# Quantum Machine Learning: from physics to software engineering

Alexey Melnikova and Mohammad Kordzanganeha and Alexander Alodjantsb and Ray-
Kuang Leec,d,e,f CONTACT Alexey Melnikov. Email<EMAIL_ADDRESS>aTerra
Quantum AG, 9000 St. Gallen, Switzerland; bITMO University, 197101 St.
Petersburg, Russia; cInstitute of Photonics Technologies, National Tsing Hua
University, Hsinchu 300, Taiwan; dDepartment of Physics, National Tsing Hua
University, Hsinchu 300, Taiwan; ePhysics Division, National Center for
Theoretical Sciences, Taipei 10617, Taiwan; fCenter for Quantum Technology,
Hsinchu 30013, Taiwan

###### Abstract

Quantum machine learning (QML) is a new, rapidly growing, and fascinating area
of research where quantum information science and quantum technologies meet
novel machine learning and artificial intelligent facilities. A comprehensive
analysis of the main directions of current QML methods and approaches is
performed in this review. The aim of our work is twofold. First, we show how
classical machine learning approach can help improve the facilities of quantum
computers and simulators available today. It is most important due to the
modern noisy intermediate-scale quantum (NISQ) era of quantum technologies. In
particular, the classical machine learning approach allows optimizing quantum
hardware for achieving desired quantum states by implementing quantum devices.
Second, we discuss how quantum algorithms and quantum computers may be useful
for solving keystone classical machine learning tasks. Currently, quantum-
inspired algorithms, which use a quantum approach to classical information
processing, represent a powerful tool in software engineering for improving
classical computation capacities. In this work, we discuss various quantum
neural network capabilities that can be implemented in quantum-classical
training algorithms for variational circuits. It is expected that quantum
computers will be involved in routine machine learning procedures. In this
sense, we are showing how it is essential to elucidate the speedup problem for
random walks on arbitrary graphs, which are used in both classical and quantum
algorithms. Quantum technologies enhanced by machine learning in fundamental
and applied quantum physics, as well as quantum tomography and photonic
quantum computing, are also covered.

###### keywords:

Quantum information and computing, machine learning, quantum technologies,
quantum and quantum-inspired algorithms, quantum walks, graph theory,
variational quantum circuits, quantum tomography, photonic quantum computing.

††articletype: REVIEW ARTICLE

##  1 Introduction

Nowadays due to exponential grows of information, computational speedup,
acceleration of information transmission and recognition, many key global
interdisciplinary problems for modern societies emerge [1]. In everyday life
we face a problem of big data everywhere. Classical information science and
relevant technological achievements in communication and computing enable to
move our society from Internet of computers to the Internet of Things (IoT),
when humans interact with spatially distributed smart systems including high
precision sensors, various recommendation systems based on huge amount of on-
line information processing and its recognition [2]. Artificial intelligence
(AI) and machine learning (ML) drive the progress in this movement of our
society. These tasks and facilities require online information recognition
that is actually possible only on the basis of the parallel information
processing.

Today, a number of areas have formed in information science, physics,
mathematics and engineering which propose to solve these problems by means of
various approaches of parallel processing of information by spatially
distributed systems. Our vision of the problem we establish schematically in
Fig. 1 that reflects content of this paper. Nowadays AI predominantly focuses
on ML approach that provides solutions for Big data problem, data mining
problem, explainable AI, and knowledge discovery. As a result, in our everyday
life we can find distributed intelligent systems (DIS) which represent
networks of natural intelligent agents (humans) interacting with artificial
intelligence agents (chatbots, digital avatars, recommendation systems, etc.
), see e.g. [3] and references therein. Such systems require new approaches to
data processing that may be described by means of cognitive computing which
possess human cognitive capabilities, cf. [4]. At the same time such a system
operates within a lot of uncertainties which may provide new complexities.
But, how about quantum approach and quantum technologies which can help us in
this way?

Certainly, quantum approach and relevant quantum technologies are one of the
drivers of current progress in development of information sciences and
artificial intelligence, which have common goals of designing efficient
computing, as well as fast and secure communication and smart IoT. The mutual
overlapping of the three seminal disciplines are bearing meaningful fruits
today. Within left halve of ellipsis in Fig. 1 we establish some crucial
topics of quantum technologies studies which are interdisciplinary right now.
In particular, quantum computing opens new horizons for classical software
engineering, see e.g. [5]. Especially it is necessary to mention quantum
inspired algorithms and quantum inspired approaches which utilize quantum
probability and quantum measurement theory for classical computing [6].

Figure 1: Interdisciplinary paradigm of quantum machine learning that is based
on current classical information, quantum technologies, and artificial
intelligence, respectively (the details are given in the text).

Quantum computers as physical systems, biological neurons and human brain are
capable for parallel information proceeding in natural way. However,
sufficient criterion for speedup information processing is still unknown in
many cases.

A qubit which is minimal tool of quantum information science, is established
by superposition of well distinguished two quantum states defined in Hilbert
space and represents indispensable ingredient for parallel information
processing [7]. Quantum algorithms (software) which are proposed many years
ago utilize qubits quantum superposition and entanglement power for achieving
so-called quantum supremacy and speedup in solution of NP-hard problems which
unattainable with classical algorithms [8]. Quantum computers (hardware) as
physical devices firstly was proposed by Richard Feynman, deals with simple
two-level systems as physical qubits performing quantum computation [9, 10].
Despite the fact that a lot of time has passed since the successful
demonstration of the first quantum gates and the simplest operations with them
(see e.g. [11]), there exists a large gap between the quantum information
theory, quantum algorithms, and quantum computers designed to execute them.
Existing quantum computers and simulators are still very far from quantum
supremacy demonstrations in solving real problems related to our daily life.
This can be partly explained by the modern noisy intermediate-scale quantum
(NISQ) era of the development of quantum technologies [12]. Currently, quantum
computers are restricted by small number of qubits, and relatively high level
of various noises which include decoherence processes that completely (or,
partially) destroy the effects of interference and entanglement. In this
regard, the problem of quantum supremacy for specific tasks represents the
subject of heated debates [13, 14, 15].

Surface codes and creation of logical qubits are purposed for significant
reduction of computation errors [16, 17]. In particular, such a codes presume
mapping of some graph of physical qubits onto the logical qubit. Typically,
special network-like circuits are designed for quantum processor consisting of
logical qubits. However, it is unclear how this mapping is unique and how such
a networks is optimal and universal for various computation physical
platforms?

As an example, minor embedding procedure is supposed for quantum annealing
computers which are based on superconductor quantum hardware [18]. Obviously,
various physical platforms examined now for quantum computation can use
different mapping procedures and relay to design of specific networks of
qubits accounting specific noises and decoherence. Thus, the choosing of
appropriate network architecture represent keystone problem for current
quantum computing and properly relays to demonstration of quantum supremacy.

Clearly, the solution of this problem is connected not only with the
properties of quantum systems, but also with the ability of networks to
parallel and robust information processing. An important example that we refer
here is a human brain as a complex network comprising from biologically active
networks which exhibit fast information processing. Noteworthy, the
architecture of such a computations is a non-von Neumann. In order, human
brain capable for pattern retrieving by means of association. A long time ago,
Hopfield introduced simple neural network model for associative memory [19].
As time evolves, neural networks have represented an indispensable tool for
parallel classical computing. Artificial intelligence and machine learning
paradigm, cognitive and neuromorphic computing, use neural network some
specific peculiarities represent vital approach proposed to explore the full
power of parallel character of computation [1, 20].

Quantum machine learning (QML) is a new paradigm of parallel computation where
quantum computing meets network theory for improving computation speed-up and
current, NISQ-era quantum technology facilities, by means of quantum or
classical computational systems and algorithms [21, 22, 23, 24]. In Fig. 2 we
represent a timeline of the appearance and development of some important
algorithms [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45] which are able to improve computational complexity,
accuracy and efficiency within various type of hardware available now. In this
work we are going to discuss most of them in details.

Figure 2: Timeline with milestones of quantum machine learning achievements

In more general, nowadays QML disciplines occur at the border of current
quantum technologies and artificial intelligence and includes all their
methods and approaches to information processing, see Fig.1. The rapidly
growing number of publications and reviews in this discipline indicates an
increasing interest in it from the scientific community, see e.g. [46, 47, 48,
49, 50, 51, 52, 53]. In particular, seminal problems in algorithm theory which
are capable to enhancement of quantum computing by means of ML approach we can
find in [21, 46, 50, 53]. Some applications of ML approach to solve timely
problems in material science, photonics, physical chemistry reader can find in
[47, 48, 49, 51, 52], respectively.

It is important to notice that ML approach is closely connecting with
knowledge discoveries in modern fundamental physics which closely connected
with a problem of big data and their recognition. In order, we talk about
automated scientific discovery which can significantly expand our knowledge of
Nature, cf. [54, 55]. In particular, it is worth to mention research of the
Large Hadron Collider (LHC), where data mining can contribute to new
discoveries in the field of fundamental physics [56]. Another important
example constitutes network researches on registration of gravitational waves
and extreme weak signals in astronomy, see e.g. [57]. Clearly, further
discoveries in this area require improvement of sensitivity of network
detectors (which are interferometers) and obtained data mining where ML
approach can significantly promote, cf. [58].

Despite the fact that previous review papers [21, 22, 46, 47, 49, 50, 51, 52,
53] theoretically substantiates and discusses the effectiveness of quantum
approaches and quantum algorithms in ML problems, in practice there are many
problems that do not allow to see quantum supremacy in experiment. Within the
NISQ era of modern quantum computers and simulators, their capabilities are
not yet enough to achieve quantum supremacy, cf. [12]. In this regard, hybrid
information processing algorithms that take into account the sharing of
quantum and classical computers come forward. Quantum-classical variational,
quantum approximate optimization (QAOA) algorithms are very useful and
effective in this case, see e.g. [59, 60, 61, 62]. In this review work we are
going to discuss various approaches which are use for QML paradigm within
current NISQ-era realities. Unlike previous work [21, 22, 46, 47, 49, 50, 51,
52, 53], below we will focus on methods and approaches of ML that can be
effective, especially for hybrid (quantum-classical) algorithms, see Fig. 2.

In its most general form, current work can be divided into two large parts
which we establish as Sections 2 and 3, respectively. In particular, in Sec. 2
we consider a variety of problems where the ML approach may be enhanced by
means of quantum technologies, as it is presented in Fig. 1. In general, we
speak here about speed-up of data processing by quantum computers and/or
quantum simulators which we can use for classical ML purposes, see Fig. 3. An
important part of these studies is devoted to optimal encoding, or embedding
of classical data set into the quantum device [63], and recognition of data
set from quantum state readout. We establish comprehensive analysis of quantum
neural networks (QNN) features as a novel models in QML whose parameters are
updated classically. We discussed how such a model may be used in timely
hybrid quantum-classical variational algorithms.

On the contrary, in Sec. 3 we establish currently developing QML hot
directions where classical ML approach can help to solve NISQ-era quantum
computing and quantum technology tasks, cf. Fig.1. In particular, it is
necessary to mention automation of quantum experiments, quantum state
tomography, quantum error correction, etc. where classical ML technique may be
applied. Especially, we note here ML algorithms which can be useful in
recognition of quantum speedup problem of random (classical, or quantum) walks
performed on various graphs. The solution of this problem proposed by us plays
essential role for both of current quantum computing hardware and software
development.

##  2 Machine learning enhanced by quantum technologies

In this section an impact that quantum technologies make in machine learning
is discussed. The outline of the topics is given in Fig. 3.

Figure 3: Quantum technologies help in improving machine learning. Sections
that discuss a particular topic are labeled.

###  2.1 Machine learning models

At its roots, machine learning is a procedural algorithm that is augmented by
the provision of external data to model a specific probability distribution.
The data could consist of only environment variables (features),
$\textbf{x}\in\mathcal{X}$ \- unsupervised learning - features and their
associated outcomes (labels), $\textbf{y}\in\mathcal{Y}$ \- supervised
learning - or environment variables and a reward for specific actions, $R(a)$
\- reinforcement learning.

####  2.1.1 Unsupervised learning

The point of unsupervised learning is to infer attributes about a series of
data points, usually to find the affinity of data points to a clustering
regime. A popular method of unsupervised learning is known as the K-means
clustering [64] where data points are assigned to a chosen number of clusters,
and the position of the centres of these clusters could be trained.
Unsupervised learning is applied to many real-world problems, from customer
segmentation in different industries [65] to criminal activity detection [66].

####  2.1.2 Supervised learning

In contrast, supervised learning endeavours to infer patterns in the provided
data. The goal of such models is to generalise this inference to previously-
unseen data points. In a linear regression setting, this is often done by
linear interpolation [67], but for an a-priory problem where some degree of
non-linearity is plausible, supervised learning can train non-linear
regression models and provide better alternatives. Supervised learning is also
used for logistical learning, where instead of a regression model, a
categorical probability distribution is to be learned. Supervised learning has
seen considerable success in many areas, from credit-rating models [68] to
scientific fields [69].

####  2.1.3 Reinforcement learning

Finally, reinforcement learning is the optimisation of a set of actions
(policy) in an environment. The environment allows actions and provides
rewards if certain conditions are met. An agent is made to explore this
environment by investigating the outcomes of certain actions given its current
state and accordingly optimise its model variables. Reinforcement learning
often attracts significant attention from the gaming industry [70] but it has
also contributed to real-life scenarios such as portfolio management [71].

####  2.1.4 Exponential growth of practical machine learning models

A recurring theme in all three modes of ML is the high complexity of their
models. This could be caused by a high-dimensional input size like classifying
a high-resolution image database [72], or a complex problem like image
segmentation [73]. A commonly-used - but known to be inaccurate [74, 75] \-
measure of complexity is the parameter-count of an ML model. It is simply the
number of trainable parameters of a model. Most familiarly in neural networks,
the parameters are the weights and biases associated with each neural layer.
For smaller problems, the parameter-count could be as small as hundreds [76,
77], but cutting-edge AI models such as DALL-E 2 [78], Gopher [79], and GPT-3
[80] are models that have tens or hundreds of billions of parameters and is
increasing [81, 82]. This level of high-dimensionality comes at a great
financial and environmental cost. Ref. [83] assessed the Carbon emission and
the financial cost of fine-tuning and training several large ML models in
2019. They found that in some cases these models emitted more CO2 than the
entire lifetime of an average American car, and could cost over $3m. In
addition to this great cost, there are concerns regarding the scalability,
namely that the exponential-growth in computing power - known as Moore’s law
[84], revisited in [85] \- is growing at a slower rate than ML research [86,
87].

####  2.1.5 Quantum-enhanced machine learning

The idea behind the field of quantum machine learning is to use the
capabilities of quantum computers to provide scalable machine learning models
that can provide machine learning capabilities beyond what classical models
can be expected to deliver, at a healthier cost.

Quantum computers offer an exponential computational space advantage over
classical computers by representing information in quantum binary digits or
qubits. Where classical computers work in the Boolean space,
$\mathbb{B}^{\otimes n}$, qubits form an exponentially growing, special
unitary space, $\mathbb{SU}(2^{n})$. This means that while a classical
register with $n$ bits can hold an $n$-digit binary string, a quantum register
of the same size holds all possible strings of such size, providing an
exponential number of terms in comparison to its classical counterpart [7].

In addition to addressing the scalability concerns, classical machine learning
models operate within the realm of classical statistical theory, which in some
cases seems to diverge from human behavioural surveys. Ref. [88] introduced
the _sure thing principle_ , which shows how unrelated uncertainties could
affect a human’s decision, which a classical statistical model would deem as
unrelated and remove. In Refs. [89, 90] it is showed that in some cases people
tend to give higher credence to two events occurring in conjunction than
either happening individually, which is contrary to the classical statistical
theory picture. In Ref. [6, 91] it is argued that that these problems could be
addressed by using a quantum statistical formulation. In addition, other
similar issues like the problem of negation [92] and others listed in Ref.[93]
are also shown to have a resolution in the quantum theory. The distributional
compositional categorical (DisCoCat) model of language [94] could be addressed
as the first theoretically successful attempt at harnessing this advantage of
quantum machine learning.

###  2.2 Quantum neural networks

For a given data provision method, e.g. supervised learning, a host of
different machine learning architectures could be considered. A machine
learning architecture has a set of trainable parameters $\theta$ that can be
realised based on an initial probability distribution. Any specific
realisation of the parameters of a machine learning architecture is a model.
The quest of machine learning is to train these parameters and achieve a
nearer probability distribution to that of the problem in question. The fully
trained version of each architecture yields a different model with different
performance, and generally, the architectures that can spot and infer existent
patterns in the data are said to be of superior performance. It is also
important to avoid models that find non-existent patterns, models that are
said to over-fit their pattern-recognition to the provided data, and when
evaluated on previously-unseen data fail to perform as well. A model that can
spot existent patterns without over-fitting to the provided data is said to
have a high _generalisation ability_. This metric establishes a platform for
model selection111 Sometimes referred to as Occam’s factor [95]. [96].

For any given problem, there are a variety of architecture classes to choose
from. Some of the most commonly-used architectures are multi-layered
perceptrons (neural networks), convolutional networks for image processing,
and graph neural networks for graphically-structured data. QML contributes to
this list by introducing quantum models such as QNN [97].

Quantum neural networks are models in QML whose parameters are updated
classically. The training pipeline includes providing data to the quantum
model, calculating an objective (loss) value, and then adapting the QNN
parameters in such a way as to reduce this objective function. The specific
approach to providing the data to the quantum model is known as the data
encoding strategy, and it can have a drastic effect on the functional
representation of the model. Sec 2.2.1 covers the various approaches to data
encoding, and Sec 2.2.4 offers a review of the theoretical advances in
exploring the analytical form of this representation. In QNNs, the objective
function is (or includes) the expectation value of a parametrised quantum
circuit (PQC) [98]. PQCs are quantum circuits that make use of continuous-
variable group rotations. Fine-tuning the architecture of the PQC can have a
direct effect on the performance of the resultant QNN model. Sec 2.2.2 reviews
the various PQC parametrisations suggested in the literature.

The consequences of the choice of the loss function are outlined in Sec 2.3.1.
After making this choice, one could evaluate the PQC, and pass the result to
the loss function to obtain a loss value. To minimise the loss value, it is
important to tune the trainable parameters in such a way to maximally minimise
this value. This is achieved - in both classical and quantum ML - by
calculating the gradient of the loss function with respect to the model
parameters222Some alternative approaches exist known as gradient-free
optimisation methods [99].. The gradient vector of a function points to the
direction of maximal increase in that function, and to maximally reduce the
loss function one could find the gradient and step in the opposite direction.
Sec 2.2.3 reviews the literature concerning QNN gradient computation.

####  2.2.1 Data encoding strategies

There are three over-arching data encoding strategies [100]:

  * •

State embedding: the features are mapped to the computational bases. This is
often used for categorical data, and as the number of bases grows, the number
of data points needs to follow the same trend, otherwise, the encoding will be
sparse [97].

  * •

Amplitude embedding: when the features of the dataset are mapped to amplitudes
of the qubit states. This embedding could be repeated to increase the
complexity of the encoding. For $n$ qubits, this method allows us to encode up
to $2^{n+1}$ features onto the quantum system.

  * •

Observable embedding: the features are encoded in a Hamiltonian with respect
to which the quantum system is measured. This encoding is typically used in
quantum native problems - see Sec 2.5.2 \- namely variational quantum
eigensolvers (VQE) [30] and quantum differential equation solvers [101, 102].

It is important to recognise that state embedding is the only discrete-
variable encoding with a strong resemblance to classical ML, whereas the other
two are continuous-variable methods and can be considered analogue machine
learning333This is subject to the input methodology. Normally, a digital
computer is used to set up the quantum circuit, in which case the learning is
still fully digital..

Amplitude embedding could be sub-divided into sub-categories: angle embedding,
state amplitude embedding, squeezing embedding, and displacement embedding
[103, 104]. Ref. [100] provides an expressivity comparison between these
encoding methods. Effective encoding strategies were analysed in [101, 105,
106, 107].

####  2.2.2 Parametrised architecture

The specific parametrisation of the network could dramatically change the
output of a circuit. In classical neural networks, adding parameters to a
network improves the model expressivity, whereas, in a quantum circuit,
parameters could become redundant in over-parametrised circuits [108].
Additionally, the architecture must be trainable, whereas it was shown that
this cannot be assumed in an a-priori setting [109] \- see Sec 2.2.5. Many
architectures have been suggested in the literature, and many templates are
readily available to choose from on QML packages [110, 111, 112].

Ref. [113] introduced a family of hardware-efficient architectures and used
them as variational eigensolvers - see Sec 2.5.2. These architectures repeated
variational gates and used CNOT gates to create highly-entangled systems.
Based on the discrete model in Ref. [114] and made continuous in Ref. [115] a
model was devised using RZZ quantum gates that was shown to be computationally
expensive to classically simulate [116, 117], named the instantaneous
polynomial-time quantum ansatz (IQP).

Another approach to creating quantum circuits is to take inspiration from
tensor networks [118]. Famous architectures in this class are the tensor-tree
network (TTN), matrix product state (MPS), and the quantum convolutional
neural networks (QCNN) [119, 120, 121, 34].

####  2.2.3 Gradient calculation

Despite its excessive memory usage [122], the most prominent gradient
calculation method in classical ML is the back-propagation algorithm [123].
This method computes the gradient of every function that trainable parameters
are passed through alongside its output and employs the chain rule to create
an automatically differentiable ML routine. The back-propagation method can
(and has been [124]) implemented for QML, but as it requires access to the
quantum state-vector, it can only be used on a simulator and not a real
quantum processing unit (QPU). As quantum advantage can only occur in the
latter setting, it is important to seek alternatives that can operate on QPUs.

The first proposed algorithm is known as the finite-difference differentiation
method [125]. As its name suggests, it calculates the gradient by using the
first principles of taking derivatives, i.e. adding a finite difference to the
trainable parameters one at a time, and observing the change that this action
makes. This method is prone to error in the NISQ era.

As an alternative, a discovery was made in [126] known as the parameter-shift
rule that suggested an exact, analytic derivative could be calculated by
evaluating the circuit twice for each trainable parameter. The suggestion was
that the derivative of a circuit with respect to a trainable parameter
$\theta$ is half of the evaluation of the circuit with $\theta$ shifted by
$\frac{\pi}{2}$ subtracted from the circuit when it is shifted by
$-\frac{\pi}{2}$. This suggestion initially worked only on trainable
parameters applied to Pauli rotations, but later works [127, 128, 129, 130,
131, 132, 133, 134] expanded to its current form, applicable to any
parametrisation. The parameter-shift rule is the state-of-the-art gradient
computation method and is compatible with QPUs, but, one of its major problems
is its scalability. As mentioned, the number of circuit evaluations for this
method increases linearly with the number of trainable parameters, and this
poses a challenge to how complex the quantum models can get. A notable effort
to mitigate this effect was by parallelising the gradient computation, which
is now natively provided when using PennyLane on AWS Braket [135].

As a transitional gradient computation method for QNNs, Ref. [136] introduced
the adjoint algorithm. Similar to the back-propagation method, adjoint can
only be run on a simulator and calculates the entire gradient vector using a
single evaluation of the circuit. However, its memory usage is superior to the
former. It works by holding a copy of the quantum state and its adjoint in the
memory, and in turn applying the gates in reverse order, calculating the
gradients wherever possible. This means that two overall evaluations of the
circuit are made, first to evaluate the output, and second to compute the
gradient.

Alternative suggestions have also been made to optimise QML models following
the geometry of their group space. Ref. [137] suggested a Riemannian gradient
flow over the Hilbert space, which through hardware implementation showed its
favourable optimisation performance.

####  2.2.4 Quantum neural networks as universal Fourier estimators

Ref. [107] explored the effects of data encoding on the expressivity of the
model. It proved that the data re-uploading technique suggested by Ref. [138]
created a truncated Fourier series limited by the number of repetitions of the
encoding gates. Ref. [139] also showed that QNNs can be universal Fourier
estimators - an analogue to the universality theorem in classical multi-
layered perceptrons [140]. Another point proven by Refs. [107, 138] was that
by repeating the encoding strategy (in amplitude embedding and more
specifically the angle embedding) more Fourier bases are added to the final
functional representation of the circuit. This was true if the repetitions
were added in parallel qubits or series. This sparked a question about the
accessibility of these Fourier bases, i.e. whether their coefficients can
independently be altered which remains an open question at the time of this
publication.

####  2.2.5 Barren plateaus and trainability issues

QNNs could suffer from the problem of vanishing gradients. This is when during
training, the gradient of the model tends to zero in all directions. This
could severely affect the efficiency of the training or even bring it to a
halt. This is known as the barren plateau (BP) problem.

BPs are not usually at the centre of attention in classical ML, but their
dominance in quantum architectures makes them one of the most important
factors in choosing a circuit. Ref. [109] showed that the expectation value of
the derivative of a well-parametrised444Formally when the variational
architecture of a quantum circuit nears a 1-design[141]. quantum circuit is
equal to zero, and that its variance decays exponentially with the number of
qubits. Ref. [142] confirmed that barren plateaus also exist in gradient-free
optimisation methods. In addition, Ref. [143] showed that in the NISQ era,
using deep circuits flattens the overall loss landscape resulting in noise-
induced BPs.These are mathematically different kinds of Barren plateaus that
flatten the landscape as a whole. The illustrations in Fig 4 summarise these
phenomena.

Figure 4: Visualisation of the barren plateau phenomenon in (a) noise-free and
(b) noise-induced settings.

These two findings painted a sobering picture for the future of QNNs, namely
that they need to be shallow and low on qubit-count to be trainable, which
contradicts the vision of high-dimensional, million-qubit QML models.

Many remedies have been proposed: Ref. [144] suggested that instead of
training all parameters at once, the training could be done layer-wise and
Ref. [145] showed that if the depth of the variational layers of the QNN is of
the order O($\log(n)$), $n$ being the number of qubits, and that only local
measurements are made, the QNN remains trainable. This was tested on circuits
with up to 100 qubits and no BPs were detected. Other remedies included
introducing correlations in the trainable parameters [146, 147] and specific
initialisations of the parameters [148] of the circuit by applying adjoint
operators of certain variational gates.

More analysis was done on specific architectures: Ref. [149] showed that under
well-defined weak conditions, the TTN architecture was free from the BP
problem; and Ref. [150] showed that the quantum convolutional neural network
architecture introduced in Ref. [34] was also free from BPs. Ref. [151]
developed a platform based on ZX calculus 555ZX calculus is a graphical
language in quantum information theory [152, 153, 154]. to analyse whether a
given QNN is subject to suffering from BPs. In addition to confirming the
results from the two earlier contributions, it also proved that the matrix
product state [155] and hardware efficient, strongly-entangling QNNs suffered
from BPs. Furthermore, Ref. [156] related the barren plateau phenomenon to the
degree of entanglement present in the system.

###  2.3 Quantum learning theory

####  2.3.1 Supervised QML methods are kernel methods

In Refs. [100, 157] the similarities between the QNNs and kernel models were
brought to focus. First introduced in Ref. [158], kernel methods are well-
established ML techniques with many applications. In conjunction with support
vector machines (SVM), the way they work is by mapping the features of a
dataset into a high-dimensional space through a function $\phi(\textbf{x})$
and then using a kernel function, $\mathcal{K}(x_{1},x_{2})$, as a distance
measure between any two data points in this high-dimensional space. This is
exactly the behaviour observed in QNNs: the features are first embedded into a
high-dimensional quantum state-vector, and by overlapping one encoded state
with another we can find the level of similarity between two points in this
space. In this high-dimensional space, one hopes to find better insight into
the data - usually expressed as a decision boundary in the form of a
hyperplane in classification tasks. Ref. [159] used this link and developed a
framework for searching for possible quantum advantages over classical models.
It also showed that large models could scatter the data so far apart that a
distance measure becomes too large for optimisation purposes, and proposed
that an intermediate step be added to map the high-dimensional space into a
lower-dimensional hyperplane to improve its training performance.

####  2.3.2 Bayesian inference

Bayesian inference is an alternative approach to the statistical learning
theory where Bayes’ theorem [160] is used to adapt an initial assumption about
the problem (prior distribution) based on newly-found data (evidence) to get a
posterior distribution. Bayesian learning is when this logic is applied to ML.
This is done by applying a distribution to every parameter in the network and
updating the distributions when training. Calculating the posterior
distribution is generally computationally expensive, but it is possible to
approximate it using a trick known as variational inference [161, 162]
successfully demonstrated an approximate back-propagation algorithm on a
Bayesian neural network (BNN), referred to as Bayes-by-backprop.

The first implementations of Bayesian QNNs were in Refs. [163, 164, 165] which
attempted to make quantum circuits into an exact Bayesian inference machine.
Ref. [166] introduced two efficient, but approximate methods - one from kernel
theory and another using a classical adversary - to use QNNs to perform
variational inference. The work consists of a quantum circuit that can be
modelled to produce the probability distribution of a phenomenon by exploiting
the probabilistic nature of quantum mechanics - known as a Born machine [167,
168] or quantum generative models [169, 155, 170, 171, 172, 173, 174]. This
could also be used later to quantify the prediction error for a single data
point, as it has been done classically in Ref. [175].

####  2.3.3 Model complexity and generalisation error bounds

Intuitively, complex phenomena require complex modelling, but quantifying the
complexity of a given model is non-trivial. There are multiple ways of
defining the model complexity: Vapnik-Chervonenkis (VC) dimension [176],
Rademacher complexity [177], and effective dimension [178]666For an exhaustive
list of complexity measures see [75]. The complexity measures are also
connected to the generalisation error because when the model becomes too
complex for the problem, the generalisation is expected to worsen.

Much work has been done to quantify the complexity and the generalisation
error of quantum neural networks: Ref. [179] explored a generalisation error
bound through the Rademacher complexity that explicitly accounted for the
encoding strategy; and Ref. [180] used the effective dimension - a measure
dependent on the sample size - to bound the generalisation error of QNNs as
well as prove their higher expressivity given the same number of trainable
parameters. Other attempts were also made to quantify the complexity (also
referred to as the expressivity) of QNNs in Refs. [181, 182, 183, 184, 185].
Notably, Ref. [186] theoretically proved that the generalisation error of a
QNN grew as $\mathcal{O}(\sqrt{T/N})$ where $T$ was the number of parametrised
quantum gates in the QNN, and $N$ was the number of data samples in the
dataset. The latter work implies that QML models are better at generalising
from fewer data points.

###  2.4 Hybrid quantum neural networks

Figure 5: An example of a hybrid quantum-classical ML model. In this case, the
inputs are passed into a fully connected classical multi-layered perceptron,
and its outputs are fed into the embedding of a quantum circuit. Depending on
the setting, some measurements of this quantum circuit are taken and then
passed into another fully-connected layer, the output of which can be compared
with the label.

Just as there are classical and quantum models, one could also combine the two
to create hybrid models – see Fig. 5. It is conceivable that in the NISQ era,
one could use the understanding of QML described in Sec. 2.3 to find a regime
where quantum models cover some bases that classical models do not. Ref. [41]
developed a platform for hybrid quantum high-performance computing (HQC) cloud
and it was deployed on the QMware hardware [187, 188]. It showed that for
high-dimensional data, a combination of classical and quantum networks in
conjunction could offer two advantages: computational speed and the quality of
the solution. The data points were, first, fed to a shallow777Apart from
performance considerations, this also provided an effective way to avoid
barren plateaus described in Sec 2.2.5. quantum circuit composed of 4 qubits,
two of which were measured and their corresponding values were passed onto a
neural network. Two classical datasets were chosen to explore the
effectiveness of a hybrid solution and to compare it when the quantum part is
removed, leaving only a classical network: the sci-kit circles’ dataset [189]
and the Boston housing dataset [190]. The former is a synthetic geometrical
dataset that consists of two concentric circles in a 2-dimensional square of
side $x=2\pi$, and the latter is about the distribution of the property value
given its population status and the number of rooms. In both cases, it was
shown that the hybrid network generalises better than the classical and this
difference is most visible at the extremes of problems with very small
training sample sizes. However, this difference became smaller as the number
of samples grew.

In continuation, Ref. [44] suggested a hyperparameter optimisation scheme
aimed at architecture selection of hybrid networks. This work also implemented
a hybrid network for training, but in two new ways: 1) using a real-world,
image recognition dataset [191], and 2) the quantum part of the hybrid network
was inserted in the middle of the classical implementation. Additionally, the
architecture of the quantum part was a subject of hyper-parameter
optimisation, namely the number of qubits used and the number of repetition
layers included were optimised. Training this network showed that the hybrid
network was able to achieve better quality solutions albeit by a small margin.
It is notable that because of this architecture optimisation, a highly
improved quantum circuit was achieved. The performance of this circuit was
theoretically measured by applying analysis methods such as ZX reducibility
[152], Fourier analysis [107], Fisher information [192, 193, 108], and the
effective dimension [178, 180].

###  2.5 Applications and realisations

QML automatically inherits all classical ML problems and implementations, as
it is simply a different model to apply to data science challenges. In
addition to this inheritance, QML research has also provided novel, quantum-
native solutions. In both cases, QML has so far been unable to provide a
definite, practical advantage over classical alternatives, and all the
suggested advantages are purely theoretical.

####  2.5.1 Solving classical problems

QML is employed in many classical applications. Some notable contributions are
in sciences [194, 195, 196, 197, 198, 199, 200], in finance [201, 202, 203,
42], pharmaceutical [43, 45], and automotive industries [44]. In many cases,
these models replaced a previously-known classical setting [204, 205, 206,
207]. Quantum generative adversarial networks were suggested in Ref. [208] and
followed by Refs. [209, 210, 211, 212, 213, 214, 215, 216]. Similarly, quantum
recurrent neural networks were investigated in [217, 218] and two approaches
to image recognition were proposed in Refs. [34, 219]. Ref. [220] looked at a
classical-style approach to quantum natural language processing. The
applications of QML in reinforcement learning were also explored in Refs.
[221, 222, 223, 38]. Finally, a celebrated application is the quantum auto-
encoder, where quantum data are compressed and then re-constructed from less
information, a notable suggestion was made in Ref. [224].

####  2.5.2 Quantum-native problems

Native problems are novel, quantum-inspired ML problems that are specifically
designed to be solved by a QML algorithm.

Perhaps the most known QML algorithm is the variational quantum eigensolver
(VQE). The problem formulation is that the input data is a Hamiltonian and we
are required to find its ground state and ground-state energy. The VQE
solution consists of preparing a PQC of trainable parameters and taking the
expectation value of the Hamiltonian. This yields the energy expectation of
the prepared state, and the idea is that by minimising this expectation value,
we can achieve the ground-state energy, at which point the prepared state will
represent the ground state of the problem. This was first implemented to find
the ground-state energy of $\text{He}-\text{H}^{+}$ [30] and was then
substantially extended in Ref. [59]. VQE remains one of the most promising
areas of QML.

Ref. [25] showed that PQCs can be used to solve a linear system of equations
(LSE). They proposed a commonly known as the Harrow, Hassidim, and Lloyd
(HHL). Refs. [225, 226, 227, 228, 229, 230, 231, 232] improved this algorithm
and Ref. [233] extended it to also include non-linear differential equations.
Ref.[102] showed that it is possible to use a quantum feature map to solve
non-linear differential equations on QNNs. This is also an exciting and
promising area of QML.

An important QML formulation is known as the quadratic unconstrained binary
optimisation (QUBO) [234, 235]. This is generally a classical problem, but
using the Ising model - see [236] \- this can be solved on a quantum computer
[237, 238]. A common demonstration of the latter is the max cut problem [239]
\- see Fig 6. There are solutions for the QUBO problem on both gate-based
quantum computers and quantum annealers [240, 241, 242], and this general
concept has seen use in many sectors [243].

Lastly, another quantum-native formulation is in natural language processing.
Ref. [94] developed a platform for turning grammatical sentences into quantum
categories using J. Lambek’s work [244]. Refs. [245, 246, 247] tested this
algorithm on quantum hardware, and later a full QNLP package was developed
[248]. The initial value proposition of QNLP in this way is that this
algorithm is natively grammar-aware, but given that large classical language
models are shown to infer grammar [80], the real advantage of this approach
could lie in other avenues, such as a potential Grover-style [249] speed-up in
text classification.

Figure 6: The max cut problem. The abstract manifestation of this problem is a
general graph, and we are interested in finding a partition of its vertices
such that the number of edges connecting the resultant graph to the
complimentary graph is maximal.

###  2.6 Open questions

####  2.6.1 Quantum advantage

Despite the theoretical findings in Sec 2.3.3, there is limited demonstrable
success in using QML in real-life problems, and this is not purely due to
hardware shortcomings. Ref. [250] showed that there exists a class of datasets
that could showcase quantum advantage, and Ref. [159] found a mathematical
formulation for where we can expect to find such an advantage. In Refs. [251,
252, 253] attempts were also made to devise a set of rules for potential
quantum advantage. However, Ref. [254] argued that a shift of perspective from
quantum advantage to alternative research questions could unlock a better
understanding of QML. The suggested research questions were: finding an
efficient building block for QML, finding bridges between statistical learning
theory and quantum computing, and making QML software ready for scalable ML.

####  2.6.2 Optimal parametrisation

In Sec 2.2.2, we encountered various QNN parametrisations with specific
properties. An open question is how to optimally parametrise a circuit to
avoid barren plateaus, be as expressive as possible, and be free of
redundancy. A potential characteristic of such parametrisation is a high level
of Fourier accessibility as mentioned in Sec 2.2.4, potentially requiring a
quantifiable measure of this accessibility.

####  2.6.3 Theory for hybrid models

Despite the successes outlined in Sec 2.4, the theoretical grounding for such
models is limited. We saw that hybrid networks performed well if the quantum
section was introduced at the beginning of the model architecture [41] or in
the middle [44]. From an information-theoretic perspective, this needs to be
investigated in more detail to shed light on the effect of hybridisation. Such
investigation could identify if there exist areas where the application of a
quantum part could complement a classical circuit by either introducing an
information bottle-neck to prevent over-fitting or by creating high-
dimensional models.

####  2.6.4 An efficient optimisation method

As mentioned in Sec. 2.2.3, the current gradient calculation methods are
either only available on simulators or require a linearly-growing number of
circuit evaluations, neither of which can accommodate a billion-parameter,
million-qubit setting. This poses a barrier to the future of QML, and thus an
efficient optimisation method is needed for the long term.

##  3 Quantum technologies enhanced by machine learning

Figure 7: Machine learning helps in solving problems in fundamental and
applied quantum physics. Sections that discuss a particular problem are
labeled.

In this section an impact that machine learning makes in quantum technologies
is discussed. The outline of the topics is given in Fig. 7. Today machine
learning is used to realize algorithms and protocols in quantum devices, by
autonomously learning how to control [35, 255, 256, 257, 258], error-correct
[259, 36, 260, 261], and measure quantum devices [262]. Given experimental
data, ML can reconstruct quantum states of physical systems [263, 264, 265,
266], learn compact representations of these states [267, 268], and validate
the experiment [269]. In this section we discuss the impact of machine
learning on fundamental and applied physics, and give specific examples from
quantum computing and quantum communication.

###  3.1 Machine learning in fundamental and applied quantum physics

Since its full development in the mid-1920s, a century later quantum mechanics
is still considered as the most powerful theory, modeling a wide range of
physical phenomena from subatomic to cosmological scales with the most precise
accuracy. Even though the measurement problem and quantum gravity had led many
physicists to conclude that quantum mechanics cannot be a complete theory, the
spooky action of entanglement in the Einstein-Podolsky-Rosen pair [270], has
provided the resources for quantum information processing tasks. With machine
learning, one may be able to model different physical systems (e.g., quantum,
statistical, gravitational) using artificial neural networks, which might lead
to the development of a new framework for fundamental physics.

Even without a precise description of a physical apparatus and solely based on
measurement data, one can prove the quantumness of some observed correlations
by the device-independent test of Bell nonlocality [271]. In particular, by
using generative algorithms to blend automatically many multilayer perceptrons
(MLPs), a machine learning approach may allow the detection and quantification
of nonlocality as well as its quantum (or postquantum) nature [272, 273, 274].

####  3.1.1 Machine learning in quantum computing

Machine learning has also became an essential element in applied quantum
information science and quantum technologies. ML that was inspired by the
success of automated designs [31], was demonstrated to be capable of designing
new quantum experiments [32].

Quantum experiments represent an essential step towards creating a quantum
computer. More specifically, for example, three-particle photonic quantum
states represent a building block for a photonic quantum computing
architecture. In Ref. [32] ML algorithm used is a reinforcement learning
algorithm based on the projective simulation model [275, 276, 277, 278, 279,
280]. An agent, the reinforcement learning algorithm, puts optical elements on
a (simulated) optical table. Each action adds an optical element to an
existing setup. In case the resulting setup achieves the goal, e.g., creates a
desired multiphoton entangled state, the agent receives a reward. The
described learning scheme is depicted in Fig. 8(a).

The initial photonic setup is an output of a double spontaneous parametric
down-conversion (SPDC) process in two nonlinear crystals. Neglecting these
higher-order terms in the down-conversion, the initial state $\ket{\psi(0)}$
can be written as a tensor product of two orbital angular momentum entangled
photons,

$\ket{\psi_{0}}=\frac{1}{3}\left(\sum_{m=-1}^{1}\ket{m}_{a}\ket{-m}_{b}\right)\otimes\left(\sum_{m=-1}^{1}\ket{m}_{c}\ket{-m}_{d}\right),$
(1)

where the indices $a,b,c$ and $d$ specify four arms in the optical setup. The
actions available to the agent consist of beam splitters (BS), mirrors (Refl),
shift-parametrized holograms (Holo), and Dove prisms (DP). The final photonic
state $\ket{\psi_{f}}$ is obtained by measuring the arm $a$, and post-
selecting the state in the other arms based on the measurement outcome in $a$.

Figure 8: A reinforcement learning algorithm that designs a quantum
experiment. An experiment on an optical table is shown as an example. (a) The
learning scheme depicts how an agent, the reinforcement learning algorithm,
learns to design quantum experiments. (b) Representation of the reinforcement
learning algorithm, projective simulation, as a two-layered network of clips.

The reinforcement learning algorithm that achieves the experimental designs is
shown in Fig. 8(b). It is the projective simulation agent represented by a
two-layered network of clips. The first layer corresponds to the current state
of the optical setup, whereas the second layer is the layer of actions. The
connectives between layers define the memory of the agent, which changes
during the learning process. The connectivities correspond to the
probabilities of reaching a certain action in a given state of a quantum
optical setup. During the learning process the agent automatically adjusts the
connectivities, and thereby prioterize some actions other the other. As shown
in Ref. [32] this leads to a variety of entangled states of improved
efficiency of their realization.

####  3.1.2 Machine learning in quantum communication

In addition to designing new experiments, ML helps in designing new quantum
algorithms [281] and protocols [282]. Designing new algorithms and protocols
has similarities to experiment design. In particular, similar to experiment
design, every protocol can be broken down into individual actions. In the case
of the quantum communication protocol, these actions are, e.g: apply $T$-gate
to the second qubit, apply $H$-gate to the first qubit, send the third qubit
to the location $B$, and measure the first qubit in the $Z$-basis. Because of
the combinatorial nature of the design problem, the number of possible
protocols grows exponentially with the number of available actions. For that
reason, a bruteforce search of a solution is impossible for an estimated
number of possible states of a quantum communication environment $0.6\times
10^{12}$ [282].

A reinforcement learning approach to quantum protocol design, first proposed
in Ref. [282], is shown to be applicable to a variety of quantum communication
tasks: quantum teleportation, entanglement purification, and a quantum
repeater. The scheme of the learning setting is shown in Fig. 9. The agent
perceives the quantum environment state, and chooses an action based on the
projective simulation deliberation process. The projective simulation network
used in this work is similar to the one in Fig. 8(b), with addition of
hierarchical skill acquisition. This skill is of particular importance in the
long-distance quantum communication setting, which has to include multiple
repeater schemes.

With the help of projective simulation, it was demonstrated that reinforcement
learning can play a helpful assisting role in designing quantum communication
protocols. It is shown that the use of ML in the protocol design is not
limited to rediscovering existing protocols. The agent finds new protocols
that are better than existing protocols in case optimal situations lack
certain symmetries assumed by the known basic protocols.

Figure 9: A reinforcement learning algorithm that designs a long-distance
quantum communication protocol. The algorithm is based on projective
simulation with episodic and compositional memory (ECM). Given the state of
the quantum communication environment, the algorithm chooses how to modify
this state by acting on the environment. The goals of the PS agent are: (a)
teleportation protocol (b) entanglement purification protocol (c) quantum
repeater protocol (d) quantum repeater protocol for long-distance quantum
communication.

###  3.2 Machine learning in random walks problems

Random walks paradigm plays important role in many scientific fields related
to the transfer of charge, energy, or information transport [283, 284, 285,
286, 287]. Random (classical) walks on graphs represent indispensable tool for
many subroutines in computational algorithms [288, 289, 290]. Quantum walks
(QW) represent generalization of classical walks to quantum domain and use
quantum particle instead of classical one [291, 292]. Resulting quantum
interference pattern which governs QW physics fundamentally differs form the
classical one [293]. For quantum information science crucially important that
quantum particle exhibits quantum parallelism which appear as a result of
various paths interference and entanglement. It was shown that quantum
particle propagates quadratically faster than classical one on certain graphs
which are line [294], cycle [295, 296], hypercube [297, 298], and glued trees
graphs [299], respectively. It is expected that algorithms based on QW should
demonstrate quadratic speedup that is $O(\sqrt{N})$. Such parallelism may be
useful for quantum information processing and quantum algorithms purposes
[300, 299, 301]. It is especially important to note that QW are explored in
quantum search algorithms which represent important tools for speedup of QML
algorithms [21, 302, 303, 304].

Noteworthy, QW speedup demonstration with arbitrary graphs represents open
problem [305]. Standard approach would be to simulate quantum and classical
dynamics on given graph, which provides an answer in which case a particle
would arrive a target vertex faster. However, this approach may be difficult
(and costly) to use in computations for the graphs possessing large number of
vertices; the propagation time scales polynomially in the size of the graph.
Second, we usually interested in a set of graphs for which obtained results of
the simulations cannot reveal some general features of quantum advantage.

In a number of works we attacked this problem by means of ML approach [39, 40,
306]. We explore a supervised learning approach to predict a quantum speedup
just by looking at a graph. In particular, we designed a classical-quantum
convolutional neural network (CQCNN) that learns from graph samples to
recognize the quantum speedup of random walks.

The basic concept of CQCNN that we use in Ref. [39, 40, 306] is shown in Fig.
10 and Fig. 11, respectively. In particular, we examined in Ref. [39, 40, 307,
308, 306] continuous-time random walks and suppose that classical random walk
representing stochastic (Markovian) process defined on a connected graph. It
starts at the time $t=0$ from initial node $i$ and hits target vertex $t$.
Unlike the classical case, a quantum particle due to interference phenomenon
will be “smeared” across all vertices of the graphs. Thus, in the quantum case
we propose an additional (sink) vertex $s$ that is connected to the target
vertex and provides localization of the quantum particle due to energy
relaxation from $t$ to $s$ vertices which happen with the rate $\gamma$. In
other words, the quantum particle may be permanently monitored in the sink
vertex. Mathematically graph is characterized by it weighted adjacency matrix
$A$ that is relevant to Hamiltonian $\mathcal{H}=\hbar A$. Notice, for chiral
QW time asymmetry may be obtained by using complex-valued adjacency matrix
elements [306]. We characterize quantum transport by means of
Gorini–Kossakowski–Sudarshan–Lindblad equation that look like (cf. [309]):

Figure 10: A schematic representation of considered in Refs. [39, 40] random
walks on (a) connected random graph, (b) cycle graph. The labels ($i$), and
($t$) specify initial and target vertices, respectively; $s$ is a sink vertex
which is require to localize and detect quantum particle. The $\gamma$ is
coupling parameter between target and sink vertices, respectively.

$\displaystyle\frac{\mathrm{d}\rho(t)}{\mathrm{d}t}=-\frac{i}{\hbar}(1-p)\left[\mathcal{H},\rho(t)\right]+p\sum_{mk}\left(L_{mk}\rho(t)L_{mk}^{\dagger}-\frac{1}{2}\left\\{L_{mk}^{\dagger}L_{mk},\rho(t)\right\\}\right)$
$\displaystyle+\gamma\left(L_{s}\rho(t)L_{s}^{\dagger}-\frac{1}{2}\left\\{L_{s}^{\dagger}L_{s},\rho(t)\right\\}\right),$
(2)

where $\rho(t)$ is time dependent density operator,
$L_{mk}=T_{mk}\ket{m}\bra{k}$ and $L_{s}=\ket{s}\bra{t}$ operators
characterize transitions from vertices $k$ to $m$ and from $t$ (target) to $s$
(sink), respectively; $\gamma$ is coupling parameter between target and sink
vertices. The parameter $p$ lies $0\leq p\leq 1$ condition and determines the
decoherence; the value $p=0$ is relevant to purely quantum transport, while
$p=1$ determines completely classical random walks.

Solution of Eq. (3.2) specifies quantum probability
$\mathrm{P^{q}}(t)\equiv\rho_{(n+1)(n+1)}(t)$ ($n$ is the total number of
vertices), which is relevant to QW on a chosen graph. The classical random
walk may be established by the probability distribution

$\mathrm{P(t)}=\mathrm{e}^{-It}\mathrm{e}^{Tt}\mathrm{P(0)},$ (3)

where $\mathrm{P}(t)$ is a vector of probabilities $\mathrm{P}_{v}(t)$ of
detecting a classical particle in vertices $v\in\mathcal{V}$ of the graph; $I$
is the identity matrix of size $n\times n$. The transition matrix $T$ is a
matrix of probabilities $T_{vu}$ for a particle to jump from $u$ to $v$. In
this case the sink vertex is not needed and we can assume $\gamma=0$. We are
interested in the probability of finding a particle in the target (or, in the
sink) vertex which is described by solutions of Eqs. (3.2) and (3).

Figure 11: Schematic representation of CQCNN approach which is used for
predicting the quantum speedup on the graphs represented in Fig. 10. (a) -
scheme of the CQCNN architecture. The neural network takes a labeled graph in
form of an adjacency matrix $A$ as an input. The $A$ then processed by
convolutional layers with filters of graph-specific “edge-to-edge” and “edge-
to-vertex”, respectively. These filters act as functions of a weighted total
number of neighboring vertices of each vertex. The convolutional layers are
connected with fully-connected layers which classify the input graph. Data and
error propagation are shown with arrows. (b) and (c) demonstrates processes of
CQCNN training and testing, respectively.

Then, one possible to compare $\mathrm{P^{q}}(t)\equiv\rho_{(n+1)(n+1)}(t)$
and $\mathrm{P^{c}}(t)\equiv\mathrm{P}_{n}(t)$ against $\mathrm{P}_{th}=1/\log
n$ that determines threshold value of probability for a given graph. If this
probability is larger than $\mathrm{P}_{th}$, we can conclude that the
particle occurs at the target. The time at which one of inequalities
$\mathrm{P^{q}}(t)>\mathrm{P}_{th}$, $\mathrm{P^{c}}(t)>\mathrm{P}_{th}$
fulfilled, is called the hitting time for quantum or classical particle,
respectively. Hence, by comparing the solutions to Eqs. (3.2) and (3), we can
define the particle transfer efficiency: it is $1$ if the quantum particle
reached the target first, and $0$ otherwise.

In Fig. 11 we schematically summarize proposed CQCNN approach for detection of
QW speedup. In order, the architecture of CQCNN is shown in Fig.11(a). It
consists of a two-dimensional input layer that takes one graph represented by
an adjacency matrix $A$. This layer is connected to several convolutional
layers, the number of which depends on the number of vertices $n$ of the input
graph. The number of layers is the same for all graph sizes. CQCNN has a
layout with convolutional and fully connected layers, and two output neurons
that specify two possible output classes. The convolutional layers are used to
extract features from graphs, and decrease the dimensionality of the input.

Empirically we find out that relevant features are in the rows and columns of
adjacency matrices matrices. The first convolutional layer comprises six
filters (or, feature detectors) which define three different ways of
processing the input graph. These three ways are marked by green, red, blue
colors in Fig. 11(a), respectively. The constructed filters are form ‘crosses’
which are shown in Fig. 11(a) and capture a weighted sum of column and row
elements. These filters act as functions of a weighted total number of
neighboring vertices of each vertex. Thus, the cross ‘edge-to-edge’ and ‘edge-
to-vertex’ filters crucially important in designed CQCNN; they are capable for
prediction of the quantum advantage by QW.

Fig. 11(b) shows schematically training procedure by using some graphs samples
which are established by adjacency matrices $A$ as an input. CQCNN made
prediction at the output determining classical or quantum classes depending on
the values of two output neurons. The predicted class is determined by means
of the index of a neuron with the largest output value
$\mathrm{class}=\mathrm{argmax}_{m}y(m)$. Having a correct label, the loss
value is computed.

The filters that we constructed in CQCNN play an essential role in the success
of learning. CQCNN learns by stochastic gradient descent algorithm that takes
the cross entropy loss function. The loss on a test example $i$ is defined
relative to the correct class $\mathrm{class}_{i}$ (classical or quantum, $0$
or $1$) of this example:

$\mathrm{loss}_{i}=-\kappa(\mathrm{class}_{i})\log\left(\frac{\mathrm{e}^{x(\mathrm{class}_{i})}}{\mathrm{e}^{x(0)}+\mathrm{e}^{x(1)}}\right),$
(4)

where we defined the values of the output neurons as $x(0)$ and $x(1)$;
$\kappa(\mathrm{class}_{i})$ is the total fraction of examples from this class
in the dataset. As we have shown in Ref. [39] CQCNN constructed a function
that generalizes over seen graphs to unseen graphs, as the classification
accuracy (which may be defined as the fraction of correct predictions) goes
up.

CQCNN testing procedure is not principally different from the training process
how it is seen in Fig. 11(c), cf. Fig. 11(b). The CQCNN does not receive any
feedback on its prediction in this case and the network is not modified.

We apply the described ML approach to different sets of graphs. In particular,
to understand how our approach works in a systematic way, we first analyze the
CQCNN on line graphs with up to $10$ vertices. CQCNN was trained over $2000$
epochs with a single batch of $3$ examples per epoch.

Then, we simulated CQCNN’s learning process for random graphs, each sampled
uniformly from the set of all possible graphs with $n$ vertices and $m$ edges.
The learning performance results in the absence of decoherence ($p=0$) are
shown in Fig. 12 for $n=15,20,25$; the $m$ is chosen uniformly from $n-1$ to
$(n^{2}-n)/2$. Our simulations shows that the loss after training is
vanishing; it is below $3\times 10^{-3}$ for all these random graphs. In Fig.
12(a) we see that both recall and precision are about $90\%$ for the
“classical” part of the set, and is in the range of $25-35\%$ for the
“quantum” one. n888 Recall quantifies the fraction of correct predictions in a
particular class, whereas precision identifies the fraction of a particular
class predictions that turned out to be correct. Thus, one can see that CQCNN
helps to classify random graphs correctly much better than a random guess
without performing any QW dynamics simulations. In Fig. 12(b)-(c) we represent
samples of correctly classified graphs.

Figure 12: (a) CQCNN learning performance. Dataset consists of random graphs
with $n=15,20$ and $25$ vertices, $1000$ examples for each $n$, and the
corresponding classical and quantum labels. CQCNN was simulated during $3000$
epochs, $100$ mini batches each with the batch size of $3$ examples. The
neural network was tested on $1000$ random graphs for each $n$. (b), (c)
establish random graph examples taken from the test set which were correctly
classified by CQCNN (initial and target vertices are marked in yellow and red,
respectively). The classical particle is faster on (b), whereas the quantum
one is faster on graph (c).

####  3.2.1 Quantum walks with decoherence

In the presence of decoherence, i.e. for $p>0$ physical picture is getting
richer. In Fig. 13 we demonstrate results of QW dynamics simulation on cycle
graph consisting of $6$-vertices; the efficiency of transport is measured
between opposite vertices of the graph as it is shown in Fig. 10(b).
Simulations performed for $1000$ randomly sampled values of the decoherence
parameter $p$ and used to train CQCNN. After the training procedure we suggest
CQCNN to predict if the QW can lead to an advantage for a new given parameter
$p$. In Fig. 13 we represent the results of the transfer efficiency
predictions as a violet line. From Fig. 13 clearly seen that at the value of
decoherence parameter $p\simeq 0.34$ abrupt crossover from quantum ($p\simeq
1$) to classical ($p\simeq 0$) regime transport occurs. Thus, one possible to
expect QW advantage in transport in domain of $p<0.34$. Physically, such an
crossover may be relevant to quantum tunneling features in the presence of
dissipation, cf. [310, 311]. Notice that the parameter $p$ is temperature-
dependent in general, cf. [312]. In this case we can recognize the established
crossover as a (second-order) phase transition from quantum to classical
(thermal activation) regime that happens for a graph at some finite
temperature.

The Fig. 13 demonstrates predictions of CQCNN which are based on the learned
values of the output neurons; they are shown in Fig. 13 as a classical (blue)
and quantum (green) classes, respectively. CQCNN made decision about the class
by using the maximum value of the output neurons activation. From Fig. 13 it
is clearly seen that the “vote” for the quantum class grows up to the maximum
value $p\simeq 0.2$, which corresponds to the highest confidence for the
quantum class. Simultaneously, the confidence in the classical class grows
with increasing of decoherence parameter $p$. Separation between classes
become more evident after the crossover point of $p\simeq 0.34$.

Figure 13: Prediction of transfer efficiency (violet curve) for a $6$-cycle
graph versus decoherence parameter $p$. The activation values of output
neurons are shown in blue and green. The results obtained by averaging of $5$
CQCNN networks. Standard deviations are marked by shaded regions.

Thus, obtained results play significant role in creation of soft- and hardware
systems which are based on the graph approach at their basis. CQCNN that we
proposed here allows to find out which graphs, and under which conditions on
decoherence, can provide a quantum advantage. This is especially relevant to
NISQ era quantum devises development.

###  3.3 Machine learning in quantum tomography

With the capability to find the best fit to arbitrarily complicated data
patterns with a limited number of parameters available, machine learning has
provided a powerful approach for quantum tomography. Here, quantum tomography
or quantum state tomography (QST) refers to the reconstruction about a quantum
state with its comprehensive information by using measurements on an ensemble
of identical quantum states [313, 314, 315, 316, 317, 318, 319]. However, the
exponential growth in bases for a Hilbert space of $N$-qubit states implies
that exact tomography techniques require exponential measurements and/or
calculations. In order to leverage the full power of quantum states and
related quantum processes, a well characterization and validation of large
quantum systems is demanded and remains important challenge [320].

Traditionally, by estimating the closest probability distribution to the data
for any arbitrary quantum states, the maximum likelihood estimation (MLE)
method is used in quantum tomography [321, 322]. However, MLE method requires
exponential-in-$N$ amount of data as well as an exponential-in-$N$ time for
processing. Albeit dealing with Gaussian quantum states, unavoidable coupling
from the noisy environment makes a precise characterization on the quantum
features in a large Hilbert space almost unattackable. Moreover, MLE also
suffers from the over-fitting problem when the number of bases grows. To make
QST more accessible, several alternative algorithms are proposed by assuming
some physical restrictions imposed upon the state in question, such as the
adaptive quantum tomography [323], permutationally invariant tomography [324],
quantum compressed sensing [325, 326, 327, 328], tensor networks [329, 330],
generative models [331], feed-forward neural networks [332], and variational
autoencoders [333].

Figure 14: Schematic of machine learning enhanced quantum state tomography
with convolutional neural network (CNN). Here, the noisy data of quadrature
sequence obtained by quantum homodyne tomography in a single-scan are fed to
the convolutional layers, with the shortcut and average pooling in the
architecture. Then, after flattening and normalization, the predicted matrices
are inverted to reconstruct the density matrices in truncation.

To reduce the over-fitting problem in MLE, the restricted Boltzmann machine
(RBM) [334, 335, 336, 337] has provided a powerful solution in QST. With the
help of two layers of stochastic binary units, a visible layer and a hidden
layer, the RBM acts as a universal function approximator. For qubits on an IBM
Q quantum computer, quantum state reconstructions via ML were demonstrated
with four qubits [338]. For continuous variables, the convolutional neural
network (CNN) has been experimentally implemented with the quantum homodyne
tomography for continuous variables [339, 265, 340].

As illustrated in Fig. 14, the time sequence data obtained in the optical
homodyne measurements share the similarity to the voice (sound) pattern
recognition [341, 342]. Here, the noisy data of quadrature sequence are fed
into a CNN, composited with $30$ convolutional layers in total. In applying
CNN, we take the advantage of good generalizability to extract the resulting
density matrix from the time series data [341]. In our deep CNN, there are
four convolution blocks used, each containing $1$ to $9$ convolution layers
(filters) in different sizes. Five shortcuts are also introduced among the
convolution blocks, in order to tackle the gradient vanishing problem. Instead
of max-pooling, average pooling is applied to produce higher fidelity results,
as all the tomography data should be equally weighted. Finally, after
flattening two fully connected layers and normalization, the predicted
matrices are inverted to reconstruct the density matrices in truncation.

Here, the loss function we want to minimize is the mean squared error (MSE);
while the optimizer used for training is Adam. We take the batch size as $32$
in the training process. By this setting, the network is trained with $70$
epochs to decrease the loss (MSE) up to $5\times 10^{-6}$. Practically,
instead of an infinite sum on the photon number basis, we keep the sum in the
probability up to $0.9999$ by truncating the photon number. Here, the
resulting density matrix is represented in photon number basis, which is
truncated to $35\times 35$ by considering the maximum anti-squeezing level up
to $20$ dB.

As to avoid non-physical states, we impose the positive semi-definite
constraint into the predicted density matrix. Here, an auxiliary (lower
triangular) matrix is introduced before generating the predicted factorized
density matrix through the Cholesky decomposition. During the training
process, the normalization also ensures the trace of the output density matrix
is kept as $1$. More than a million data sets are fed into our CNN machine
with a variety of squeezed ($\rho^{sq}$), squeezed thermal ($\rho^{sq}_{th}$),
and thermal states ($\rho_{th}$) in different squeezing levels, quadrature
angles, and reservoir temperatures, i.e.,

$\displaystyle\rho^{sq}=\hat{S}\rho_{0}\hat{S}^{\dagger};$ (5)
$\displaystyle\rho^{sq}_{th}=\hat{S}\rho_{th}\hat{S}^{\dagger}.$ (6)

Here, $\rho_{0}=|0\rangle\langle 0|$ with the vacuum state $|0\rangle$,
$\rho_{th}=\sum_{n}P(n)\,|n\rangle\langle n|$ with the probability
distribution function
$P(n)=\frac{1}{\bar{n}+1}(\frac{\bar{n}}{\bar{n}+1})^{n}$, defined with the
mean-photon number $\bar{n}=\frac{1}{\text{exp}[\hbar\omega/k_{B}T]-1}$ at a
fitting temperature $T$, and
$\hat{S}(\xi)=\text{exp}[\frac{1}{2}\xi^{\ast}\hat{a}^{2}-\frac{1}{2}\xi\hat{a}^{{\dagger}2}]$
denotes the squeezing operator, with the squeezing parameter $\xi\equiv
r\,\text{exp}(i\phi)$ characterized by the squeezing factor $r$ and the
squeezing angle $\phi$. All the training is carried out with the Python
package tensorflow.keras performed in GPU (Nvidia Titan RTX).

The validation of ML-enhanced QST is verified with simulated data set, through
the average fidelity obtained by MLE and CNN by calculating the purity of
quantum state, i.e., $\text{purity}\equiv\text{tr}(\rho^{2})$. Compared with
the time-consuming MLE method, ML-enhanced QST keeps the fidelity up to $0.99$
even taking $20$ dB anti-squeezing level into consideration. With prior
knowledge in squeezed states, such a supervised CNN machine can be trained in
a very short time (typically in less than one hour), enabling us to build a
specific machine learning for certain kinds of problem. When well-trained, an
average time of about $38.1$ milliseconds (by averaging $100$ times) costs in
a standard GPU server. One unique advantage of ML-enhanced QST is that we can
precisely identify the pure squeezed and noisy parts in extracting the
degradation information. By directly applying the singular value decomposition
to the predicted density matrix, i.e.,
$\rho=\sigma_{1}\,\rho^{sq}+c_{1}\,\rho^{sq}_{th}+d_{1}\,\rho_{th}$, all the
weighting ratio about the ideal (pure) squeezed state, the squeezed thermal
state, and thermal state can be obtained. With this identification, one should
be able to suppress and/or control the degradation at higher squeezing levels,
which should be immediately applied to the applications for gravitational wave
detectors and quantum photonic computing.

Toward a real-time QST to give physical descriptions of every feature observed
in the quantum noise, a characteristic model to directly predict physical
parameters in such a CNN configuration is also demonstrated [343]. Without
dealing with a density matrix in a higher dimensional Hilbert space, the
predicted physical parameters obtained by the characteristic model are as good
as those generated by a reconstruction model. One of the most promising
advantages for ML in QST is that only fewer measurement settings are needed
[344]. Even with incomplete projective measurements, the resilience of ML-
based quantum state estimation techniques was demonstrated from partial
measurement results [345]. Furthermore, such a high-performance, lightweight,
and easy-to-install supervised characteristic model-based ML-QST can be easily
installed on edge devices such as FPGA as an in-line diagnostic toolbox for
all possible applications with squeezed states.

In addition to the squeezed states illustrated here, similar machine learning
concepts can be readily applied to a specific family of continuous variables,
such as non-Gaussian states. Of course, different learning (adaptation)
processes should be applied in dealing with single-photon states,
Schrödinger’s cat states [346, 347], and Gottesman-Kitaev-Preskill states for
quantum error code corrections [348]. Alternatively, it is possible to use
less training data with a better kernel developed in machine learning, such as
the reinforce learning, generative adversarial network, and the deep
reinforcement learning used in the optimization problems [264, 349, 350, 351,
352, 353, 354, 355]. Even without any prior information, informational
completeness certification net (ICCNet), along with a fidelity prediction net
(FidNet), have also been carried out to uniquely reconstruct any given quantum
state [356].

Applications of these data-driven learning and/or adaptation ML are not
limited to quantum state tomography. Identification and estimation of quantum
process tomography, Hamiltonian tomography, and quantum channel tomography, as
well as quantum phase estimation, are also in progresses [357, 358, 359, 360].
Moreover, ML in quantum tomography can be used for the quantum state
preparation, for the general single-preparation quantum information processing
(SIPQIP) framework [361].

###  3.4 Photonic quantum computing

In addition to classical information processing, photonic quantum computing is
also one of the possible technologies to demonstrate quantum advantage [14,
362, 363], i.e., in which a quantum system has been shown to outperform a
classical one on some well-defined information processing task. Even though
the computational task on the implementation of photonic Boson sampling is
non-universal [364], meaning that it cannot perform arbitrary quantum
operations, whether any useful applications exist within the heavily
restricted space of non-universal photonic systems is an open question.
Nevertheless, the advantages of photonics as a quantum technologies platform
compared to other platforms are a high degree of integration with mature
classical photonic technologies, and the fact that the photonic circuits
involved can be operated at room temperature [365, 366].

Reviews on the recent advances of machine learning, in particular deep
learning, for the photonic structure design and optical data analysis, as well
as the challenges and perspectives, can be found in Refs. [48, 367, 368, 369].
Inverse designs and optimization on the photonic crystals, plasmonic nano-
structures, programmable meta-materials, and meta-surfaces have been actively
explored for high-speed optical communication and computing, ultrasensitive
biochemical detection, efficient solar energy harvesting, and super-resolution
imaging [370]. By utilizing adaptive linear optics, quantum machine learning
is proposed for performing quantum variational classification and quantum
kernel estimation [371].

Towards the goal of realization of on-chip quantum photonics, silicon-based
materials have been actively explored due to their compatibility with
conventional CMOS fabrication processes. Instead of using photons or entangled
photons pairs from optical parametric process, which suffers from the low
probabilities and rare success events in post-selection, quantum optical
microresonator on a chip employs the Kerr nonlinearity has the advantage over
the photonic qubit-based approach [372, 373, 374, 375, 376]. Unlike all gate-
based quantum computing, which are based on single-qubit and entangling gates,
photonic quantum computing does not rely on any physical gates during the
computation, but on the preparation of quantum modes, e.g., cluster states,
with the time-domain and/or frequency-domain multiplexings [377, 378].

Regarding the integrated quantum photonics, as well as the progress of the
hybrid quantum devices, machine-learning methodology has also been widely
applied in order to offer an efficient means to design photonic structures, as
well as to tailor light–matter interaction [379]. For the emerging field of
machine learning quantum photonics, we need not ML for automated design of
quantum optical experiments [380], such as the quantum walk with graph states
illustrated in Sec. 3.2, but also quantum technologies enhanced by machine
learning, such as ML in quantum tomography illustrated in Sec. 3.3 to have ML-
assisted quantum measurements. Along this direction, quantum optical neural
network (QONN) was introduced to leverage advances in integrated quantum
photonics [381]. With the newly developed protocols such as quantum optical
state compression for quantum networking and black-box quantum simulation,
many thousands of optoelectronic components should be integrated in
monolithically integrated circuits. It is expected to see the combination of
quantum measurements, quantum metrology, and optimization of on-chip and free-
space meta-devices for photonic quantum computing as a promising route for
automatization of quantum experiments.

##  4 Conclusions

Figure 15: Quantum machine learning ecosystem for the next decade

Artificial intelligence and machine learning are currently key factors in the
progress of modern society in the world. Now it is quite difficult to find an
area of our life where achievements in the field of artificial intelligence
would not be used. However, this success is largely ensured by the development
of classical information technologies in terms of hardware, which possess
natural limitations. The creation of quantum computers and quantum networks
can bypass these limitations in different fields. Quantum machine learning is
a rapidly developing new field of research at the border of artificial
intelligence, quantum information science and quantum technology.

The ecosystem of QML, which has developed to date and will be in demand for
the next decade, is depicted in Fig.15. Here we proceed from the fact that in
the near future, the main role in our daily life will be played by various
(tensor) network systems of transmission, processing, and intelligent
recognition of large amounts of information. In this regard, we are confident
that practically significant quantum computers will be embedded into large
distributed intelligent systems environment that will surround us everywhere.

In this review, we outlined the hot topics of interplay between promising
artificial intelligence methods and modern quantum computing. These topics are
predominantly associated with the limited capabilities of NISQ era quantum
computers and use variety of variational algorithms such as variational
eigensolver and quantum approximate optimisation algorithm, respectively. A
special place in our review is given to so-called quantum neural networks
which represent new QML models whose parameters are updated classically and
may be used within quantum-classical training algorithms for variational
circuits. In this sense we discussed promising hybrid information processing
methods that use both classical ML algorithms and quantum devices. The
training procedure proposes data providing to the quantum model, calculating
an objective (loss) value, and then adapting the QNN parameters. Thus, the
whole procedure represent hybrid quantum-classical algorithm. In this work we
discuss another possible application for hybrid computation, that may use
classical-quantum convolutional neural network designed for resolving speedup
of random walks on chosen graphs. Our approach is based on training CQCNN,
which learns to extract feature vectors from graph adjacency matrices combined
with a decoherence parameter. We have shown that even without any decoherence
in generally the speedup of random walks essentially depends on topological
properties of the graph, i.e. on adjacency matrix peculiarities. Our findings
open new perspectives in quantum-classical algorithms which explore random
walks as subroutines.

##  5 Acknowledgements

A.P.A acknowledges support from the Goszadanie No. 2019-1339 project of
Ministry of Science and Higher Education of Russian Federation. R.-K.L. is
partially supported by the National Science and Technology Council of Taiwan
(No. 110-2123-M-007-002).

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