the framework to existing applications proposed in the literature in Section
IV and draw conclusions in Section V.

##  II State of Quantum Computing in Finance

###  II-A Current State of quantum computing research

Quantum algorithms and hardware are an active area of research. In 2019, more
than 100 academic groups and government-affiliate laboratories worldwide were
researching how to design, build, and manipulate quantum bits (also called
qubits) [4]. However, to build a quantum computer which can execute arbitrary
quantum algorithms requires the ability to adaptively, reliably and precisely
prepare, control and read out a large number of qubits. The necessary
requirements are described by the DiVincenzo criteria [5]:

  * •

A scalable physical system with well-characterized qubits

  * •

The ability to initialize the state of the qubits to a simple fiducial state

  * •

Long relevant decoherence times

  * •

A "universal" set of quantum gates

  * •

A qubit-specific measurement capability

###  II-B Hardware

Various types of physical systems that could provide these properties are
being researched. To build gate-based qubits the following five types are
often considered [6]: photons, trapped atoms, nuclear magnetic resonance,
quantum dots, and superconductors. There is also a concept to use anyons (a
type of quasiparticles) to build quantum computers [7]. Recent demonstrations
of the power of quantum computational advantage over classical computers have
been constructed using superconducting qubits [8] and photonic qubits [9].
However, many of the physical systems on which current quantum computers are
based cannot be scaled easily. For quantum computing to find a broader usage,
more qubits are needed and the errors of actual hardware need to be reduced
[10]. Besides, the physical quantum computer architectures generally have
constraints on qubit connectivity such as only nearest-neighbor couplings.
Additionally, one of the most critical aspects of quantum computers is the
preservation of the quantum state of the qubits, i.e. the coherence of the
qubits. This means that while qubits are under the control of the programmer,
they must otherwise be isolated from the rest of the universe. This requires,
for example, large cooling devices [11]. In summary, the central challenge in
building quantum computers is to maintain the simultaneous ability to control
quantum systems, measure them, and at the same time preserve their strong
isolation from uncontrolled parts of their environment [6]. All these points
render it uncertain whether the current leading quantum technologies used to
create early demonstration systems will be used to create future large-scale
quantum computers.

###  II-C Simulators

Since quantum computers are still at an early stage, quantum simulators have
been developed. These are essentially software programs that run on classical
computers and allow quantum programs to be executed and tested in an
environment that predicts how qubits respond to operations. In this way,
circuits of qubits can be tested to some degree without noise and with
arbitrary connectivity. There are a variety of software libraries that can be
used to simulate the behavior of qubits for different purposes.888There is an
extensive list of tools for simulating the behavior of qubits, which can be
found on [12] [12].

In this context, it is important to discuss the exponential overhead
associated with the simulation of quantum computers with classical computers.
While a classical bit is limited to a value of either $0$ or $1$, a qubit
could be in a combination of $0$ and $1$ states, a phenomenon which is known
as quantum superposition. This superposition of states together with the
postulates of quantum mechanics for composite systems implies that, while
describing the state of a register with $n$ bits requires $n$ bits, describing
the state of a quantum register with $n$ qubits requires $2^{n}$ complex
numbers. It follows that simulating the behavior of a quantum computer on a
classical computer is exponentially expensive in terms of the number of qubits
[13] and only feasible for small systems.

###  II-D Software

Some of the hardware constraints, such as qubit connectivity or errors, can be
circumvented by software. For example, swap operations [14] can be used to
apply gates between arbitrary qubits using the available (limited)
connections. A swap gate simply interchanges the state of two qubits and can
therefore be used to move the relevant logical qubits to two connected
physical qubits where the desired gate can be applied. However, performing
swaps increases the number of operations, which has two main drawbacks. First,
it slows the algorithm down, and second, since all gates on a practical
quantum computer are noisy, it leads to more overall noise which can even
result in the algorithm becoming unusable in practice.

In a similar way, the challenge that quantum gates are noisy and quickly lead
to calculations which are unreliable can be tackled using quantum error
correction [15]. This comes at the expense of needing even more qubits. Once
quantum computers pass the size needed to implement error correction, they can
correct the errors during a calculation, which will in turn allow even larger
computations to be made.

Further development of quantum software could reduce the hardware requirements
of quantum computers and, in turn, lead to earlier applications of quantum
computers in the financial services industry. While it can be a struggle to
build the hardware that could support ambitious algorithms, the theoretical
speed-up predicted for these algorithms is also a reminder of the potential of
quantum computing. Algorithm research and hardware development therefore
incentivize each other [16]. Experimental progress is necessary to unlock the
full potential of quantum computing. However, it is possible that noisy
intermediate-scale quantum computers will find interesting applications far
before general-purpose, fault-tolerant quantum computing is available [17].

Another factor that needs to be mentioned when discussing quantum computer
development is the availability of open source software projects that lower
the barrier to learn quantum computing and thus accelerate its use.
Understanding, creating, and executing complicated mathematical models on
quantum computer hardware has become easier with the development of open
source software projects [18]. The next ten years are expected to be the
decade of quantum systems and the emergence of a real hardware ecosystem that
will provide the foundation for improving quantum computing [19].

Although more qubits are needed to solve more complex problems and quantum
coherence (the time during which a quantum state is preserved) and topology
are limited (e.g. logical operations cannot be applied to every qubit), noisy
intermediate-scale quantum computers are already available to the public.
Furthermore, there is a range of open source software and quantum computer
platforms publicly accessible to connect a personal computer to a quantum
computer over the cloud; see, e.g., [18] for a review. To run an algorithm on
a real quantum computer requires the usage of a software with a quantum
computer back end. In addition, a full-stack library allowing for the
compilation/embedding, simulation, and execution of quantum instructions on a
quantum computer is needed to implement a specific use case. footnote 10 shows
the software platforms as of the end of January 2021, using a similar
representation as [20].

Figure 1: The nodes in light blue show software platforms that can be
installed on the user’s personal computer. White nodes show simulators run
locally, i.e. on the user’s local device. In this case, the number of qubits
that can be simulated is dependent on the performance of the user’s local
computer. The quantum computing company resources are shown in clouds. The red
triangles and squares represent the requirements to get access to the
corresponding resource. The quantum simulators and usable quantum computers
provided are shown in dark green. For example, to connect to IBM’s cloud
(light green) and use one of their quantum computers, one needs to download
and install the quantum programming software QISKit locally, register on IBM’s
website to get an API key, and then request access to the quantum
device.101010Qiskit is an open source software development kit for working
with quantum computers at the level of pulses, circuits, and application
modules. For further information, see https://qiskit.org/.The number of qubits
of the quantum computers are written in brackets, where the corresponding
information is publicly available.

###  II-E Fields of quantum computing in finance

Although there are still hurdles to overcome in terms of the development of
quantum computer hardware and software, there are already multiple
publications that suggest potential applications of quantum computing in the
financial industry. In [1] the following three main fields of application are
discussed:

  * •

Optimization

  * •

Machine learning

  * •

Monte Carlo methods

In [2] the same categorization is employed, but the more general term
"Simulation" instead of "Monte Carlo" is used. In this paper, the authors also
provide examples for concrete use cases. Furthermore, they point to three
areas of financial services, where problems currently arise that pose a
challenge for classical computers. These areas include asset management,
investment banking, and retail and corporate banking. There are also numerous
other potential future fields of application related to quantum technology
such as quantum simulation, quantum money and quantum cryptography [1]. In
[21], the authors see potential applications in features selection, product
recommendations, arbitrage, portfolio management, derivative pricing, risk
analysis, credit scoring, fraud detection, anti-money laundering, and
forecasting financial crashes.

Hence, there is a wide variety of proposed applications for quantum computing
in the financial industry, even though the technology is still in its infancy.
This underscores that quantum computing could have a significant impact on the
financial industry in the future.

###  II-F Impact of quantum computing on the financial industry

The impact of quantum computing on society is discussed in [22]. The author
writes that many areas such as secure digital communication, money transfer,
and signature of digital documents are based on concepts of cryptography.
Using a quantum computer to break cryptography would have a major impact on
our economy and society. It is easy to follow his conclusion that a world
without reliable electronic payments and banking transactions would grind to a
halt. As for faster search/optimization and simulation, from a utilitarian
perspective, increased efficiency in areas such as planning, resource
allocation, machine learning, and development of new drugs and materials can
benefit society as a whole.

The literature on the impacts of quantum computers on the financial industry
or an estimate of the potential monetary benefits is scarce. It was suggested
in [21] that applying quantum computing in finance results in enhancing
investment gains, reducing capital requirements, creating new investment
opportunities, and improving risk management and compliance. It was stated in
[23] that application of quantum computing could lead to an increase in cyber-
security, accelerate high-frequency trading, or lead to greater accuracy in
the simulation of customers’ purchasing preferences, resulting in an improved
data-driven customer relation management.

The authors in [1] write that the race in the field of quantum computing is
largely motivated by the disruption the technology is expected to bring,
resulting in a complete transformation of the financial services industry.
However, the authors do not elaborate on the expected degree of disruption or
impact. The high expectations might be explained by the theoretical speed-up
compared to classical computers. However, it is still very difficult to
estimate when the technology will be available to exploit the potential of
these algorithms. For instance, in [24] it is suggested that we stand at least
five years away from quantum computing significantly impacting the financial
services landscape. This is in line with the authors of [19] who state that
widely-adopted commercial applications may remain several years away. Also,
there are no estimates as to how expensive maintaining the infrastructure and
thus running a quantum computer will be.

##  III Research Methodology

In this section, we propose a framework to systematically review the quantum
computing applications proposed in the literature for the financial industry.
In the first subsection, we discuss the selection of relevant literature. In
the second subsection, we propose the framework that is applied to the
selected literature and the use cases described in it.

###  III-A Literature Selection

As shown in Section II, various potential areas for quantum computing in the
financial industry have already been identified. To analyze specific use cases
in the financial industry in more detail, we first search for relevant
literature on the main source: Google Scholar. With this search, a broad
selection of academic literature on the topic can be covered and thus, the
existing research be identified. For the search itself, papers are excluded
that provide panel introductions; papers that are not available in English nor
German; and teaching cases and pedagogical research papers. In a next step,
the selection of the final sample is performed. Only papers that describe a
concrete application of quantum computing for the financial industry or its
implementation on a corresponding device are considered. In addition, a
requirement is that the paper must be peer-reviewed and published by 2021 at
the latest in order to be considered. In total, 13 papers that meet all
requirements are identified and are analyzed in the next subsection.

###  III-B Layered Framework Analysis

The selected literature proposes a variety of applications of quantum
computers for the financial industry. However, the proposed implementations
and expected advantages of these applications differ in many aspects. Due to
these differences, it is difficult to compare them directly. We therefore
develop a framework to compare the applications of quantum computing for the
financial industry. We neglect operational aspects such as manufacturer-
specific Application Programming Interfaces (API) and software needed for the
interaction with the quantum computer, but instead focus on a conceptional
point of view. The framework divides each proposal into four different layers,
as shown in Figure 2. While the upper layers are closely tied to the task from
the financial industry which shall be solved, the lower layers are related to
the technical aspects of implementation of quantum computers.

The top layer describes the use case in finance and the second layer contains
the corresponding methodology like optimization, machine learning, and Monte
Carlo simulation. Most applications described in the literature are based on a
small set of existing quantum algorithms and use its speed-up to improve the
results (speed, precision etc.) for the use case in the financial industry.
Therefore, the third layer describes the quantum algorithm used. The last
layer is about the quantum hardware for which an implementation has already
been developed. It also includes an assessment of whether the algorithm has
already been run on appropriate hardware and, if so, with what result.

Note that the four layers are not completely independent: the same way that
the methodologies must be selected according to the use case, there is not one
quantum algorithm which fits all methodologies. Some of the algorithms are
more suitable for near-time prototype quantum computers and others require the
implementation of larger quantum computers, but come with the advantage of
theoretically proven speed-ups (see also [25]). However, this separation into
layers allows us to see the differences between the various use cases of
quantum computing in finance.

The specific focus of each of the four layers is described in more detail in
the following paragraphs. For each layer, we include examples to illustrate
its meaning.

Use CaseMethodologyQuantum AlgorithmQuantum Hardware Figure 2: Main layers for
the use of quantum computing in finance.

####  III-B1 Use Case

The use case layer describes the specific problem for which quantum computing
can be useful in finance and what the concrete benefits are. The benefit
should result, for example, in lower costs, more revenue, or lower risk, and
ultimately in higher profitability of existing business areas or the
development of new business opportunities. This can be achieved by methods
such as calculating the price of a financial product more quickly, which can
open up an arbitrage opportunity. A benefit can also result if reputational
and financial risks in connection with, e.g., fraud can be minimized.

##### Portfolio Optimization

An important use case is portfolio optimization, where the aim is to optimize
asset allocation optimizing a certain objective function, e.g. maximizing the
Sharpe ratio [26]. In [25] it is emphasized in that although portfolio
optimization in its simplest form is relatively easy to solve, as it can be
reduced to solving a single system of linear equations, the computational
difficulty of this problem increases when more realistic constraints are added
to the problem.

##### Transaction Settlement

Transaction clearing is one further use case of quantum computing. The
clearing house’s goal is usually to settle as many transactions as possible or
to maximize the total value of settled transactions, but this is a difficult
optimization problem due to a combination of legal frameworks and additional
options. However, if more transactions could be cleared by using a quantum
computer, this increases the efficiency of markets (for an overview see [27]).

##### Estimating Risk Measures

A quantum computer could also be used to calculate risk metrics such as Value-
at-Risk (VaR), Conditional VaR (CVaR) or credit risk. The benefit of using a
quantum computer is that these measures can be calculated faster or more
accurately. This brings great advantages for financial institutions, such as
optimizing risk-bearing capital or higher profits through lower credit
defaults.

##### Pricing Financial Derivatives

Another application is in the pricing of financial derivatives, which can be
complicated functions of market prices. A significant short-term advantage for
first movers is to price actively traded derivatives and bonds faster than
other market participants. In addition, pricing bonds and derivatives using a
quantum computer instead of classical Monte Carlo simulation could contribute
to a more accurate valuation, especially for less liquid traded derivatives
and bonds with complex payout structures and contract terms. The reason for
this is that more price-driving factors could be taken into account compared
to simulations with classical computers. An advantage arises especially at the
stage when only a limited number of players have exclusive access to faster
calculation capabilities.

##### Predicting Financial Crashes

Quantum computers could make better predictions of financial crises. The topic
is highly relevant from both micro- and macroeconomic perspectives and is a
central issue in the study of macroeconomic stability. The predictability of
financial crises is widely considered to be low and very complex (for an
overview see [28]). The many financial actors and the hardly-predictable
market psychology (behavioral finance) are only two factors contributing to
this fact. Quantum computing could help to calculate complex models that take
into account many actors with their complex decision-making processes and
their interrelationships.

####  III-B2 Methodology

The literature discussed in Section II mainly refers to three areas in which
quantum computing can be useful for the financial industry. In our framework,
we classify the relevant literature identified (Section III-A) according to
the same areas.

##### Optimization

Optimization problems are the core of many financial problems for portfolio
optimization or rebalancing. These kinds of problems are thus seen as
potential use cases of quantum computing.

##### Monte Carlo (simulation)

Monte Carlo (simulation) can be used for modeling complex systems such as the
value of an asset. Using a quantum computer instead of traditional Monte Carlo
(simulation) could contribute to a more accurate/quicker solution of complex
problems.

##### Machine learning

The use of machine learning concepts, which are already used widely in the
financial sector, are expected to be used even more in the future and
represent another promising application field for quantum computing.

####  III-B3 Quantum Algorithms

From a theoretical point of view, it can be shown that a quantum computer can
compute anything a classical computer can compute [14]. Of special interest
are, of course, quantum algorithms which give an advantage over classical
algorithms. Which problems can or cannot be solved efficiently on a quantum
computer is still an open question in theoretical computer science, just as it
is for classical computers. Research in quantum computing has also spurred
progress in classical algorithms, for instance as described in [29], [30] and
[31]. Therefore, for most problems it remains open whether quantum computers
have an advantage over classical computers.

From a more practical point of view, a second question arises: for which
problems are current and near-term quantum computers useful? Such an analysis
must take into account the hardware constraints of _real_ quantum computers,
e.g. time it takes to implement specific physical transformations (i.e. ‘clock
speed’ of the quantum computer), their size (i.e. number of qubits), noise
level, transformation of initial inputs from classical computers to quantum
hardware, etc. This question has also lead to the development of algorithms
which are better suited to run on present hardware, such as hybrid quantum-
classical algorithms for optimization problems. Though selectively there are
results showing supremacy of quantum algorithms, to the best of our knowledge
there is no rigorous end-to-end consideration to paint the full picture.

Below, we describe the quantum computing algorithms often discussed in the
literature in connection with quantum computing in finance. The reader is
referred to Ref. [32] or [33] for a concise and more comprehensive list of
quantum algorithms. In this paper, we distinguish between the following
algorithms:

##### Searching and counting

[34] [34] presents a search algorithm that needs quadratically fewer queries
in an unstructured search compared to any classical algorithm and illustrates
it with searching a specific phone number in a phone book. More precisely, a
phone book with $n$ unsorted entries could be searched in $O(\sqrt{n})$ steps
on a quantum computer as opposed to $O(n)$ steps on a classical computer.

[35] [35] present generalizations of Grover’s search algorithm, called
"Quantum Amplitude Estimation (QAE)" and "approximate counting", which can be
used to estimate the number of items in a set fulfilling a certain condition
and are core subroutines in quantum computation for various applications. The
traditional approach for amplitude estimation is to use the Quantum Phase
Estimation (QPA) algorithm, which consists of many controlled amplification
operations followed by a quantum Fourier transform [14, 36].

Recently, several improved variants of Quantum Amplitude Estimation have been
introduced. One variant does not rely on QPE but is only based on Grover’s
algorithm, which reduces the required number of qubits and gates [37]. Another
approach [36] allows the quantum amplitude estimation algorithm to be
implemented without the use of expensive controlled operations.

Approximate counting is especially useful to estimate statistical properties
of a sample such as it is ‘traditionally’ often done by Monte Carlo methods.

##### Solving linear equation systems

[38] [38] propose an algorithm for solving systems of linear equations. More
precisely, the algorithm calculates the approximate result of a function of
the solution $x$ to the linear equation $Ax=b$. However, there are various
restrictions on the exact type of problems this algorithm can solve [39, 25]
which might explain why there are only very few papers using this method in
the context of finance [40].

Nevertheless, solving linear equations is a fundamental building block for
many algorithms, including machine learning algorithms. The Harrow-Hassidim-
Lloyd (HHL) algorithm has lead to several propositions for general-purpose
fast machine learning using quantum algorithms (e.g., [41] [41], [42] [42],
[43] [43] or [44] [44]), including a quantum version of principal component
analysis [45] (qPCA).

The speed-up achieved by the quantum machine learning algorithms depends on
the type of problem to be solved, but may be polynomial or even
superpolynomial [33].111111The proposed fast algorithms for recommendation
systems running on quantum computers have lead researchers to ask the question
whether there could be better classical algorithms for this problem. This has
lead to the discovery of classical algorithms ([30]) which are much faster
than the previously known ones. To apply any of the above quantum machine
learning algorithms requires, however, for the problem to be formulated in a
very specific form. Bringing a realistic problem into this form can be
challenging and may ruin the speed-up which could be gained from the quantum
machine learning algorithm [39, 46].

##### Optimization

Improvements to solve optimization problems such as semi-definite programs via
a quantum version of the interior point method (QIPM) have been proposed by
[47, 48, 49]. These algorithms are based on Gibbs sampling [50] and may reach
a quadratic speed-up [33]. Due to the generality of semi-definite programming
problems, which include e.g. second order cone programs (SOCPs) or linear
programs (LPs) possible applications of improvement are vast.

##### Approximate Optimization

In our analysis, each layer represented in Figure 2 is considered in
isolation. However, a completely isolated view is only partially possible for
quantum algorithms. While the algorithms discussed above are suitable for
gate-based quantum computations, the Ising model forms the basis for quantum
annealing (see Paragraph III-B4). This model can be used to solve
combinatorial constraint satisfaction problems [51, 52].

For many combinatorial optimization problems it is even hard to find an
approximate solution which is close to the optimum [53]. On the other hand,
there often exist algorithms which are known to find reasonably good
approximations. The Quantum Approximate Optimization Algorithms [51, 52, 54]
serve to find approximate solutions faster than the currently known classical
algorithm by exploiting quantum fluctuations [55], i.e., tunnelling effects
between local minima. Amongst the most widely used algorithms for this type of
procedure are the Variational Quantum Eigensolver (VQE) [56] and the Quantum
Approximate Optimization Algorithm (QAOA) [57]. Both of these algorithms are
classical-quantum hybrid algorithms, i.e., they combine the use of a classical
computer with a (smaller) quantum computer which can further improve the
applicability of these algorithms on current quantum computing devices [58].

A special form of combinatorial optimization problems are so-called Quadratic
Unconstrained Binary Optimization (QUBO) problems. They unify a variety of
combinatorial optimization problems that are often encountered in the
financial industry. QUBO can be mapped to the Ising model, which allows the
application of quantum algorithms such as VQE or QAOA [59].

Note that in the meantime there have been improvements in _classical_
approximation techniques for the constraint satisfaction problems where QAOA
are typically used [60]. The exact quantum advantage for these problems
therefore remains an open research question.

##### Further algorithms

The above discussion includes only a small selection of all proposed quantum
algorithms. A current list with many more algorithms can be found on [33, 61].

We have not discussed any of the algorithms for the hidden subgroup problem,
which includes Simon’s algorithm [62] and Shor’s algorithm [63]. Shor’s
algorithm is among the best-studied quantum algorithms, since it allows for
fast integer factorization and taking discrete logarithms — thereby
threatening many of the currently used crypto systems [64, 65]. We are not
aware of any results which directly apply these algorithms in finance.

####  III-B4 Quantum Hardware

The quantum hardware being considered actually for problems in finance can be
classified into gate-based quantum computers and quantum annealers.

##### Gate-based quantum computers

Most of the algorithms discussed above are intended for gate-based quantum
computers. A gate-based quantum computer system relies on the construction of
reliable qubits on which a sequence of elementary quantum logical gates can be
executed after each other. This is similar to the operations of a classical
computer where a computation consists of applying elementary operations such
as NOR, AND, OR or NOT. The elementary gates can be assembled into an
arbitrary sequence to execute a quantum algorithm. The input is the initial
state of the quantum register, comprising of all the qubits of the quantum
computer. At the end the final state of the quantum register, i.e. the result
of the computation, is measured to obtain a classical result [14].

This class of quantum computers is difficult to scale in practice [66].
Current gate-based quantum computers still have a very limited number of
qubits and are error-prone. While error correction is possible [67], this
introduces a large overhead (see, e.g., [68] for an introduction).
Nevertheless, most of the quantum computers depicted in Figure 10 are gate-
based and the recent experiments showing an advantage were achieved on this
type of quantum computer.

##### Quantum annealers

Quantum annealing is based on a general result of quantum mechanics known as
the adiabatic theorem (quantum annealing is also known as adiabatic quantum
computation [69]). The idea behind this theorem is that, when starting from a
physical system in its ground state (the state with minimum energy) and slowly
(!) modifying it, the system will remain in the ground state. The ground state
of the initial system could be easily prepared while the ground state of the
final system is the solution that is looked for. This is due to the fact that
the final physical system has been tuned in such a way its ground state
represents a solution to an actual problem of interest. Quantum annealing is
closely related with, but not limited to, optimization problems.

Even though the way computation is represented in adiabatic quantum computing
seems very different from gate-based quantum computing, the two can be shown
to be equivalent from a theoretical perspective [70]. Adiabatic quantum
computing is hoped to be more easily scalable and less sensitive to errors
than gate-based quantum computers.

##  IV Results

In this section we present the 13 selected papers and the results of the
analysis according to the layer framework. In the first subsection we give an
overview of the properties in the different layers. In the following
subsections, we then discuss the results of the corresponding papers in
detail.

###  IV-A Overview

The papers are presented in the morphological box in Figure 3. The attributes
of the different layers are shown horizontally. Each paper is represented by a
blue line. If identical properties are present for several papers, the
corresponding line depicted is superimposed and therefore appears darker. In
addition, all papers are shown in tabular form in Table I. This helps to get a
rough overview. In the following (Section IV-B to IV-D), we then discuss the
papers in detail.

Use Case |  Portfolio optimisation Portfolio optimisation |  Transaction settlement Transaction settlement |  Predicting financial crashes Predicting financial crashes |  Estimating risk measure(s) Estimating risk measure(s) |  Derivative pricingDerivative pricing
---|---|---|---|---|---

Methodology |  OptimisationOptimisation |  Monte CarloMonte Carlo |  Machine learningMachine learning
---|---|---|---

Quantum
algorithm |  Approximate Optimization (VQE QAOA, QUBO) Approximate Optimization (VQE QAOA, QUBO) |  Optimization (QIPM) Optimization (QIPM) |  Solving linear equation systems (HHL, qPCA) Solving linear equation systems (HHL, qPCA) |  Searching and counting (QAE, QPA, QPE) Searching and counting (QAE, QPA, QPE)
---|---|---|---|---

Hardware |  Quantum annealerQuantum annealer |  Gate-based quantum computerGate-based quantum computer
---|---|---

Figure 3: Morphological box representing paper selection

####  IV-A1 Use Cases

The papers propose use cases for quantum computing in finance in five
different areas. These include portfolio optimization, transaction settlement,
financial crash prediction, derivatives pricing and risk measure estimation.
The breakdown of the 13 papers across these areas is shown in Figure 4. It is
noticeable that five papers focus on the calculation of optimal portfolios
with the help of a quantum computer. Furthermore, there is one cluster with
three papers discussing the pricing derivatives with quantum computers.

Number0481216 5Portfolio optimisation1Transaction settlement2Predict. financ.
crashes2Estimat. risk measure(s)3Derivative pricing  Figure 4: Breakdown of
quantum computing applications for finance according to the "use case" layer.

####  IV-A2 Methodologies

As shown in Figure 5, eight of the 13 papers considered relate to
optimization, six to Monte Carlo simulation and none to machine learning. The
lack of papers focusing on machine learning is surprising as such methods are
already applied today using traditional computers. The reason for the
different number of papers in the three different methodologies can be
explained by the hardware requirements [25]. They described that it has been
shown that full-scale fault-tolerant (gate-based) quantum computers provide a
significant speed-up by drastically reducing the number of samples required
compared to classical Monte Carlo methods [71, 72]. The main limitation for
optimization algorithms (for gate-based quantum computer) is their dependence
on quantum linear system solvers [73, 74, 75]. These algorithms require large
depth circuits and remain beyond the reach of near and intermediate term
devices. One of the main challenges for bringing quantum optimization closer
to reality is finding ways to redesign quantum linear systems solvers, and
quantum linear algebra more generally, with reduced hardware requirements. In
contrast, quantum algorithms for machine learning require full-scale quantum
computers and also impose additional requirements (such as the availability of
quantum random access memory (QRAM), which is a quantum analogue of classical
random access memory (RAM) and could be difficult to construct) [25].

Number0481216 8Optimisation5Monte Carlo0Machine learning  Figure 5: Breakdown
of quantum computing applications for finance according to the "use case"
layer.

####  IV-A3 Quantum Algorithms

Figure 6 shows the breakdown of the quantum algorithm layer. The application
in the field of Monte Carlo simulation, namely the estimation of risk measures
or the pricing of derivatives most often use searching and counting
algorithms. However, there is one paper using a quantum algorithm for solving
linear equation systems. Most applications related to optimization or
portfolio optimization are based on approximate optimization. This is
explained by the fact that quantum annealing is mainly used for optimization
problems and uses Ising models. However, there are two publications each that
use a quantum algorithm to solver linear equation systems and quantum
optimization algorithm to find solutions to find optimal portfolios. Note that
the related algorithms discussed have been covered in detail in subsubsection
III-B3.

Number0481216 4216Searching and counting(QAE, QPA, QPE)Solving linear
equationsystems (HHL, qPCA)Optimization(QIPM)Approximate optimization(VQE,
QAOA, QUBO) Figure 6: Breakdown of quantum computing applications for finance
according to the "quantum algorithm" layer.

####  IV-A4 Hardware

Four of the use cases and their implementations relate to quantum annealer
devices and the other nine to gate-based quantum computers, as shown in Figure
7. The quantum annealer devices are mainly related to implementations for
optimization-related use cases and the Ising model as quantum algorithm. This
is not surprising, since quantum annealers are particularly well-suited for
optimization problems. Gate-based quantum computers, on the other hand, cover
both the optimization and Monte Carlo use cases and are mostly based on QPE or
QAE algorithms.

Number0481216 9Gate-based QC4Quantum annealer  Figure 7: Breakdown of quantum
computing applications for finance according to the "quantum hardware" layer.

###  IV-B Portfolio optimization

In the following, we will explain in more detail the results of the five
papers describing portfolio optimization with a quantum computer. In
particular, we will explain the domains of the considered use cases and
summaries the results.

All papers examine portfolio optimization under classical [76] portfolio
theory [76]. The problem of identifying the portfolio that maximizes return
for a given level of risk (formulated as the standard deviation of returns) is
a central task of classical portfolio theory. The three papers by [77] [77],
[78] [78] and [79] [79] discuss portfolio optimization without constraints,
which is basically a quadratic optimization problem. Note that constraints
are, for example, no short selling or budget constraints. The main difference
between the approach of [79] [79], [77] [77] and [78] [78] is that the former
allow any fraction of an asset to be held in a portfolio and is intended for
gate-based quantum computers, while the other implementations only allow
assets to either be held in the portfolio or not and is intended for quantum
annealers.

[79] [79] use the HHL algorithm and its variations to describe the theoretical
implementation to identify the minimum risk portfolio for a given desired
return. Because there are no inequality constraints, the problem becomes a
linear least squares problem for which there is a closed form solution. The
approach of [79] [79] is intended for implementation on a gate-based quantum
computer. The algorithm can in principle be efficient in the sense that it has
a running time proportional to the number of qubits involved in the
computations, i.e. the running time is $poly(log(N))$ versus $poly(N)$ with
classical algorithms. The hardware requirements are $O(N)$ for storing and
accessing the data.

[77] [77] and [78] [78] restrict to a binary optimization problem, i. e.
assets can either be included in the portfolio or not. This type of
optimization problem is also called Quadratic Unconstrained Binary
optimization (QUBO) problem. [77] [77] describe the implementation of the
problem on a quantum annealer and its hybrid solver. They try to find the
optimal portfolio of two main stock indices, the Nikkei225 and the S&P500
indices. The results are compared with conventional tools. The results show
that for problems of the size of the example used, certain quantum annealers
in its current, still limited scope, are already able to approach the
performance of the commercial solvers. [78] [78] create a hybrid algorithm
(classical and quantum) and use real financial data statistics as [77] [77]
and formulate the case also as a QUBO problem. The optimized reverse annealing
protocol turns out to be more than 100 times faster than the corresponding
forward quantum annealing (as described by [77] [77]) on average. However, it
is not clear what increase in speed could theoretically be achieved compared
to a classical approach. Another question that needs to be clarified is
whether the implementation of QUBO can be adapted so that assets in a
portfolio can also be weighted arbitrarily, which is of more practical
importance.

The papers discussed above describe promising speed-ups, both based on
theoretical as well as experimental results. In practice, however, the
portfolio optimization problem often comes with constraints. For example, one
can add constraints to ensure that the portfolio is diversified by adding
restrictions on the amount of investment in certain asset classes. [80] [80]
describe the case of positive portfolio optimization (no short selling) and
budget constraints. They emphasize that when there are no inequality
constraints (as described by [79] [79]) the Markowitz portfolio optimization
problem [76] becomes a linear least squares problem for which there is a
closed form solution. However, if there are constraints there is no closed
form solution. [80] [80] reduce this problem to Second Order Cone Programs
(SOCPs), which is a more general family of optimization problems and is the
main contribution of their work. The problem can be further solved by
considering the quantum SOCP interior-point method of [81] [81]. Theoretical
analysis shows that the running time of this algorithm scales better than
classical complexity. However, this speed-up depends on problem-dependent
parameters.

The last of our five papers relating to portfolio optimization is by [82] [82]
and deals with portfolio rebalancing which is of special relevance for
institutional investors. Portfolio rebalancing is the process of maintaining
the original risk and return characteristics of a portfolio over time, as
defined, for example, by [83]. [82] [82] define portfolio rebalancing as a
periodic asset management process in which traders maintain the net value of
an institutional portfolio. It is unclear why they refer to the net value of
the portfolio rather than the risk and return characteristics. However, as
[82] [82] write, rebalancing can be achieved through a combination of
financial instruments, including long and short positions in assets and their
derivatives. They also write that because of the computational load of
rebalancing, it is typically done overnight (when markets are closed).
However, if it can be calculated more quickly, it could enable investors to be
more agile in response to changes in markets. They formulated two approaches,
one with soft constraints and the Quantum Approximate optimization Algorithm
(QAOA), and another with hard constraints and the quantum alternating operator
ansatz. The quantum alternating operator ansatz, is the consideration of
general parameterized families of unitaries rather than only those
corresponding to the time evolution under a fixed local Hamiltonian for a time
specified by the parameter [84]. The results of the experiments from running
the algorithms on an ideal simulator highlight in particular the importance of
scaling the input data for the QUAO approach and the performance advantages of
the quantum alternating operator approach over it. However, further
investigation is needed at several critical points to show the path from the
QUAO and/or quantum alternating operator ansatz in this use case to a
potential quantum advantage. In particular, the potential, i. e. theoretical
speed-up of these approaches compared to a classical approach is not yet
clear.

Finally, it is worth mentioning the index tracking problem, which is a variant
of the portfolio optimization problem. The index tracking problem aims to
select a few assets so that the resulting portfolio’s return is as close as
possible to a given reference index. For instance, [85] [85]121212This paper
is not included in our sample because the algorithm is not yet implemented on
quantum hardware. discuss a hybrid approach to solve optimization problems
with constraints. They describe the application of quantum optimization
algorithms to index tracking. They numerically simulate the application of
quantum optimization algorithms such as QAOA and VQE to this problem, and
compare the performance of different quantum variational optimization
algorithms in their pruning method.

In summary, the relevance of faster portfolio optimization is confirmed by all
authors. There are several research directions and algorithms within portfolio
optimization. However, more research is needed to find a clear benefit for the
financial industry.

###  IV-C Transaction Settlement

One use case that is closely related to combinatorial optimization is
transaction settlement and described by [86] [86]. They focus on securities
settlement in the capital markets. By transaction settlement, they refer to
the process by which securities (tradable financial assets such as shares,
bonds and derivatives) are delivered in exchange for payment. This exchange
between parties can be facilitated by a clearing house, which also mitigates
counterparty risk. The clearing house’s goal is usually to clear as many
transactions as possible or to maximise the total value of cleared
transactions, but this is a difficult optimization problem due to a
combination of legal frameworks and additional optionalities.

[86] [86] emphasise that transaction settlement is a binary quadratic
optimizations problem. As discussed in Section IV-B, algorithms such as the
Variational Quantum Eigensolver (VQE) and the Quantum Approximate optimization
Algorithm (QAOA) exist for solving quadratic unconstrained binary optimization
problems (QUBO) with quantum computers. However, many relevant problems in
practice are Mixed Binary optimization (MBO) problems, with discrete and
continuous variables or with constraints (e. g. inequality constraints) that
cannot be modelled as part of a QUBO problem. Since the transaction settlement
problem also has corresponding constraints, it also belongs to MBO problems.

The main contribution of [86] [86] is the introduction of an approach to
extend the existing quantum methods (VQE and QAOA) to more general Mixed
Binary Optimization (MBO) problem classes. They use a hybrid quantum/classical
heuristics to address MBO problems which allows one to model optimization with
inequality constraints with a quantum computer. Furthermore, they apply this
approach to the transaction settlement problem and demonstrate the performance
of the corresponding algorithm on concrete instances using cloud-based quantum
hardware. Their algorithm presents only part of the transaction settlement
problem. However, the algorithm lays the basis for further development and
also allows for the use of problem-specific variational forms, such as those
proposed by QAOA, which may further improve the performance.

###  IV-D Derivative Pricing

The price of derivatives depends on the price of a single asset or multiple
assets (e.g. stocks) and other parameters (e.g. interest rate curve). As
elaborated by [87] [87], due to the stochastic nature of the underlying
parameters to which derivatives relate, calculating their fair value can be a
difficult task. While analytical models exist for the simplest types of
options, the simplifying assumptions about market dynamics required for the
models to provide closed-form solutions often limit their applicability.
Therefore, in most cases, numerical methods must be used for option pricing,
with Monte Carlo being one of the most popular methods due to its flexibility
and ability to handle stochastic parameters generically [87].

In this context, it is important to discuss the difference between path-
independent and path-dependent derivatives, which is also highlighted by [88].
[88]. Path-independent derivatives have a payoff function, i.e. the function
that defines the payoff structure of a financial derivatives at the exercise
date that depends on an underlying asset at a single point in time. Therefore,
the price of the asset up to the exercise date of the derivative is irrelevant
for the price of it. In contrast, the payoff structure of a path-dependent
derivative depends on the development of the asset’s price and its history up
to the exercise date. Derivatives that are path-independent and relate to a
single asset are the easiest to value, and in most cases the numerical
calculation is straightforward and would, according to [88] [88], probably not
benefit from the use of a quantum computer. Path-independent derivatives,
which relate to multiple assets, are only slightly more difficult to value
because the probability distributions must take into account correlations
between assets, but they can usually also be valued quite efficiently on
classical computers. Path-dependent derivatives, on the other hand, are much
more difficult to price than path-independent derivatives because they require
an often computationally expensive payoff calculation at multiple points in
time on each path, so minimizing the number of paths required for this step
would lead to a significant advantage in the pricing process. It is in this
last case that the greatest impact of quantum computing becomes apparent
according to [88] [88].

[40] [40] discuss the Black-Scholes-Merton model [89, 90], which considers the
pricing of options, which are a fundamental subset of financial derivatives.
They show how the relevant probability distributions can be prepared on a
gate-based quantum computer in quantum superposition, the payoff functions can
be implemented via quantum circuits, and the price of options can be extracted
via quantum measurements. Their most important contribution is the use of the
amplitude estimation (AE) algorithm, which can theoretically achieve a
quadratic speed-up in the number of steps required to estimate the
corresponding price compared to the Monte Carlo approach. In other words, a
quadratic speed-up in the number of samples required to estimate the price of
the derivative to a given error: If the desired error is $\epsilon$, then
classical methods show $1/\epsilon$ dependence in the number of samples, while
the quantum algorithm shows $1/\epsilon^{2}$ dependence. Another promising
point about this approach is that it can, in principle, be applied to any
derivative whose payoff is a function that can be efficiently decomposed into
elementary arithmetic operations within a quantum circuit.

[88] [88] extend the approach of [40] [40], i.e. of using AE algorithms to
price options, and place a strong emphasis on implementing the algorithms in a
gate-based quantum computer. In detail, they show the implementation of the
quantum circuits required to generate the input states and operators needed by
AE to price the different option types, and apply a simple error mitigation
scheme. Furthermore, they investigate the performance of their approach using
a quantum device. The results are affected by readout errors and errors that
occur during the execution of the circuits. However, the accuracy of the
calculated option price was dramatically increased by the simple error
mitigation scheme: it reduced the error averaged over the initial spot price
from 62% to 21%. They came to the conclusion that much larger quantum hardware
capable of running deeper quantum circuits with more qubits than currently
available quantum computers is needed to evaluate typical portfolios in the
financial industry. Moreover, the results suggest that developments in quantum
hardware, such as error corrections, are needed to perform practical-relevant
experiments and achieve speed-up compared to current applied methods.

As mentioned above, there are many derivative financial instruments for which
there is no general analytical solution and which are solved in practice with
numerical simulations. For these, however, there is a trade-off between the
number of stochastic factors considered and the computational time to run a
numerical simulation. The multi-factor Heath-Jarrow-Morton (HJM) model [91] is
widely used in finance for the valuation of interest rate derivatives.
However, the main problem in building an HJM model to describe the development
of the yield curve is the large number of degrees of freedom. One possible
selection of factors is based on principal component analysis (PCA). Besides,
there is also factor analysis or selection based on economic intuition [92].
[87] [87] show how to apply quantum principal component analysis (qPCA) to
reduce the number of stochastic factors to find the fair price of a interest
rate derivative. Moreover, they implement this algorithm in a quantum
computer. However, the experiments exhausted the computational power offered
by the current quantum processors in terms of gate fidelity, connectivity and
number of qubits. Nevertheless, their work is a first step towards the
development of a general quantum algorithm to fully simulate the HJM model for
pricing interest rate derivatives on quantum computers.

In summary, the three contributions already describe in detail the possible
use of quantum computers for the pricing of different types of derivatives.
Furthermore, by using the AE algorithm, a theoretical quadratic speed-up is
shown compared to classical Monte Carlo simulations. Moreover, simple
experiments have already been implemented on real quantum computer devices.
However, these experiments also show that extensive hardware developments are
still necessary for a practical application of quantum computer for derivative
pricing. However, it was also shown that improvements can be achieved with
error corrections and thus a practical application on near-term noisy
intermediate-scale quantum computers could exist.

###  IV-E Estimating Risk Measure(s)

Two papers discuss how risk measures can be estimated with quantum computers.
In this context, it is important to note that there are many different risk
measures for different types of risks. However, one of the most popular
methods for quantitatively assessing the risk of portfolios consisting of
assets such as stocks, credits, or fixed income securities is the Value at
Risk (VaR). In short, it is a potential loss that will not be exceeded with a
certain probability, defined as the confidence level, over a given time period
and for a given portfolio of financial assets [93]. While the basic concept of
VaR is simple, many complications can arise in practice. One important
complication is the non-linearity in the portfolio payoff structure caused,
for example, by derivatives such as options [94]. The Monte Carlo method is
one of the methods to overcome this problem [95].

[10] [10] present in their paper a quantum algorithm that analyzes risk
measures such as VaR. Specifically, they show how to use quantum AE to
evaluate the risk measures VaR and Conditional VaR (CVaR) on a gate-based
quantum computer. [10] [10] has implemented also a small example of two assets
on a real quantum-computer device. However, this example is of very low
complexity and no nonlinear payoff structures were considered. For more
industry relevant examples and to achieve a practical relevant speed-up, more
qubits are needed and the errors of the current hardware must be reduced. In
addition, they make another important point, namely that Monte Carlo
simulations can be massively parallelised in practice, which pushes the bound
for quantum advantage even higher.

In a follow-up paper, [96] [96] build on this work and use a very similar
approach, and thus also show a theoretical quadratic speed-up to estimate the
difference between the VaR and the expected loss value. This risk measure is
also known as the economic capital requirement. They also estimate the circuit
depth of applications of this method for practically relevant use cases. The
requirements for such use cases are shown to be well beyond the capabilities
of current gate-based quantum hardware. However, as already discussed above by
other authors, they concluded that research in algorithms can help reduce the
number of qubits as well as the circuit depth so that quantum computers will
find applications in finance sooner.

###  IV-F Predicting Financial Crashes

Two papers demonstrate the prediction of financial crises using a quantum
computer. However, both papers address the same approach. In the paper by [17]
[17] the basis is elaborated. The later paper by [97] [97] then shows the
implementation of this method on a quantum annealer and corresponding results.

The financial network (banks, companies, individuals) holds a set of assets as
well as some of the other institutions in the network. The question is what
change in the value of the assets could cause a massive drop in the market
value of the institutions, i.e., a financial crash. According to [17] [17],
this type of problem is extremely difficult to solve even for simple models,
making it difficult and almost unsolvable for classical computers.
Nevertheless, this problem is of great importance from a regulatory and
macroeconomic perspective. [17] [17] propose a nonlinear network model for
financial markets to predict such financial crashes. Their main contribution
is the efficient construction of a QUBO formula (see also Section IV-B)
representing the problem. They also show how to find a solution using a
quantum annealer.

[97] [97] then implemented this proposed algorithm on a quantum annealer.
According to the authors, this work can be considered the first proof that
quantum computation can be used to study quantitative finance and help
institutions anticipate risks. However, a next-generation quantum processor is
needed to implement larger networks and therefore relevant computations for
the financial industry. This result could be sooner achieved by developing a
problem-specific quantum annealer.

Although the approach sounds promising, the prediction of financial crashes is
generally considered very difficult (see also subsubsection III-B1).
Therefore, besides the improvement of quantum hardware, it has to be proven
whether the chosen approach of the network model can indeed predict financial
crashes with high confidence. This could be shown, for example, with back-
testing using historical data.

##  V Conclusion

In this paper, we discussed the current state of quantum computing in the
financial sector and systematically analyzed potential use cases for quantum
computing in finance. For this purpose, we conducted an extensive literature
search and designed a multi-layered framework to enable a structured analysis
of the available literature and the use cases described. The corresponding use
cases focus on the areas of optimization and Monte Carlo simulation. The first
area in particular has significant potential in the context of investment
portfolio construction. The use of quantum computing could be beneficial here
in that optimal portfolios can be found more quickly. In the area of Monte
Carlo methods, applications are described primarily with regard to the pricing
of derivatives and the estimation of risk measures such as VaR or credit
default risk. In particular, faster or more accurate pricing of derivatives
could greatly improve the liquidity of financial markets, while more accurate
estimation of risk measures could lead to more stable financial institutions.
However, no paper has been found that describes a use case for quantum-based
machine learning in finance. This is somewhat surprising as machine learning
already has many applications in traditional finance.

There are already many different use cases for quantum computing in different
areas of finance, but they differ in many aspects. Some of them have already
been implemented on a real quantum device. Nevertheless, it needs to be
mentioned that breakthroughs in the development of quantum computing hardware
are necessary to realize the full potential of quantum-based algorithms for
finance. For example, more qubits are needed and the errors of current
hardware need to be reduced. The development of quantum computers is highly
complex which is one of the reasons why little literature exists on the impact
on the financial industry. However, existing publications point to the
increase in investment profits, reduction in capital requirements, creation of
new investment opportunities, and improvement in risk management and
compliance. To unlock these potentials, quantum computing must be operated
economically and cost effectively.

Table I: Overview of proposed quantum algorithm for use cases in finance. Author(s) |  Year |  Title |  Use Case |  Methodology |  Quantum algorithm |  Hardware
---|---|---|---|---|---|---
[79] |  rebentrost2018quantum |  [79] |  Determine the optimal portfolio more quickly/accurately. |  Optimisation |  Solving linear equation systems (HHL) |  Gate-based quantum computer
[78] |  venturelli2019reverse |  [78] |  Determine the optimal portfolio more quickly/accurately. |  Optimisation |  Approximate optimization (QUBO) |  Quantum annealer
[86] |  braine2021quantum |  [86] |  Settle as many transactions as possible and/or maximise the total value of the settled transactions. |  Optimisation |  Approximate optimization (VQE and QAOA) |  Gate-based quantum computer
[77] |  phillipson2020portfolio |  [77] |  Determine the optimal portfolio more quickly/accurately. |  Optimisation |  Approximate optimization (QUBO) |  Quantum annealer
[97] |  ding2019prediction |  [97] |  Prediction of financial crashes in a complex financial network. |  Optimisation |  Approximate optimization (QUBO adapted) |  Quantum annealer
[82] |  hodson2019portfolio |  [82] |  More quickly/accurately discrete portfolio optimisation under constraints. |  Optimisation |  Approximate optimization (QAOA) |  Gate-based quantum computer
[80] |  kerenidis2019quantum |  [80] |  More quickly/accurately portfolio optimisation under constraints. |  Optimisation |  Optimization (QIPM for SOCPs) |  Gate-based quantum computer
[17] |  Orus2019 |  [17] |  Forecasting financial crashes with quantum computing. |  Optimisation |  Approximate optimization (QUBO) |  Quantum annealer
[40] |  Rebentrost_2018 |  [40] |  Price derivatives more quickly/accurately |  Monte Carlo |  Searching and counting (QAE) |  Gate-based quantum computer
[87] |  martin2021toward |  [87] |  Pricing interest-rate financial derivatives with the Heath-Jarrow-Morton model more quickly |  Monte Carlo |  Optimization (qPCA) |  Gate-based quantum computer
[96] |  egger2019credit |  [96] |  Estimate credit risk more efficiently. |  Monte Carlo |  Searching and counting (QAE) |  Gate-based quantum computer
[10] |  WE2019 |  [10] |  Evaluate risk measures such as Value at Risk and Conditional Value at Risk more quickly/accurately |  Monte Carlo |  Searching and counting (QAE) |  Gate-based quantum computer
[88] |  Stamatopoulos_2020 |  [88] |  Price options such as vanilla options, multi-asset options, barrier options and path-dependent options more quickly/accurately |  Monte Carlo |  Searching and counting (QAE) |  Gate-based quantum computer

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