# Quantum Machine Learning for Finance

Marco Pistoia, Syed Farhan Ahmad, Akshay Ajagekar, Alexander Buts, Shouvanik
Chakrabarti, Dylan Herman,
Shaohan Hu, Andrew Jena, Pierre Minssen, Pradeep Niroula, Arthur Rattew, Yue
Sun, Romina Yalovetzky  Future Lab for Applied Research and Engineering,
JPMorgan Chase Bank, N.A.

###### Abstract

Quantum computers are expected to surpass the computational capabilities of
classical computers during this decade, and achieve disruptive impact on
numerous industry sectors, particularly finance. In fact, finance is estimated
to be the first industry sector to benefit from Quantum Computing not only in
the medium and long terms, but even in the short term. This review paper
presents the state of the art of quantum algorithms for financial
applications, with particular focus to those use cases that can be solved via
Machine Learning.

###### Index Terms:

Quantum Computing, Machine Learning, Artificial Intelligence, Algorithms,
Finance

##  I Introduction

The computational power of a quantum computer grows exponentially with its
number of qubits. For this reason, quantum computers are expected to surpass
the computational capabilities of classical computers and achieve disruptive
impact on numerous industry sectors, such as global energy and materials,
pharmaceuticals, telecommunication, travel and logistics, and finance.
Finance, in particular, is estimated to be the first industry sector to
benefit from quantum computing not only in the medium and long terms, but even
in the short term due to the large number of financial use cases that lend
themselves to quantum computing and their amenability to be solved effectively
even in the presence of approximations [1]. This is especially important for
taking advantage of today’s Noisy Intermediate-Scale Quantum (or NISQ) devices
[2], which are characterized by their low quantum bit (or qubit) counts, short
coherence time, and high operation noise.

This review paper presents the state of the art of quantum algorithms for
financial applications, focusing in particular on those use cases that can be
solved via Machine Learning (ML). The applicability of ML to finance has
become increasingly more significant as highly efficient ML algorithms have
evolved over time to support different data types and scale to larger data
sets. ML operations applicable to finance include regression for asset
pricing, classification for portfolio optimization, clustering for portfolio
risk analysis and stock selection, generative modeling for market regime
identification, feature extraction for fraud detection, reinforcement learning
for algorithmic trading, and Natural Language Processing (NLP) for risk
assessment, financial forecasting and accounting and auditing. Deep learning
is often used for image recognition and text classification, as well as in any
use case characterized by large unstructured datasets.

Given the complexity of the algorithms involved, and the size of the data
being analyzed, ML has been identified as one of the most important domains of
applicability of Quantum Computing. This has become even more evident with the
discovery of new quantum algorithms for linear algebra, which offer the
potential for executing linear-algebra computations on a quantum computer more
efficiently and accurately than their corresponding classical counterparts
[3]. Under certain conditions, the quantum speedup can be even exponential
[4], modulo some caveats [5]:

  * •

Efficiently loading classical data onto quantum computers and reading out
classical outputs resulting from quantum computations is still the field of
ongoing research. The majority of the quantum algorithms devised so far is
based on the existence of a Quantum Random Access Memory (QRAM) for accessing
the classical data [6]. The realization of a QRAM has been theoretically
proven, but concrete hardware implementations are still undergoing.
Alternatively, classical data can be loaded into a quantum states via
specialized circuits [7].

  * •

It is not always possible to apply a quantum linear-algebra algorithm out of
the box to solve a specific financial use case; several conditions must be met
and customizations are often necessary to address unique use-case-dependent
requirements [5]. Furthermore, multiple classical and quantum algorithmic
components are usually involved in the end-to-end solution of a financial use
case, with the potential for any such component to become the bottleneck and
negate the overall quantum advantage. The task of computing the quantum
speedup of the solution of a specific use case is, therefore, not always
intuitive.

As of today, no end-to-end application of quantum ML with exponential speedup
over its classical counterpart has been discovered, but several promising
directions have been proposed. Meanwhile, a large body of research and
engineering work has been successfully dedicated to the realization of quantum
algorithms with significant polynomial speedups in their data-processing
subroutines, if not in the data loading and output extraction.

##  II Regression

A central task in supervised learning is regression, or the problem of
training a simple model to approximate real-valued functions. This is an
extremely important routine in experimental sciences, and has recently started
to be employed on massive datasets. Computationally, this reduces to
minimizing a loss function that captures the quality of the fit on training
data; the common choices are the $L_{\infty}$ norm for worst-case error, and
the $L_{1}$ or $L_{2}$ norms for average-case error. The smoothness of the
$L_{2}$ norm makes it attractive to optimization algorithms, leading to the
ubiquity of least squares regressions, where the minimization problem, given
$N$ $d$-dimensional feature vectors, can be expressed as follows:

$\arg\min_{\vec{\beta}}\sum_{i=1}^{N}w_{i}\left(y^{(i)}-\vec{\beta}^{T}\vec{x}^{(i)}\right)^{2},$

which is a convex quadratic minimization problem and reduces to solving an
$N\times d$ system of linear equations.

A quantum speedup for this task is first considered by Wiebe et al. [8] based
on quantum algorithms for solving systems of linear equations. Their algorithm
requires access to the entries of the data matrix and weight vector in
superposition, and outputs a quantum state that encodes the output in time
$\tilde{O}\left(\log(N)s^{3}\kappa^{6}/\epsilon\right)$, where
$\tilde{O}(\cdot)$ neglects polylogarithmic factors in the complexity. While
the run time of this algorithm is exponentially smaller than classical
equivalents, there are two main caveats. First, the construction of a data
structure allowing superposition access to the data points may in general
require $O(N\log(N))$ time, via the construction of a QRAM [6]. Second, $\sim
O(d)$ copies of the output state are required to obtain a full description of
the output.

Some attempts have been made to address this issue, albeit with smaller
speedups. Wang [9] gives an algorithm for $L_{2}$ regression that obtains a
classical solution in time $\mathrm{poly}(\log(N),d,\kappa,1/\epsilon)$. A
second approach is to use algorithms that do not output a full description of
$\vec{\beta}$, but rather $\vec{\beta}^{T}\vec{x}$ for a new data point
$\vec{x}$. Schuld et al. [10] obtain such an algorithm given a quantum state
encoding of the training data and $\vec{x}$ that takes time
$O(\log(N)\kappa^{2}/\epsilon^{3})$. A third approach is to consider special
quantum data structures for accessing the training data. These _efficient_
QRAM data structures are different from a general QRAM, in that the cost of
inserting, updating or deleting a single entry is $O(\mathrm{poly}(\log(n)))$.
Such data structures can be useful when there is a large initial corpus of
data, but the regression task must be performed repeatedly, and each updates
to the data set is small (or constant). The construction of the original
dataset thus has cost $O(N\log(N))$, but future upkeep and regression tasks
each cost $O(\mathrm{poly}(\log(n,d)))$. Thus, the cost of the initial
construction is effectively amortized over the lifetime of the deployment.
This setting can be used to obtain $L_{2}$-regression algorithms with
complexity $\tilde{O}\left(\kappa\mu/\epsilon(\mathrm{poly}(\log(Nd)))\right)$
[11, 12].

There also exist quantum-inspired classical algorithms [13] based on the last
approach that use data structures providing sampling access to the data and
obtain algorithms with cost $\tilde{O}\left(\lVert X\rVert_{F}^{6}\lVert
X\rVert^{2}/\epsilon^{4}\right)$. $L_{1}$ and $L_{\infty}$ regression problems
are not smooth and are therefore often solved via smooth relaxations that are
problem-specific.

A heuristic approach to regression can be used by simply training
parameterized quantum circuits (PQC) as function approximators. A common
application of classical regression techniques is in learning time series via
Recurrent Neural Networks (RNNs) (often using Long Short Term Memory (LSTM)
units). These techniques are commonly used to make predictions about evolving
processes from historical data. Quantum versions of RNNs have been proposed
that use PQCs as the function model [14, 15, 16] that empirically anticipate
improvements in convergence or error rate.

With the above discussion on various quantum regression algorithms, we next
look at several financial applications that take advantage of these
techniques.

###  II-A Asset Pricing

Asset Pricing is the task of assigning prices to various categories of
financial instruments, such as stocks, bonds, and derivatives. There are
several economic models used to assign these prices based on a sequence of
instantaneous (or spot) prices, including general equilibrium pricing and
arbitrage-free pricing [17]. The common approach to predicting these spot
prices is to model them as functions of simple underlying stochastic
processes, such as Brownian or Geometric Brownian motion. Historical financial
data can then be used to determine the parameters of these stochastic models.
More generally, however, predicting spot prices (as well as many other time-
varying quantities of financial interest) can be modeled as a _time series_
learning problem. Specifically, given a sequence of historic prices up to time
$t$, can accurate predictions be made for prices in the future? Since this
reduces to predicting real values based on training data, it is best modeled
as a supervised-regression problem. Stochastic pricing models with
historically calibrated parameters can be viewed as ad hoc solutions based on
domain knowledge.

The increasing success of deep RNNs for time-series prediction—especially
those that leverage LSTM—has led researchers to consider using these general-
purpose algorithms for asset pricing. Gu et al. [18] and Chen et al. [19]
investigate the use of LSTM-based deep-learning methods for asset pricing,
obtaining promising results. Specifically, Gu et al. [18] show that ML
forecasts on the S&P 500 achieve an out-of-sample annualized Sharpe Ratio of
0.77 versus the 0.51 of a buy-and-hold investor. They also find that a value-
weighted long-short decile spread strategy based on neural network forecasts
of stock prices achieve an annualized Sharpe Ratio of 1.35, nearly double the
state-of-the-art classical regression approaches. Chen et al. further show
that refined models based on LSTM forecasts can achieve out-of-sample
annualized Sharpe Ratios much larger than naive deep-learning forecasts as
well as classical approaches, including the Fama-French five-factor model.

The biggest challenge in deploying deep-learning methods is that training
complex neural networks can often be a much more computationally intensive
process than the simple parameter calibration required by classical
approaches. PQCs may offer advantages over classical variational regression
models in terms of expressivity, training complexity and prediction
performance. In 2020, the use of PQCs to formulate RNNs has been described
[14], along with proposals of quantum LSTM models [15]. Both approaches show
potential empirical improvements over classical neural networks for particular
functions, although the applicability to asset pricing is yet to be
investigated.

###  II-B Multi-Asset Trend Following Strategies

Regression models can be used to predict $1$-day returns of a multi-asset
class portfolio. Each financial asset class (e.g., equity, bond, cash or
commodities) might have different internal dynamics. Nevertheless, a
regression model might be able to encompass the global dynamic. An example of
global dynamic is the following: if the equity market is bearish, an investor
might prefer safer investments, say, bonds, and therefore cause a rise in bond
prices. Indeed, there has been a negative correlation between equity and bond
since the beginning of the century. However, this correlation was positive
between 1970 and 2000 [20]. Therefore, the relationship between asset classes
evolves and should not be taken as general truth.

To predict returns daily, one can use historical prices from various time
points (e.g., 1 month, 3 months, etc.) to introduce trend information in the
input data. However, this causes an increase in the number of features in the
data, and could result in over-fitting. Consequently, one would rather use a
Lasso regression [21] than a vanilla regression. Indeed, by adding an $L_{1}$
penalty term $\lambda\|\vec{\beta}\|_{1}$ to the cost function of the $L_{2}$
regression problem, the model will select a subset of relevant features. As
with any regression method, one can treat it as a classification model. The
classes here would be buy, hold or sell. The $L_{1}$ norm may not be the most
attractive regularization term for quantum implementation. Nevertheless, Du et
al. [22] provide the closest known implementation with a differentially
private Lasso estimator.

###  II-C Implied-Volatility Estimation

The Implied volatility metric captures the financial market’s view of the
likelihood of changes in a given security’s price. The analysis of volatility
is crucial for risk management, portfolio hedging and option pricing. A
precise notion of the market’s expectation of volatility is required [23].
Portfolios have a sensitivity with respect to volatility changes. For
instance, it has been proven that implied volatilities—such as those of oil,
gold and the US stock market—play a role on the returns of the equity sector.
Particularly, their impact on the prices and returns of the ten most
representative US equity sectors has been quantified [24].

A quantum approach for learning implied volatilities has been proposed [25].
This uses the deep quantum neural networks firstly introduced by Beer et al.
[26]. Given $N$ options, the input data is its strike prices
$K_{1},K_{2},\ldots,K_{N}$, and the output is the implied volatilities
$\sigma_{K_{1}},\sigma_{K_{2}},\ldots,\sigma_{K_{N}}$. A sigmoid function is
used to convert the strike prices to numbers in $[0,1]$, which are then
represented as quantum states $\ket{\phi_{n}^{\mathit{in}}}$ for
$n=1,2,\ldots,N$. The network consists of one input neuron, one output neuron,
and one hidden layer with two neurons. The output of the network is a density
matrix
$\rho_{n}^{\mathit{out}}=\ket{\phi_{n}^{\mathit{out}}}\bra{\phi_{n}^{\mathit{out}}}$.
The implied volatility of each of the $N$ options is then calculated using its
respective element in the density matrix.

##  III Classification

The goal of classification in ML is to predict the labels for new data points
using a model that is fit by a labeled dataset. Well-known traditional
classification algorithms include Linear Classification, Nearest Centroid and
Support Vector Machines (SVMs). More recently, neural-network-based methods
have seen tremendous success. Once a neural network has had its weights
trained via a labeled dataset, it can be used to perform inference on unseen
data instances. It has been empirically demonstrated that neural networks
achieve better performance than traditional methods, especially on large
datasets [27]. Neural networks, on the other hand, fall short in terms of
transparency and interpretability, which could be desirable when it comes when
making decisions or performing tasks that involve handling sensitive
information [28].

#### Linear Classification

A linear classifier allows for classifying an object in a dataset based on the
value of a linear combination of that object’s characteristics, known as
feature values and stored in a feature vector. The common algorithm used for
this is the perceptron method [29], which finds a $\gamma$ margin classifier
given $n$ $d$-dimensional data points in time $O(nd/\gamma^{2})$. Quantum
algorithms have been shown to be able to provide speedup, bringing the running
time down to $O(\sqrt{n}d/\gamma^{2})$ [30] or $O(\sqrt{nd}/\gamma)$ [31]. It
was later discovered that the optimal classical algorithm for training
classifiers with constant margin runs in $\tilde{O}(n+d)$ [32], and that a
corresponding optimal quantum classification algorithm can bring about
quadratic speedup, leading to running time $\tilde{O}(\sqrt{n}+\sqrt{d})$
[33].

#### Distance-based Classifiers

Distance-based classifiers predict the label of a new data point based on its
distance to reference points according to some metric. Typical examples are
the nearest-centroid and $k$-nearest-neighbors ($k$-NN) classifiers [34].

The nearest-centroid algorithm is a good baseline classifier that offers
interpretable results. This algorithm takes as input a number of labeled data
points, where each data point belongs to a specific class. The model fitting
consists of computing the centroids, which are the barycenters of data points
that cluster together in space. Once the centroids are found, a new data point
is classified by finding its closest centroid in terms of Euclidean distance.
A quantum version of the nearest centroid algorithm was used to perform
classification on the Modified National Institute of Standards and Technology
(MNIST) handwritten-digit dataset [28]. The experiments were executed on the
IonQ trapped-ion quantum computer. This approach utilizes novel quantum
procedures for loading the classical data onto quantum states and estimating
distances between these states. Its accuracy matches that of the classical
nearest-centroid algorithm. The authors, however, do not claim any quantum
speedup in terms of time complexity.

Another distance-based classifier is the $k$-NN algorithm, which predicts the
label of a data point based on a majority vote among $k$ closest training
samples according to a metric, such as the Euclidean distance. Several quantum
approaches for $k$-NN have been proposed. In particular, [35] utilizes an
algorithm for computing Hamming distances in superposition, and the quantum
minimum-finding method [36] to find the neighbors with the smallest Hamming
distances to a sample. If feature vectors lie in a low-dimensional space, the
algorithm can classify a new sample with worst-case time complexity
$O(\sqrt{M}\log(M))$, where $M$ is the training set size. Another quantum
$k$-NN approach achieves a query complexity of $O(\sqrt{kM})$ [37]. The
authors design an oracle that encodes the fidelity between two states into a
quantum register, which allows for the usage of a quantum algorithm for
finding $k$-minima [38].

#### Support Vector Machines

An SVM [39] consists of solving a convex quadratic optimization problem to
find the hyperplane that results in the maximum margin between two classes of
data. The dual problem

$\max_{\alpha_{i}}\sum_{i=1}^{N}\alpha_{i}-\frac{1}{2}\sum_{i=1}^{N}\sum_{k=1}^{N}\alpha_{i}\alpha_{k}y_{i}y_{k}K(x_{i},x_{k})\mid\alpha_{i}\geq
0\\\ $ (1)

is the one usually solved.
$K:\mathbb{R}^{m}\times\mathbb{R}^{m}\mapsto\mathbb{R}$ is symmetric and
positive semi-definite; a common example is the _Radial Basis Function_. $K$
induces the, potentially _non-linear_ , feature map
$\phi(x)=K(\cdot,x)=K_{x}$. The Hilbert Space built from such maps is called
the _Reproducing Kernel Hilbert Space_ (RKHS) with reproducing kernel $K$. The
optimal classifier in RKHS is one that is a linear combination of $K_{x}$’s
over a subset of the training data [34].

One of proposed quantum enhancements to the SVM is based on evidence that
universal quantum computation, most likely, cannot be efficiently simulated on
a classical computer [40]. Thus, one should be able to construct a quantum
circuit for the map $\vec{x}\mapsto\mathcal{U}_{\Phi{(\vec{x})}}\ket{0}$;
$\mathcal{U}_{\Phi{(\vec{x})}}$ is a unitary operation applied to the
computational basis state consisting of all qubits in the $\ket{0}$ state,
such that this operation is not classically feasible. This is called a
_quantum feature map_ and maps classical data $\vec{x}$ into a quantum Hilbert
Space that is exponentially large in the dimension of $\vec{x}$. A potential
quantum kernel is

$K(x_{i},x_{j})=\lVert\bra{0}\mathcal{U}^{\dagger}_{\Phi{(\vec{x_{i}}})}\mathcal{U}_{\Phi{(\vec{x_{j}})}}\ket{0}\rVert^{2}$

which is symmetric and positive semi-definite. In this case the RKHS is
spanned by the functionals, $K(\cdot,x)$, that are constructed from quantum
circuits. The coefficients $\alpha_{i}$ of the decision function in RKHS can
be computed by a convex optimizer running on a classical computer; the quantum
computer is used to evaluate the kernel. This hybrid model is called QSVM
[40]. There is potential quantum advantage in the expressability of the
feature map, as long as the associated kernel is infeasible for a classical
device to compute. This kernel can be computed on a quantum device utilizing
either the _destructive SWAP_ [41] or the _controlled-SWAP_[42] tests. The
latter has better asymptotic complexity, but is not as feasible on NISQ
devices. While the former’s asymptotic scaling is prohibitive, it can
efficiently be implemented on small quantum computers; this allows for
experimentation in the near-term.

The class of _Instantaneous Quantum Polynomial_ (IQP) circuits has been
suggested as a potential candidate for $\mathcal{U}_{\Phi{(\vec{x})}}$ [40]:

$\begin{array}[]{r@{}l}\mathcal{U}_{\Phi{(\vec{x})}}&{}=U_{\Phi(\vec{x})}H^{\otimes
n}U_{\Phi(\vec{x})}H^{\otimes n}\\\
U_{\Phi(\vec{x})}&{}=\exp\left(i\sum_{S\subseteq[n]}\phi_{S}(\vec{x})\prod_{i\in
S}Z_{i}\right)\end{array},$ (2)

where the functions $\phi_{S}$ represent classical preprocessing. If the IQP
circuit is deep enough, it is believed that computing inner products of states
resulting from these embeddings is #P-Hard [40], thereby potentially out of
reach of classical devices.

A secondary consideration is whether quantum algorithms can be used to
accelerate the training of classical SVMs. Algorithms of complexity
$\tilde{O}(\sqrt{n}+\sqrt{d})$ has been proposed for training kernel
classifiers and $\ell_{2}$ margin SVMs [33]. These algorithms are optimal and
provide a quadratic speedup over corresponding optimal classical algorithms
[32]. However these algorithms have complexity polynomial in the inverse of
the error $(1/\epsilon)$. There exist classical algorithms with
$O(\mathrm{poly}(\log(1/\epsilon)))$ complexity using interior-point methods;
Kerenidis et al. [43] propose a quantum algorithm to speed up these methods in
terms of the dimension $d$. While the quantum run time depends on terms that
are difficult to bound directly, for random instances, the quantum algorithm
can indeed provide a speedup, leading to a $O(n^{2.59})$ complexity, compared
to the $O(n^{3.11})$ classical algorithm’s complexity.

In SVM, an optimal hyperplane is obtained that divides the dataset into
multiple classes, with a time complexity of $O(\log(1/\epsilon){\rm
poly}(N,M))$, where $N$ represents the feature space dimension, $M$ the number
of input points, and $\epsilon$ the accuracy. QSVMs have mathematically been
proven to have a run time of $O(\log(NM))$ [44].

#### Variational Quantum Classifiers

Variational Quantum Classifiers (VQCs) are hybrid quantum-classical ML
architectures meant for classification tasks that utilize the quantum state
space as a feature space to potentially obtain a quantum advantage. A VQC
circuit mainly consists of a quantum embedding, a PQC for processing the
quantum data, a measurement routine, and a classical optimization loop for
updating the parameters of the PQC. First, classical input data $\vec{x}$ is
mapped to a quantum state non-linearly using the feature-map circuit,
$\mathcal{U}_{\Phi(\vec{x})}$, defined in Equation 2. Applying
$\mathcal{U}_{\Phi(\vec{x})}$ to $\ket{0}^{n}$ results in the state
$\ket{\Phi(\vec{x})}$.

Next, a PQC, $W(\vec{\theta})$, is constructed with parameters $\vec{\theta}$.
An example of such a PQC is one made from compositions of single qubit
rotations and entangling gates. PQC architectures have been discussed where
descriptors, such as the entangling capability and expressibility, are used to
characterize the performance of the PQCs [45].

In case of a binary-classification problem, a measurement routine is used to
get a binary output. This is accomplished by measuring state
$W(\vec{\theta})\mathcal{U}_{\Phi(\vec{x})}\ket{0}^{n}$ in the Pauli Z-basis
and mapping the output bit-string to a function with binary outcome
$f:\\{0,1\\}^{n}\rightarrow\\{+1,-1\\}$. The probability of obtaining an
outcome, $y=\pm 1$, is

$p_{y}(\vec{x})=\sum_{i\in
f^{-1}(y)}\lVert\bra{i}W(\vec{\theta})\ket{\Phi(\vec{x})}\rVert^{2}.$

We repeat this step for $R$ measurement shots, which gives an empirical
distribution, $\hat{p}_{y}(\vec{x})$.

Then, a classical cost function is formulated to enable optimizing the
parameters $(\vec{\theta},b)$, where $b\in[-1,+1]$ is an added bias parameter.
Once the classifier is trained on the training data set using a classical
optimizer, the trained circuit can now be used to assign labels to unlabelled
data. Several optimizers have been proposed and used, both gradient based,
such as ADAM and SPSA [46, 47], and gradient-free ones, such as COBYLA [48].

VQCs have some limitations, and solving these drawbacks is an active area of
research. Barren plateaus occur in optimization algorithms of quantum ML when
the parameter search space turns flat once the optimizer is run [49, 50].
Architecture design problems, such as choosing the correct cost functions and
initializing the parameters, is a very complex process that has not been
completely understood yet [51]. Additionally, a given variational quantum
circuit with fixed form may not be able to capture all of the necessary states
in the Hilbert space in its parameterization, and as a result, work on
adaptive variational quantum algorithms, such as the Evolutionary Variational
Quantum Eigensolver (EVQE), may be applicable to VQC [52].

There is a connection between the QSVM and VQC formulations [40, 53, 54]
similar to the connection between classical Neural Networks and SVMs [55].
There are various discussions on how data encoding affects VQCs [54, 56], such
as repeatedly encoding the inputs [57]. In addition, efficient methods were
presented for encoding categorical features [58]. Lastly, there has been
research into the expressiveness of PQCs [56, 59].

Support vector machines have been used to predict stock prices for over two
decades, but also to predict financial distress and company’s credit rating
[60].

Next we look at a few example financial applications where the aforementioned
quantum classifications techniques have been applied.

###  III-A Prediction of Binary Options

SVM can be used to predict the outcome of exotic options. The double no-touch
is a binary option [61] with a constant payout and is earned if and only if
the underlying asset price remains between a predefined lower and upper bound
until expiration. Unlike other options, such as a vanilla call, the payoff is
not continuous, but all-or-nothing. Therefore, one can use SVM to separate the
two classes corresponding to the binary option outcome. As these classes are
not linearly separable, one needs a kernel to predict the outcome. This type
of exotic option is often used in foreign exchange. The features selected to
train the model could be the average directional index and the ratio between
realized volatility over implied volatility.

###  III-B Financial Forecasting

Financial forecasting is a planning tool that helps businesses to adapt to
uncertainty based on predictions. Particularly, an algorithm to forecast
annual earnings is of interest to any company. Such an algorithm has been
proposed [62] that leverages $k$-NN. It matches a company’s recent trend in
annual earnings to historical earning sequences of other firms that are
similar—known as neighbor firms. Some of the features taken into account to
find such neighbors include matches based on industry, size and past accruals.

###  III-C Credit Scoring

Credit scoring is a method to evaluate the credit risk of loan applications.
It helps credit analysts to decide whether the applicants are worthy of
credit. Based on past experience, credit scoring is the prediction of future
behavior. An algorithm for this has been proposed using weighted $k$-NN [63].
The credit applicants are classified into one of two groups: a group whose
members are likely to repay their debts and another group that should be
denied credit because of high likelihood of defaulting.

##  IV Clustering

Clustering consists of identifying groups of data points that are close to
each other according to certain metrics. The feature space in which the data
is encoded and the grouping metric are proxies for the actual similarities and
differences of the data points. Inspired by quantum mechanics and suitable for
high-dimensional data, Quantum Clustering (QC) [64] is an algorithm that
belongs to the family of density-based clustering algorithms, where clusters
are defined by regions of higher density of data points. The basic idea of QC
is to map each data point to a Gaussian distribution centered at that sample.
An analytical form computed from the Schödinger equation is used to determine
the potential that gives rise to a mixture of these Gaussian as its ground
state. The minima in the system’s potential energy function are used to
identify clusters and are found via gradient-descent methods. Other points are
assigned to clusters in a similar way. Dynamic Quantum Clustering (DQC) [65],
an improvement of QC, adopts the time-dependent Schrödinger Equation in order
to study evolution of quantum states associated with data points and the
structure of the potential energy function. Being data-agnostic, DQC can be
applied in a wide range of fields, especially finance, for example on S&P 500
data [66].

Classical-algorithm-inspired quantum-clustering techniques have also been
proposed. For example, $k$-means is a well-known classical clustering
algorithm that identifies, among all data points, the $k$ most significant
clusters and their representative centroids. Inspired by $k$-means, and
providing the same robustness guarantees against some level $\delta$ of noise
as the classical $\delta$-$k$-means algorithm, the quantum $q$-means algorithm
[67] has time complexity that is poly-logarithmic in the size of the dataset,
and can be implemented using distance estimation and quantum matrix
multiplication. A quantum spectral clustering algorithm for data represented
as a graph has also been proposed [43]. To overcome the potentially huge
time/space overhead of loading large datasets onto a quantum device, coresets
have been proposed [68], which are small datasets combined with weight
functions to sufficiently summarize original datasets. If small enough and
still a faithful representation of the original dataset, a coreset could be
used to enable execution on a NISQ computer [69, 68, 70, 71].

Next we briefly discuss several use cases of these quantum clustering
algorithms in the financial sector.

###  IV-A Fraud Detection

Clustering techniques can be used to perform anomaly detection by learning,
from existing data, the normal mode(s), and then using this information to
identify if a new data point is normal or otherwise anomalous [72]. Clustering
can improve learning from imbalanced datasets, which oftentimes is the case
for fraud data [73]. Clustering can also be combined with additional feature-
selection and extraction techniques. For example, in time series data, a
series could be hiking abnormally fast but still stay in a normal value range.
Adding derivatives into the clustering algorithm can help detecting such an
anomaly [74].

###  IV-B Stock Selection

Cluster analysis has also been used by investors for maximizing profit and
minimizing loss. Stock returns are likely to be similar in a region thanks to
geographic and macroeconomic features. Identification of stock clusters allows
one to track those with similar returns but different risks. Once stocks are
grouped by cluster analysis, informed investors can use the output for
guidance. They will, for instance, look for same-return stocks and then choose
to minimize risks. Alternatively, they will pick a cluster of same-risk stocks
and high return [75].

###  IV-C Exchange Rate Regimes

In 1999, Levy-Yeyati and Sturzenegger [76] wanted to exhibit the inconsistency
between the self-reported de jure classification from the International
Monetary Fund (IMF) and the actual behavior shown in the data. In order to
overcome bias, the authors proposed to use $k$-means to perform cluster
analysis for exchange rate regimes. This led to a de facto classification,
that has then been widely used as well as tested against prior methodologies
[77].

###  IV-D Hedge Fund Clustering

Due to the variety of hedge fund—and, therefore, investing strategies—it can
be hard for investors to classify such investment vehicles. Moreover, hedge
funds tend to reveal less information than other type of funds as they do not
fall under the same disclosure requirements. To classify hedge funds,
predefined classes would not be able to manage correctly future type of hedge
funds. Hence, clustering methods, such as $k$-means, have been used to
overcome this issue [78]. The features considered are based on available
characteristics of hedge funds, such as asset classes, size, fees, leverage
and liquidity.

##  V Generative Modeling

A Generative Model learns a probability distribution over data [79]. In
supervised learning, where the model is provided as a set of input/label pairs
$\\{(x_{i},y_{i})\\}$, the model learns $P(X,Y)$, the joint probability
distribution of inputs and labels [80]. In unsupervised learning, these models
can be used to generate new data given only samples [81]. Since measuring a
quantum state naturally results in a probability distribution over the
outcomes, it makes sense to see if quantum computation can be utilized for
generative modeling.

#### Boltzmann Machine

The Boltzmann Machine [82] is defined by a collection of visible (observed)
and hidden (marginalized out) random variables, and an undirected graph of
conditional dependencies among them. It originates from thermodynamics where
the nodes represent a system of correlated classical spins, $s_{i}$, under an
external magnetic field. The classical Ising Hamiltonian

$\mathcal{H}=-\sum_{i,j}J_{ij}s_{i}s_{j}-\sum_{i}h_{i}s_{i}$

represents the energy of the system. Probabilistic inference is performed by
sampling from the steady-state distribution—a Gibbs state—over the visible
nodes. This is usually done utilizing Markov Chain Monte Carlo (MCMC) methods
[82]. In most cases, the graph is restricted to being bipartite to make
sampling feasible, resulting in the Restricted Boltzmann Machine (RBM) [83].

To formulate the _quantum_ Boltzmann Machine, we quantize the Ising
Hamiltonian by making the replacements $s_{i}\mapsto\sigma^{z}_{i}$, where
$\sigma^{z}_{i}$ is the Pauli $Z$ spin operator for the $i$-th qubit. This
results in a quantum Hamiltonian, and thus nodes are associated with qubits,
and sampling is performed by projective measurements on the visible qubits.

One potential quantum method to sample from the visible nodes of the Gibbs
state is to utilize _Quantum Annealing_ (QA) [84, 83, 85]. For example, QA can
be performed using the D-Wave devices [86].

Alternatively, we can prepare the quantum Gibbs state for this system by
performing Imaginary Time Evolution (ITE) [87]. If the initial state is
maximally mixed, performing ITE according to a quantum Hamiltonian will result
in the associated Gibbs State. ITE can be performed variationaly, via
McLachlan’s principle, on a gate-based quantum computer [88]. Interestingly,
the model introduced by Zoufal et al. [87] can be utilized to formulate a
Boltzmann Machine without restricted connections that is tractable on a
quantum device.

#### Generative Adversarial Learning

As another prominent architecture for modeling probability distributions,
Generative Adversarial Networks (GANs) [89], operate by simultaneously
training a generator network $\mathcal{G}_{\theta}$ and a discriminator
network $\mathcal{D}_{\phi}$ against each other through adversarial games, for
which $\mathcal{G}_{\theta}$ tries to fool $\mathcal{D}_{\phi}$ by generating
fake data samples that are non-distinguishable from the ones drawn from the
real distribution, whereas $\mathcal{D}_{\phi}$ tries to tell them apart and
not be fooled by $\mathcal{G}_{\theta}$. Quantum GANs (qGANs) have since been
proposed [90, 91] and experimentally tested, for example, on superconducting
quantum computers [92]. Either of qGAN’s generator or discriminator, or both,
can be in the form of quantum circuits. In addition to the original GAN’s
cross entropy, other distance metrics, such as Wasserstein [93], have also
been proposed to improve the adversarial training on NISQ devices.

#### Quantum Born Machine

Closely related to quantum Boltzmann Machines and qGANs, Quantum Born Machines
[94, 95] are another class of methods based on PQCs that have been studied for
performing distributed-learning tasks. For example, Coyle et al. [95] propose
using maximum mean discrepancy, the Stein discrepancy, and the Sinkhorn
divergence, to improve the training of a subclass of quantum circuit Born
machines.

Having discussed several quantum generative modeling techniques, we next look
at sample use cases in the finance domain where these techniques can be
applied.

###  V-A Fraud Detection

Quantum versions of the Boltzmann Machine have be utilized for generative-
learning and discriminative-learning tasks [83, 85]. Specifically for fraud
detection, a Variational ITE Boltzmann Machine methodology has been utilized
to classify anomalous credit-card transactions [87]. The system Hamiltonian is
represented by a sum of Pauli strings whose coefficients are functions of
trained parameters and input features. As mentioned earlier, this formulation
is not restricted to the Ising Hamiltonian typically utilized by Boltzmann
Machines. Predictions are performed by sampling from a single visible qubit
indicating whether the transaction was fraudulent.

qGANs were combined with a framework for Generative Adversarial Anomaly
detection, AnoGAN [96]. The generator was a PQC; the continuous output from
the expectation of Pauli Z operators on each qubit was fed into a classical
affine upscaling layer to achieve the full input feature dimension. The goal
of the generator was to model the distribution of non-fraudulent transactions.

###  V-B Probability Distribution Preparation

One crucial step for achieving quantum advantage in many financial
applications is the efficient preparation of input probability distributions.
qGANs [97, 98] and quantum Born Machines [94, 95] have both been utilized to
learn PQCs for loading probability distributions. Upon convergence, the
quantum circuit, as an efficient representation of the underlying
distribution, can for example be used in amplitude estimation to perform
derivative pricing tasks [99], with a theoretical quadratic speedup compared
to classical Monte Carlo simulations. Additional techniques have been explored
for the general creation of continuous distributions [100, 101]. Additional
techniques exploring the creation of certain families of continuous
distributions include the work of Rattew et al. [102] for the preparation of
normal distributions.

##  VI Quantum-Assisted Feature Extraction

Feature extraction refers to the set of techniques used to identify attributes
of a dataset potentially helpful in ML tasks such as classification and
regression. A quantum algorithm may help in feature extraction by computing
properties of the dataset that a classical computer would fail to identify, or
would take a very long time to do so. By encoding a data onto a quantum state,
we can map a low-dimensional classical data to a much higher dimension in the
Hilbert space. The expanded dimensionality of the quantum representation may
be used to identify features invisible to a classical algorithm [103]. The
growing interest in quantum kernels [104, 105], used in conjunction with
Support Vector Machines, has also culminated an experimental demonstration
[106].

A widely used algorithm to extract low-dimensional features out of a high-
dimensional data is the Principal Component Analysis (PCA). In PCA, a large
feature space is analyzed to identify attributes with the highest variance.
Classical PCA takes time that is polynomial in the dimension or number of
features in the original dataset. If such a classical data is mapped to a
quantum density matrix, the quantum version of the algorithm can perform PCA
exponentially faster, that is in time polynomial in the logarithm of the
dimension [107].

Extracting features is particularly challenging while analyzing images where a
large number of pixels have to be analyzed to identify image attributes. For
these applications, a quantum computer may help in edge detection in images
[108].

In finance, feature extraction may be used in detecting anomalies in
transactions. As an example use case, graph theoretic tools are used to study
bidding markets to identify colluding communities or cartels [109]. Quantum-
aided graph kernel methods [103, 110] have been proposed to detect non-trivial
features, such as communities [111], in a graph, which may represent, for
instance, a network of financial parties that frequently transact with each
other. When working with graph representations of data, we often want to
measure the similarity between two graphs. In fact, Gaussian Boson Sampling
can be used to check if two graphs are isomorphic to each other [112].
Moreover, Gaussian Boson Sampling can be used to construct kernel vectors
representing the similarity between any two graphs [56].

Feature selection consists of choosing from a subset of the available features
to pass to the model [34]. This contrasts with methods, such as PCA, that
perform a transformation on the features. Feature selection can be formulated
as a combinatorial minimization problem with binary decision variables
designating whether to select a feature or not. Such binary optimization
problems can be solved utilizing QA [84].

Below we present examples of how these techniques can be applied to financial
use cases.

###  VI-A Model Reduction

PCA is a widely used method for dimensionality reduction that can be seen from
the perspective of singular value decomposition. With the matrix decomposition
$A=U\Sigma V$, where $\Sigma$ is a rectangular diagonal matrix, the
$k$-principal components are the first $k$ columns of $U\Sigma$.

In 2014, Lloyd et al. [107] described a quantum PCA with exponential speedup
over its classical counterpart. This theoretical speedup is realizable under
certain conditions as it is based on HHL [3]. The algorithm can be used in
finance to ease model tuning: as market conditions evolves, models needs to be
tuned in order to match the implied volatility—volatility estimated by the
model—with the market volatility. By using PCA, one reduces the number of
components and, consequently, the number of parameters, thereby easing the
model tuning.

For example, in a product based on foreign exchange, the input parameters are
various and can range from global market data, such as risk-free interest
rate, to asset specifics parameters, such as the spot price. As a consequence,
the model tuning becomes computationally expensive due to the high number of
inputs. However, as just the top three principal components can oftentimes
explain over 95% of the output variations, one can tune the model faster and
still accurately by using only these three components.

A variation of quantum PCA has been implemented on hardware [113] to solve a
similar problem by reducing the volatility factor dimension of the Heath-
Jarrow-Morton model [114] in order to estimate forward rates.

###  VI-B Combinatorial Feature Selection for Credit Score Classification

As mentioned earlier, feature selection can be cast to a combinatorial
optimization problem. In the case of supervised learning, it important to
select features that are independent and relevant to the learning task. More
specifically, for classification, the correlation coefficients between label
and features can represent the relevance. The correlation matrix of the
features can be used to represent the dependence between features. This can be
formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem,
where the quadratic terms are the entries of the correlation coefficients
between features, and the linear terms are correlations between the features
and the label. QA [84] can be used to solve the QUBO utilizing heuristics
provided by quantum mechanics. This exact formulation, solved with a Quantum
Annealer, was applied to reduce the number of features used for assessing the
credit worthiness of applicants [115].

##  VII Reinforcement Learning

Reinforcement learning (RL) [116] is a ML technique where an agent attempts to
learn through interactions with the environment. Classical RL has demonstrated
remarkable capabilities in areas such as video games [117], board games [118,
119], robotics [120] and self-driving vehicles [121].

Classical RL is often formulated as a Markov Decision Process (MDP). MDPs
enable the modeling of environments where actions are non-deterministic—that
is, where taking a given action may probabilistically lead to one of multiple
possible outcomes. As such, MDPs are useful for modeling many real-world
problems where RL agents are exposed to inherent uncertainty. An MDP is
characterized by a set of states $s\in S$, a set of actions $a\in A(s)$
available at each state $s$, transition dynamics specifying the probability of
obtaining state $s_{j}$ upon taking action $a$ at state $s_{i}$, and a reward
function $R(s_{i},a,s_{j})$. Of importance, an agent selects actions according
to a _policy_ which is maintained as a probability distribution over the
actions available at any given state. The objective of an RL agent is to learn
an optimal policy (one which selects actions maximizing the expected
cumulative rewards) given that both the transition dynamics and the reward
function of the environment are unknown a priori.

Utilizing quantum computers to perform RL was first discussed by Dong et al.
in 2005 [122], with a follow-up in 2008 [123]. In their approach, the possible
actions at any given state in the environment are maintained in a quantum
superposition, and amplitude amplification is used to increase the probability
of measuring a good action at any given state. In 2017, Dunjko et al.
published a framework for quantum RL, where they expand upon the amplitude-
amplification approach, which assumes access to an oracle representing the
environment [124]. Furthermore, they introduce more general techniques for
learning model meta-parameters, and additionally observe that there is
significant potential for quantum advantage in luck-favoring task environments
(i.e., environments where a lucky agent finds good sequences of actions much
sooner than an unlucky agent) following from quantum search-based speedups. In
a 2021 paper, Wang et al. derive a quantum RL algorithm with quadratic
performance improvements in various parameters over corresponding classical
algorithms for the evaluation of an optimal policy, state-values, and state-
action pair values (q-values) in an MDP [125]. They explain that this work is
applicable to any RL problem where the environment may be classically
simulated, as a classical circuit implementing the simulator may be
efficiently turned into a quantum circuit. Additionally, recent studies have
explored the use of variational PQCs to implement both RL and Deep RL (DRL) in
continuous action spaces [126, 127].

Next, we present some use cases showing how quantum RL techniques can be
utilized in the finance domain.

###  VII-A Algorithmic Trading

The process of executing trades of financial instruments systematically by
accounting for market variables with limited or no human intervention is
referred to as algorithmic or automated trading. Generally, algorithmic
trading is performed by predictions in a supervised manner followed by
obtaining optimal trading decisions under uncertainty associated with the
corresponding predictions and market volatility. RL bypasses the need for
predictions by casting algorithmic trading as a sequential decision-making
problem wherein trading decisions are obtained directly that maximize the
cumulative returns over a finite time horizon [128]. The domain of RL, and
more specifically DRL, has demonstrated huge applicability for algorithmic
trading [129]. However, such RL approaches for automated trading operate under
certain strong assumptions and may benefit from quantum ML techniques for
improved time and model complexity.

Algorithmic trading can be cast to a multi-period portfolio-selection problem
that involves re-balancing the portions of capital invested in selected assets
at each stage. There have been attempts to solve this multi-stage optimization
problem with a QA device to obtain an optimal trading trajectory [130].
However, this approach does not adopt any RL technique based on policy or
value function approximation. Due to the hardware limitations of the current
quantum devices, quantum RL approaches have not been directly applied yet to
automated trading. Nevertheless, components of algorithmic trading can
certainly benefit from quantum advantages offered by quantum RL. For instance,
the LSTM neural network architecture used as q-value estimator [131] could be
potentially replaced with quantum LSTM [15] for improved performance. Also,
variational quantum circuits [127] can be used for different DRL components
applied to decision-making in algorithmic trading.

###  VII-B Market Making

Market makers have an important role in financial markets as they increase the
liquidity of exchanges, thereby facilitating transactions and investment [132,
133]. A market maker is responsible for maintaining a set of sell orders
(asks) and buy orders (bids) at various quantities and prices. When incoming
market orders are made on a security held by the market maker, they are
required to transact. As such, they inherently assume risk, as a position they
are forced to acquire can subsequently depreciate. Market makers profit by
taking advantage of the gap, called spread, between the lowest ask and highest
bid. For instance, assuming an incoming market order is made to sell security
$X$, the market maker will fulfill the order purchasing it at their bid price.
If another market order is immediately made to purchase security $X$, the
market maker fulfills the order by selling it at their ask thereby price,
obtaining a profit equal to the spread.

Market making is amenable to quantum RL, where the problem can be modelled
with an agent state, taking into account attributes such as inventory and
risk-tolerance, and an environment state where the agent only has partial
information, which may not necessarily be Markovian [134].

##  VIII Natural Language Processing

Natural Language Processing (NLP) is the field concerned with automated text
and language analysis. A drawback with most search engines that use classical
NLP is that they understand separate words and not a grammatical structure.
This has triggered research in distributional compositional semantics (DisCo).
A particular DisCo model is the Coecke, Sadrzadeh and Clark (CSC) model [135,
136], based on tensor-product composition inspired by quantum theory.

In modern classical NLP, the vector space model [137] is used to compute the
meaning of individual words. Given an individual word $w$ in a text, its
meaning is computed by first setting up basis words (i.e., the most common
words in the text) and then, for each of $w$’s nearby basis words, counting
its frequency through the text. The proximity of two words is measured by the
similarity between them and it is calculated, for example, using the inner
product of their normalized representative vectors. These are called
distributional methods and cannot be extended to find the meaning of long
sentences as two sentences are not typically repeated. In contrast, algorithms
based on compositional semantics derive the meaning of a sentence from known
meanings of component words. The DisCo model combines both approaches to
introduce grammatical understanding to the composition of word vectors.

In the CSC model, each grammatical type in the text is assigned a tensor
product space based on some grammar (e.g., the Lambek’s pregroup grammar
[138]). For instance, a transitive verb takes a subject noun as a left
argument and an object noun as right argument. The meaning of a noun is
calculated as in the distributional model; its vector space is denoted as
$\mathscr{N}$. Therefore, the meaning of a transitive verb is a tensor in the
space $\mathscr{N}\otimes\mathscr{L}\otimes\mathscr{N}$, where $\mathscr{L}$
is the meaning space for the sentences. An important feature of this model is
the use of diagrammatic notation for vectors, tensors and linear maps. This
model has the computational challenge of large tensor product spaces. Even
thought there exist classical approaches—such as dimensionality reduction
[139]—to avoid the calculation of the full tensor product, they make certain
assumptions that are not always necessarily met.

The recent development of encoding classical data on quantum hardware using
variational PQCs enables quantum NLP to be particularly suitable for NISQ
devices. In particular, the quantum CSC model can encode linguistic structures
faster in comparison to its classical counterpart. Its quantum speedup stems
from the quantum nearest-neighbor algorithm that is employed for the sentence
similarity calculations in the DisCo framework. If certain conditions are met,
for the $N$-dimensional noun meaning space there is a quantum algorithm
capable of classifying any CSC model sentence composed of $n$ tensors into $M$
classes with time $O(\sqrt{MN}\log(M))$, an improvement over classical
methods’ $O(NM)$ complexity [140].

Below are a few potential applications of the discussed quantum NLP techniques
in the financial sector.

###  VIII-A Risk Assessment

Banks can quantify the chances of a successful loan payment based on a credit
risk assessment. Usually, the payment capacity is calculated based on previous
spending patterns and past loan payment history. However, this information is
not always available, especially for underbanked applicants. NLP techniques
can be applied to solve this problem, by using multiple data points to assess
credit risk. For instance, NLP can measure attitude and an entrepreneurial
mindset in business loans. Similarly, it can also point out incoherent data
and take it up for more scrutiny. Even more, the subtle aspects, such as the
lender’s and borrower’s emotions during a loan process, can be incorporated
with the help of NLP [141, 142].

###  VIII-B Financial Forecasting

Financial forecasting is based on many macroeconomic factors, which are
unstructured and scattered across different sources. This is the reason why
NLP techniques are frequently employed [143]. For example, NLP has been
proposed for classification of news articles as significant or non-significant
from the financial point of view [144]. In addition, sentiment analysis, which
plays an important role in decision-making by traders, has also been carried
out with the help of NLP techniques [145].

###  VIII-C Accounting and Auditing

Another application of NLP is accounting and auditing [142], whose objective
is the detection and prevention of fraud via evaluation of accounting systems,
monitoring of internal controls, assessment of fraud risk, and interpretation
of financial data for anomalous trends. NLP has been proposed for the creation
of semantic knowledge bases or trees for financial accounting standards. Also
auditors can detect anomalies in financial statements by applying NLP
techniques.

##  IX Conclusion

In this paper, we presented an introduction of quantum ML techniques and their
applications in the financial services sector. We identified seven machine
learning tasks, for which several quantum algorithms have been previously
proposed in the literature: regression, classification, clustering, generative
learning, feature extraction, sequential decision-making, and Natural Language
Processing. We analyzed the speedups offered by various quantum ML techniques,
and discussed the financial applications that could benefit from such quantum
acceleration. Moreover, where the literature for finance-specific quantum ML
techniques remains sparse, we provide insights into applying state-of-the-art
general quantum ML techniques to specific financial use cases. Additionally,
we consider the realities of implementing quantum computing techniques in the
financial sector, for example, by considering the challenges imposed by
hardware limitations. In summary, this article serves as a road map towards
enriching the finance industry with quantum ML techniques in the NISQ era and
beyond.

## Disclaimer

This paper was prepared for information purposes by the Future Lab for Applied
Research and Engineering (FLARE) group of JPMorgan Chase Bank, N.A.. This
paper is not a product of the Research Department of JPMorgan Chase & Co. or
its affiliates. Neither JPMorgan Chase & Co. nor any of its affiliates make
any explicit or implied representation or warranty and none of them accept any
liability in connection with this paper, including, but limited to, the
completeness, accuracy, reliability of information contained herein and the
potential legal, compliance, tax or accounting effects thereof. This document
is not intended as investment research or investment advice, or a
recommendation, offer or solicitation for the purchase or sale of any
security, financial instrument, financial product or service, or to be used in
any way for evaluating the merits of participating in any transaction.

## References

  * [1] Alexandre Menard, Ivan Ostojic, Mark Patel and Daniel Volz  “A game plan for quantum computing”  In _McKinsey & Co._, 2020
  * [2] John Preskill  “Quantum computing in the NISQ era and beyond”  In _Quantum_ , 2018
  * [3] Aram W. Harrow, Avinatan Hassidim and Seth Lloyd  “Quantum Algorithm for Linear Systems of Equations”  In _PRL_ , 2009
  * [4] András Gilyén, Yuan Su, Guang Hao Low and Nathan Wiebe  “Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics”  In _STOC_ , 2019
  * [5] Scott Aaronson  “Read the fine print”  In _Nature Phys_ , 2015
  * [6] Vittorio Giovannetti, Seth Lloyd and Lorenzo Maccone  “Quantum random access memory”  In _PRL_ APS, 2008
  * [7] John Cortese and Timothy Braje  “Loading Classical Data into a Quantum Computer”  In _arXiv:1803.01958_ , 2018
  * [8] Nathan Wiebe, Daniel Braun and Seth Lloyd  “Quantum Algorithm for Data Fitting”  In _PRL_ , 2012
  * [9] Guoming Wang  “Quantum algorithm for linear regression”  In _PRA_ , 2017
  * [10] Maria Schuld, Ilya Sinayskiy and Francesco Petruccione  “Prediction by linear regression on a quantum computer”  In _PRA_ , 2016
  * [11] Shantanav Chakraborty, András Gilyén and Stacey Jeffery  “The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation”  In _arXiv:1804.01973_ , 2018
  * [12] Iordanis Kerenidis and Anupam Prakash  “Quantum gradient descent for linear systems and least squares”  In _PRA_ , 2020
  * [13] Nai-Hui Chia, Han-Hsuan Lin and Chunhao Wang  “Quantum-inspired sublinear classical algorithms for solving low-rank linear systems”  In _arXiv:1811.04852_ , 2018
  * [14] Johannes Bausch  “Recurrent Quantum Neural Networks”  In _NeurIPS_ , 2020
  * [15] Samuel Yen-Chi Chen, Shinjae Yoo and Yao-Lung L Fang  “Quantum long short-term memory”  In _arXiv:2009.01783_ , 2020
  * [16] Yuto Takaki et al.  “Learning temporal data with a variational quantum recurrent neural network”  In _PRA_ , 2021
  * [17] John H Cochrane  “Asset pricing: Revised edition”  Princeton university press, 2009
  * [18] Shihao Gu, Bryan Kelly and Dacheng Xiu  “Empirical Asset Pricing via Machine Learning”  In _The Review of Financial Studies_ , 2020
  * [19] Luyang Chen, Markus Pelger and Jason Zhu  “Deep Learning in Asset Pricing”  In _arXiv:1904.00745_ , 2021
  * [20] Jack Fan and Marci Mitchell  “Equity-bond correlation: A historical perspective”  In _GCM Research Note_ , 2017
  * [21] Robert Tibshirani  “Regression shrinkage and selection via the lasso”  In _JRSSB_ , 1996
  * [22] Yuxuan Du et al.  “Quantum differentially private sparse regression learning”  In _arXiv:2007.11921_ , 2020
  * [23] Matthias R Fengler, Wolfgang Härdie and Peter Schmidt  “The analysis of implied volatilities”  In _Applied Quantitative Finance_ Springer, 2002
  * [24] Wasim Ahmad, Jose Arreola Hernandez, Seema Saini and Ritesh Kumar Mishra  “The US equity sectors, implied volatilities, and COVID-19: What does the spillover analysis reveal?”  In _Resources Policy_ Elsevier, 2021
  * [25] Takayuki Sakuma  “Application of deep quantum neural networks to finance”  In _arXiv:2011.07319_ , 2020
  * [26] Kerstin Beer et al.  “Training deep quantum neural networks”  In _Nat. Comm._ , 2020
  * [27] Guoqiang Peter Zhang  “Neural networks for classification: a survey”  In _TSMCPC_ IEEE, 2000
  * [28] Sonika Johri et al.  “Nearest centroid classification on a trapped ion quantum computer”  In _arXiv:2012.04145_ , 2020
  * [29] Marvin Minsky and Seymour A Papert  “Perceptrons: An introduction to computational geometry”  MIT press, 2017
  * [30] Seth Lloyd  “Quantum Machine Learning for Data Classification”  In _Physics_ , 2021
  * [31] Ashish Kapoor, Nathan Wiebe and Krysta Svore  “Quantum Perceptron Models”  In _NIPS_ , 2016
  * [32] Kenneth L Clarkson, Elad Hazan and David P Woodruff  “Sublinear optimization for machine learning”  In _JACM_ , 2012
  * [33] Tongyang Li, Shouvanik Chakrabarti and Xiaodi Wu  “Sublinear quantum algorithms for training linear and kernel-based classifiers”  In _ICML_ , 2019
  * [34] Jerome Friedman, Trevor Hastie and Robert Tibshirani  “The elements of statistical learning”  Springer, 2001
  * [35] Jing Li, Song Lin, Yu Kai and Gongde Guo  “Quantum K-nearest neighbor classification algorithm based on Hamming distance”  In _arXiv:2103.04253_ , 2021
  * [36] Christoph Durr and Peter Hoyer  “A quantum algorithm for finding the minimum”  In _arXiv:9607014_ , 1996
  * [37] Afrad Basheer, A Afham and Sandeep K Goyal  “Quantum $k$-nearest neighbors algorithm”  In _arXiv:2003.09187_ , 2020
  * [38] Kohei Miyamoto, Masakazu Iwamura and Koichi Kise  “A Quantum Algorithm for Finding $k$-Minima”  In _arXiv:1907.03315_ , 2019
  * [39] Bernhard E Boser, Isabelle M Guyon and Vladimir N Vapnik  “A training algorithm for optimal margin classifiers”  In _COLT_ , 1992
  * [40] Vojtěch Havlíček et al.  “Supervised learning with quantum-enhanced feature spaces”  In _Nature_ , 2019
  * [41] Kosuke Mitarai and Keisuke Fujii  “Methodology for replacing indirect measurements with direct measurements”  In _Phys. Rev. Research_ , 2019
  * [42] Harry Buhrman, Richard Cleve, John Watrous and Ronald De Wolf  “Quantum fingerprinting”  In _PRL_ , 2001
  * [43] Iordanis Kerenidis, Anupam Prakash and Dániel Szilágyi  “Quantum algorithms for second-order cone programming and support vector machines”  In _Quantum_ , 2021
  * [44] Patrick Rebentrost, Masoud Mohseni and Seth Lloyd  “Quantum Support Vector Machine for Big Data Classification”  In _PRL_ , 2014
  * [45] Sukin Sim, Peter D. Johnson and Alán Aspuru‐Guzik  “Expressibility and Entangling Capability of Parameterized Quantum Circuits for Hybrid Quantum‐Classical Algorithms”  In _QUTE_ , 2019
  * [46] Diederik P Kingma and Jimmy Ba  “Adam: A method for stochastic optimization”  In _arXiv:1412.6980_ , 2014
  * [47] Qi Li, Isao Sato and Yutaka Murakami  “Simultaneous perturbation stochastic approximation algorithm for automated image registration optimization”  In _IGARSS_ , 2006
  * [48] M. Powell  “A View of Algorithms for Optimization Without Derivatives”  In _Mathematics TODAY_ , 2007
  * [49] Marco Cerezo, Akira Sone, Lukasz Cincio and Patrick Coles  “Barren Plateau Issues for Variational Quantum-Classical Algorithms”  In _APS_ , 2020
  * [50] Kunal Sharma, Marco Cerezo, Lukasz Cincio and Patrick J Coles  “Trainability of dissipative perceptron-based quantum neural networks”  In _arXiv:2005.12458_ , 2020
  * [51] Marco Cerezo et al.  “Cost function dependent barren plateaus in shallow parametrized quantum circuits”  In _Nat. Comm._ , 2021
  * [52] Arthur G Rattew et al.  “A domain-agnostic, noise-resistant, hardware-efficient evolutionary variational quantum eigensolver”  In _arXiv:1910.09694_ , 2019
  * [53] Maria Schuld  “Supervised quantum machine learning models are kernel methods”  In _arXiv:2101.11020_ , 2021
  * [54] Hsin-Yuan Huang et al.  “Power of data in quantum machine learning”  In _Nat. Comm._ , 2021
  * [55] Arthur Jacot, Franck Gabriel and Clément Hongler  “Neural Tangent Kernel: Convergence and Generalization in Neural Networks”  In _arXiv:1806.07572_ , 2020
  * [56] Maria Schuld, Ryan Sweke and Johannes Jakob Meyer  “Effect of data encoding on the expressive power of variational quantum-machine-learning models”  In _PRA_ , 2021
  * [57] Adrián Pérez-Salinas, Alba Cervera-Lierta, Elies Gil-Fuster and José I. Latorre  “Data re-uploading for a universal quantum classifier”  In _Quantum_ , 2020
  * [58] Hiroshi Yano, Yudai Suzuki, Rudy Raymond and Naoki Yamamoto  “Efficient discrete feature encoding for variational quantum classifier”  In _QCE_ , 2020
  * [59] Amira Abbas et al.  “The power of quantum neural networks”  In _Nat. Comp. Sci._ , 2021
  * [60] Bingyu Cao, Deping Zhan and Xianbin Wu  “Application of SVM in Financial Research”  In _CSO_ , 2009
  * [61] Alex Nekritin  “Binary Options: Strategies for Directional and Volatility Trading”  John Wiley & Sons, 2012
  * [62] Peter D Easton et al.  “Forecasting Earnings Using k-Nearest Neighbor Matching”  In _SSRN_ , 2020
  * [63] MA Mukid, T Widiharih, A Rusgiyono and A Prahutama  “Credit scoring analysis using weighted k nearest neighbor”  In _Journal of Physics: Conference Series_ , 2018
  * [64] David Horn and Assaf Gottlieb  “Algorithm for data clustering in pattern recognition problems based on quantum mechanics”  In _PRL_ , 2001
  * [65] Marvin Weinstein and David Horn  “Dynamic quantum clustering: a method for visual exploration of structures in data”  In _Phys. Rev. E_ , 2009
  * [66] Marvin Weinstein et al.  “Analyzing big data with dynamic quantum clustering”  In _arXiv:1310.2700_ , 2013
  * [67] Iordanis Kerenidis, Jonas Landman, Alessandro Luongo and Anupam Prakash  “q-means: A quantum algorithm for unsupervised machine learning”  In _arXiv:1812.03584_ , 2018
  * [68] Teague Tomesh, Pranav Gokhale, Eric R Anschuetz and Frederic T Chong  “Coreset Clustering on Small Quantum Computers”  In _arXiv:2004.14970_ , 2020
  * [69] Sumsam Ullah Khan, Ahsan Javed Awan and Gemma Vall-Llosera  “K-Means Clustering on Noisy Intermediate Scale Quantum Computers”  In _arXiv:1909.12183_ , 2019
  * [70] Samuel S Mendelson, Robert W Strand, Guy B Oldaker IV and Jacob M Farinholt  “Quantum-assisted clustering algorithms for NISQ-era devices”  In _arXiv:1904.08992_ , 2019
  * [71] Esma Aïmeur, Gilles Brassard and Sébastien Gambs  “Quantum clustering algorithms”  In _ICML_ , 2007
  * [72] Charu C Aggarwal and S Yu Philip  “An effective and efficient algorithm for high-dimensional outlier detection”  In _VLDB_ Springer, 2005
  * [73] Naman D Singh and Abhinav Dhall  “Clustering and learning from imbalanced data”  In _arXiv:1811.00972_ , 2018
  * [74] M Sathyapriya and V Thiagarasu  “A Cluster Based Approach for Credit Card Fraud Detection System using Hmm with the Implementation of Big Data Technology”  In _IJAER_ , 2019
  * [75] Newton Da Costa Jr, Jefferson Cunha and Sergio Da Silva  “Stock selection based on cluster analysis”  In _Economics Bulletin_ , 2005
  * [76] Eduardo Levy-Yeyati and Federico Sturzenegger  “Classifying exchange rate regimes: Deeds vs. words”  In _European Economic Review_ Elsevier, 2005
  * [77] Barry Eichengreen and Raul Razo-Garcia  “How reliable are de facto exchange rate regime classifications?”  In _IJFE_ Wiley Online Library, 2013
  * [78] Nandita Das  “Hedge fund classification using k-means clustering method”  In _CEF_ , 2003
  * [79] Ian Goodfellow, Yoshua Bengio and Aaron Courville  “Deep Learning”  MIT Press, 2016
  * [80] Andrew Y Ng and Michael I Jordan  “On discriminative vs. generative classifiers: A comparison of logistic regression and naive bayes”  In _NIPS_ , 2002
  * [81] Alec Radford, Luke Metz and Soumith Chintala  “Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks”  In _arXiv:1511.06434_ , 2016
  * [82] Daphne Koller and Nir Friedman  “Probabilistic graphical models: principles and techniques”  MIT press, 2009
  * [83] Mohammad H. Amin et al.  “Quantum Boltzmann Machine”  In _PRX_ , 2018
  * [84] Edward Farhi, Jeffrey Goldstone, Sam Gutmann and Michael Sipser  “Quantum computation by adiabatic evolution”  In _arXiv:0001106_ , 2000
  * [85] Vivek Dixit et al.  “Training Restricted Boltzmann Machines With a D-Wave Quantum Annealer”  In _Front. Phys._ , 2021
  * [86] Richard Harris et al.  “Experimental investigation of an eight-qubit unit cell in a superconducting optimization processor”  In _Phys. Rev. B_ , 2010
  * [87] Christa Zoufal, Aurélien Lucchi and Stefan Woerner  “Variational quantum Boltzmann machines”  In _QMI_ , 2021
  * [88] Xiao Yuan et al.  “Theory of variational quantum simulation”  In _Quantum_ , 2019
  * [89] Ian J. Goodfellow et al.  “Generative Adversarial Networks”  In _arXiv:1406.2661_ , 2014
  * [90] Seth Lloyd and Christian Weedbrook  “Quantum Generative Adversarial Learning”  In _PRL_ American Physical Society, 2018
  * [91] Pierre-Luc Dallaire-Demers and Nathan Killoran  “Quantum generative adversarial networks”  In _PRA_ American Physical Society, 2018
  * [92] Ling Hu et al.  “Quantum generative adversarial learning in a superconducting quantum circuit”  In _Science Advances_ American Association for the Advancement of Science, 2019
  * [93] Shouvanik Chakrabarti et al.  “Quantum Wasserstein GANs”  In _arXiv:1911.00111_ , 2019
  * [94] Song Cheng, Jing Chen and Lei Wang  “Information Perspective to Probabilistic Modeling: Boltzmann Machines versus Born Machines”  In _Entropy_ , 2018
  * [95] Brian Coyle, Daniel Mills, Vincent Danos and Elham Kashefi  “The Born supremacy: quantum advantage and training of an Ising Born machine”  In _npj Quantum Information_ , 2020
  * [96] Daniel Herr, Benjamin Obert and Matthias Rosenkranz  “Anomaly detection with variational quantum generative adversarial networks”  In _Quantum Science and Technology_ IOP Publishing, 2021
  * [97] Christa Zoufal, Aurélien Lucchi and Stefan Woerner  “Quantum generative adversarial networks for learning and loading random distributions”  In _npj Quantum Information_ Nature Publishing Group, 2019
  * [98] Haozhen Situ et al.  “Quantum generative adversarial network for generating discrete distribution”  In _Information Sciences_ , 2020
  * [99] Nikitas Stamatopoulos et al.  “Option pricing using quantum computers”  In _Quantum_ , 2020
  * [100] Thomas Häner, Martin Roetteler and Krysta M Svore  “Optimizing quantum circuits for arithmetic”  In _arXiv:1805.12445_ , 2018
  * [101] Lov Grover and Terry Rudolph  “Creating superpositions that correspond to efficiently integrable probability distributions”  In _arXiv:0208112_ , 2002
  * [102] Arthur G Rattew, Yue Sun, Pierre Minssen and Marco Pistoia  “The Efficient Preparation of Normal Distributions in Quantum Registers”  In _arXiv:2009.06601_ , 2020
  * [103] Maria Schuld et al.  “A quantum hardware-induced graph kernel based on Gaussian Boson Sampling”  In _arXiv:1905.12646_ , 2019
  * [104] Rupak Chatterjee and Ting Yu  “Generalized coherent states, reproducing kernels, and quantum support vector machines”  In _arXiv:1612.03713_ , 2016
  * [105] Xinbiao Wang, Yuxuan Du, Yong Luo and Dacheng Tao  “Towards understanding the power of quantum kernels in the NISQ era”  In _arXiv:2103.16774_ , 2021
  * [106] Karol Bartkiewicz et al.  “Experimental kernel-based quantum machine learning in finite feature space”  In _Scientific Reports_ Nature Publishing Group, 2020
  * [107] Seth Lloyd, Masoud Mohseni and Patrick Rebentrost  “Quantum principal component analysis”  In _Nature Physics_ Nature Publishing Group, 2014
  * [108] Ri-Gui Zhou, Han Yu, Yu Cheng and Feng-Xin Li  “Quantum image edge extraction based on improved Prewitt operator”  In _QIP_ Springer, 2019
  * [109] Johannes Wachs and János Kertész  “A network approach to cartel detection in public auction markets”  In _Scientific Reports_ Nature Publishing Group, 2019
  * [110] Lu Bai et al.  “Quantum kernels for unattributed graphs using discrete-time quantum walks”  In _Pattern Recognition Letters_ Elsevier, 2017
  * [111] Ruslan Shaydulin et al.  “Network community detection on small quantum computers”  In _QUTE_ Wiley Online Library, 2019
  * [112] Kamil Brádler et al.  “Graph isomorphism and Gaussian boson sampling”  In _Special Matrices_ , 2021
  * [113] Ana Martin et al.  “Toward pricing financial derivatives with an ibm quantum computer”  In _Phys. Rev. Research_ , 2021
  * [114] David Heath, Robert Jarrow and Andrew Morton  “Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation”  In _Econometrica_ , 1992
  * [115] Andrew Milne, Maxwell Rounds and Phil Goddard  “Optimal feature selection in credit scoring and classification using a quantum annealer”  In _White Paper 1Qbit_ , 2017
  * [116] Richard S Sutton and Andrew G Barto  “Reinforcement learning: An introduction”  MIT press, 2018
  * [117] Volodymyr Mnih et al.  “Human-level control through deep reinforcement learning”  In _Nature_ Nature Publishing Group, 2015
  * [118] David Silver et al.  “Mastering the game of Go with deep neural networks and tree search”  In _Nature_ Nature Publishing Group, 2016
  * [119] David Silver et al.  “Mastering the game of go without human knowledge”  In _Nature_ Nature Publishing Group, 2017
  * [120] Jens Kober, J Andrew Bagnell and Jan Peters  “Reinforcement learning in robotics: A survey”  In _The International Journal of Robotics Research_ SAGE Publications Sage UK: London, England, 2013
  * [121] Ahmad EL Sallab, Mohammed Abdou, Etienne Perot and Senthil Yogamani  “Deep reinforcement learning framework for autonomous driving”  In _Electronic Imaging_ Society for Imaging ScienceTechnology, 2017
  * [122] Daoyi Dong, Chunlin Chen and Zonghai Chen  “Quantum reinforcement learning”  In _ICNC_ , 2005
  * [123] Daoyi Dong, Chunlin Chen, Hanxiong Li and Tzyh-Jong Tarn  “Quantum reinforcement learning”  In _IEEE SMC_ , 2008
  * [124] Vedran Dunjko, Jacob M Taylor and Hans J Briegel  “Advances in quantum reinforcement learning”  In _IEEE SMC_ , 2017
  * [125] Daochen Wang et al.  “Quantum algorithms for reinforcement learning with a generative model”  In _ICML_ , 2021
  * [126] Samuel Yen-Chi Chen et al.  “Variational quantum circuits for deep reinforcement learning”  In _IEEE Access_ IEEE, 2020
  * [127] Shaojun Wu, Shan Jin, Dingding Wen and Xiaoting Wang  “Quantum reinforcement learning in continuous action space”  In _arXiv:2012.10711_ , 2020
  * [128] Zihao Zhang, Stefan Zohren and Stephen Roberts  “Deep reinforcement learning for trading”  In _JFDS_ Institutional Investor Journals Umbrella, 2020
  * [129] Tidor-Vlad Pricope  “Deep Reinforcement Learning in Quantitative Algorithmic Trading: A Review”  In _arXiv:2106.00123_ , 2021
  * [130] Gili Rosenberg et al.  “Solving the optimal trading trajectory problem using a quantum annealer”  In _IEEE JSTSP_ , 2016
  * [131] Yang Li, Wanshan Zheng and Zibin Zheng  “Deep robust reinforcement learning for practical algorithmic trading”  In _IEEE Access_ IEEE, 2019
  * [132] Marco Avellaneda and Sasha Stoikov  “High-frequency trading in a limit order book”  In _Quantitative Finance_ Taylor & Francis, 2008
  * [133] Olivier Guéant, Charles-Albert Lehalle and Joaquin Fernandez-Tapia  “Dealing with the inventory risk: a solution to the market making problem”  In _Mathematics and Financial Economics_ Springer, 2013
  * [134] Thomas Spooner, John Fearnley, Rahul Savani and Andreas Koukorinis  “Market making via reinforcement learning”  In _arXiv:1804.04216_ , 2018
  * [135] Stephen Clark, Bob Coecke and Mehrnoosh Sadrzadeh  “A compositional distributional model of meaning”  In _QI_ , 2008
  * [136] Bob Coecke, Mehrnoosh Sadrzadeh and Stephen Clark  “Mathematical foundations for a compositional distributional model of meaning”  In _arXiv:1003.4394_ , 2010
  * [137] Hinrich Schütze  “Automatic word sense discrimination”  In _Computational Linguistics_ , 1998
  * [138] Joachim Lambek  “From Word to Sentence: a computational algebraic approach to grammar”  Polimetrica sas, 2008
  * [139] Tamara Polajnar, Luana Fagarasan and Stephen Clark  “Learning type-driven tensor-based meaning representations”  In _arXiv:1312.5985_ , 2013
  * [140] William Zeng and Bob Coecke  “Quantum algorithms for compositional natural language processing”  In _arXiv:1608.01406_ , 2016
  * [141] Lynnette Purda and David Skillicorn  “Accounting variables, deception, and a bag of words: Assessing the tools of fraud detection”  In _Contemporary Accounting Research_ Wiley Online Library, 2015
  * [142] Ingrid E Fisher, Margaret R Garnsey and Mark E Hughes  “Natural language processing in accounting, auditing and finance: A synthesis of the literature with a roadmap for future research”  In _ISAFM_ Wiley Online Library, 2016
  * [143] Frank Z Xing, Erik Cambria and Roy E Welsch  “Natural language based financial forecasting: a survey”  In _AI Review_ Springer, 2018
  * [144] Savaş Yıldırım, Dhanya Jothimani, Can Kavaklıoğlu and Ayşe Başar  “Classification of “Hot News” for Financial Forecast Using NLP Techniques”  In _Big Data_ , 2018
  * [145] Kostadin Mishev et al.  “Evaluation of sentiment analysis in finance: from lexicons to transformers”  In _IEEE Access_ IEEE, 2020