Quantum Phase Estimation (QPE) technique for GSP and GSEE have been explained,
some of these algorithms will be briefly mentioned. Alternatively, the
Variational Quantum Eigensolver (VQE) algorithm will also be reviewed because
it is an alternative, although less efficient, method to the QPE-based
algorithms.

The concept of variational algorithms for QML problems arises from the
creation of a variational algorithm applied to quantum chemistry, the VQE. The
Variational Quantum Eigensolver (VQE) is built upon the concept that it is
possible ton prepare the ground state by variationally determining a state
that minimizes the ground state energy.

In the variational algorithms scheme, a quantum ansatz and a classical
optimizer are needed. In VQE, the classical optimizer is still a gradient
descent, and the original quantum ansatz was Unitary Coupled-Cluster. Later,
other ansatz derived from the original such as Qubit Coupled-Cluster was
chosen. In VQE, as in all variational algorithms, the selection of the ansatz
is key for the algorithm to find a solution close to the optimum. Otherwise,
the barren plateaus phenomenon appears and taking a step requires an
exponential number of intermediate operations.

The Quantum Phase Estimation (QPE) technique allows to know the ground state
of a quantum system and its energy with greater precision than other classical
and quantum techniques. However, it is very expensive to implement. Some
methods that implement it and that have been analyzed in the TFermion software
are: Sparsity low-rank [127], Linear T [128] or Taylor naive [129].

####  8.3 TFermion library

As seen above, quantum algorithms applied to chemistry offer great advantages
over classical algorithms. In fact, the only reason why they are not currently
used is because there are no quantum devices large enough in number of gates
to be executed. Therefore, it is interesting to know what is the real cost in
gates of the execution of certain algorithms and to understand if this cost
can be reduced by introducing certain optimizations.

In order to obtain detailed data from the analysis of the main state-of-the-
art quantum algorithms of GSP and GSEE, a classical software tool has been
created following the open source scheme called TFermion. This tool contains
the implementation of the T-gate cost of the analyzed algorithms. In this way,
the user enters a molecule to be analyzed and the method among those
available, and the software returns the cost in T gates.

This gate cost is calculated according to the formulas detailed in the
subsection 8.1 and an optimized margin of error according to parameters that
can be set by the user in the configuration file. In addition, there are other
parameters that can be adjusted, such as the basis functions, the conversion
between plane and Gaussian waves, the chemical accuracy, etc.

The effort in quantum computing is divided between an attempt to create
algorithms with practical quantum advantage and the ability to run those
algorithms on quantum hardware. That leads to lines of research that are more
focused on running the algorithms even though there is noise on NISQ-type
devices and trying to improve those algorithms. On the other hand, there are
lines more focused on developing error correction and fault-tolerant
computers. This dichotomy provokes the discussion of how far it is to run
certain quantum algorithms on fault-tolerant computers or, seen in another
way, what resources a quantum device must have to be suitable for running
quantum algorithms.

In that vein, TFermion focuses on making a reliable estimate of how many
quantum bridges are needed to run the quantum algorithms that are best
positioned to achieve practical quantum advantage, the quantum chemistry
algorithms. Software tools such as TFermion are needed to size resource needs,
compare lines of research and make technology estimates. Right now, in quantum
computing, the quality of the solution, in this case the accuracy with which
it simulates a quantum chemistry system, is almost as important as the
resources used for it. A high-precision solution may be so expensive to run
that it makes no sense to continue its development, or even if it is
inexpensive with simple molecules, its scalability is abysmal. All of these
metrics can be applied using TFermion.

TFermion can be useful when developing new algorithms and applying them to
specific problems. For example, a scientist who wants to implement a quantum
algorithm to obtain the ground state energy of a molecule will be able to
decide which encoding or which mapping to use based on the results of other
algorithms applied to that molecule. Once implemented, he will be able to see
how his method compares with the state-of-the-art and know where he has
advantages and where he needs to improve. Before TFermion, this possibility
did not exist.

In order for TFermion to have continuity and remain valid as research
progresses, it has been developed and published following the open-source
philosophy. Thus, it is publicly available on GitHub. It is a modular software
that is easy to update. Thus, when a scientist develops a new algorithm and
wants it to be in TFermion, he can create an issue and ask for it to be
included or, directly, include it in the code. Then, when the user goes to run
TFermion, he will see that the list of available methods has been increased.

One of the TFermion use cases to be analyzed in the next chapter 9 is the
application of the library to the development of electric batteries. In this
case, TFermion allows for a detailed analysis of the cost of the algorithms
for a specific problem, of great complexity and with numerous industrial
applications. As before, a specific algorithm for the development of a system
with quantum properties is analyzed.

In order to follow a fully modular philosophy and to facilitate external
collaborations on the TFermion code, each implemented method has been split
into different python files. In this way, if a scientist wants to include his
own algorithm, he only has to create a concrete class that, respecting a basic
structure, calculates the count of T gates of his algorithm. To do this, he
must analyze the cost of each of the steps required for its implementation.
The extraction of the system configuration to be analyzed, the error
optimization, and other technical aspects are automatically performed by
TFermion. Once the code is implemented, an administrator is asked to verify
it, and, if approved, the new quantum algorithm is incorporated into TFermion.

The gate count for TFermion must be expressed in T gates. T-gates are single-
qubit gates that, in combination with other gates, can do what is called
universal quantum computation. Therefore, a given result in T gates is as
general as possible and serves as an upper bound and can be optimized by using
other gates. To understand this, it is necessary to explain some concepts
beforehand.

DiVincenzo’s criteria [130] assert that for a quantum computer to be scalable,
it must demonstrate the ability to execute a universal set of quantum gates.
This comprehensive set should encompass all the gates essential for conducting
any quantum computation, ensuring that any computation can be expressed as a
finite sequence of these universal gates. Fundamentally, a quantum computer,
as per these criteria, should not only manipulate individual qubits through
single-qubit gates, but also introduce entanglement in the system, a
functionality achieved through multi-qubit gates.

Following this definition, two sets of gates can be distinguished, Clifford
and non-Clifford. The Clifford gate set consists of Hadamard, phase S, and
CNOT gates. This gate set is considered minimal because excluding any one gate
renders the inability to implement certain Clifford operations. Specifically,
omitting the Hadamard gate eliminates the capability to represent powers of
$\frac{1}{\sqrt{2}}$ in the unitary matrix, excluding the phase gate disallows
the presence of $i$ in the unitary matrix, and removing the CNOT gate
restricts the controlled operations. Given that all Pauli matrices can be
constructed from the phase and Hadamard gates, each Pauli gate is inherently
an element of the Clifford group.

In addition, a quantum circuit consisting only of gates of the Clifford
ensemble can be simulated efficiently. Therefore, to achieve quantum advantage
it is necessary to use gates of the non-Clifford set, such as the T-gate.

Therefore, measuring the complexity of running an algorithm on T-gates better
represents the bottleneck that exists when running it on quantum hardware.
Clifford type gates are more dependent on the topology of the chip and
therefore a more variable result. Another important point is that the real
limitation of today’s quantum computers is not in the number of qubits they
have, but in the number of gates, circuit depth, that they can run in quantum
coherence and without noise effects.

The gate count performed by TFermion is exclusively adjusted to the operations
performed by the quantum algorithm to calculate the ground state energy,
leaving aside auxiliary operations such as ground state preparation or the
execution by some algorithms of quantum Fourier transform techniques.

Each of the algorithms analyzed in TFermion has a set of parameters that can
be varied to obtain the final gate cost. The algorithms allow a chemical
accuracy value to be set at which the energy is calculated. In this way,
certain parameters of the algorithms can be optimized so that, although the
result is always within the desired accuracy, the cost of running the
algorithm in gates is minimized. For this purpose, TFermion uses different
optimizers that allow to calculate the minimum gate cost T of the algorithm
when it is executed with a given accuracy.

In order to test the results of TFermion with a molecule that was used as a
standard by other methods and to compare their results, FeMoCo was chosen. It
is a molecule with well-known and widely studied properties. Since other
authors have also analyzed the cost of applying some of the TFermion
algorithms on FeMoCo, this molecule can also be used to validate the TFermion
results. FeMoCo was the molecule used to compare the cost of the quantum
algorithms used in TFermion.

####  8.4 Results

  * ✓

The use of QROM techniques in the plane wave naïve Taylorization method makes
it particularly efficient, and there are indications that employing plane
waves could be more effective than Gaussian for the same Taylorization
techniques in isolated molecules.

  * ✓

TFermion has been created as a software tool to help make decisions about
quantum algorithms for quantum chemistry simulations.

  * ✓

The main state-of-the-art quantum algorithms for GSP and GSEE have been
analyzed, making an exhaustive comparison between them.

  * ✓

A quantitative comparison, in terms of implementation cost, between different
encoding, mapping and representation methods has been established.

  * ✓

A line of hybrid algorithms has been developed following the model of using
classical computation as benchmark and assistance of quantum algorithms.

  * ✓

It has been shown that the most promising Hamiltonian simulation methods are,
on the one hand, to use qubitization, gaussian basis and rank factorization
and, on the other hand, Qubitization or Interaction Picture with a plane wave
basis and an encoding in first quantization.

  * ✓

Algorithm implementation costs have been provided to understand the gate count
requirements of quantum algorithms, which is useful in guiding the development
of fault-tolerant quantum devices.

  * ✓

As shown in figure 3 of 8.4, using TFermion different molecules have been
analyzed to each of the studied methods. Initially, simple molecules such as
$H_{2}$ and $HF$ were analyzed just to test the algorithm. Then, more complex
molecules such as $NH_{4}$, $CO_{2}$ or $NaCl$ were analyzed to show that
TFermion is useful.

See pages - of papers/publication_3.pdf

###  9 Applications to Electric Batteries Design

T

he electric battery industry includes any process related to the manufacture,
marketing, use or repair of a battery. This chapter is especially focused on
manufacturing. A key process in the manufacture of an electric battery is the
selection and design of its materials. For this, simulation of the behavior of
materials is a key process and can be achieved using quantum chemistry
algorithms.

Problems such as the design of stronger, lighter, or cheaper materials, the
chemical reaction to achieve propulsion, or the architecture of new electric
batteries with better properties are concrete examples of the challenges faced
by manufacturing companies. These are problems of enormous complexity in which
the most advanced computational techniques must be used to obtain results. For
this reason, these companies always look to the research field with great
interest in the new techniques it can offer them. These techniques include
high-performance computing, deep learning, and quantum computing.

One of the main questions companies in the electric battery sector, as in
other sectors, are asking themselves is when the technology needed to run
quantum computing algorithms will become available. The interest is twofold,
first because they want to make breakthroughs of their own, second because no
company wants to lag behind its competitors’ advances. At this point, TFermion
can become a fundamental tool because it allows estimating the quantum
hardware resources needed to run high-impact algorithms in battery design
problems.

In many cases, for companies, it is not so important what is created, but
rather when it is created and what is needed to create it. In order to get
ahead of the future era of quantum computing battery design, companies are
interested in sizing the amount of resources that running quantum algorithms
can cost them. This can be used to make strategic decisions in the medium
term, based on whether or not quantum algorithms will exist with practical
advantage, deciding whether to purchase a quantum computer, depending on the
amount of resources needed, it may be more efficient to outsource a quantum
computer to the cloud, etc.

####  9.1 Introduction to electric batteries design

The current situation in electric battery manufacturing is especially changing
with the arrival of many electric devices, as laptops, smartphones, autonomous
vehicles, etc. For example, an electric car is a major paradigm shift, not
only for the car itself, but also for its needs. For example, if the vehicle
fleet is mostly electric, gas stations will see their business dwindle and
will have to convert to charging points. Therefore, any tool that provides
information on the timescale of this change is very useful.

The optimization of lithium-ion battery performance is pivotal for the
advancement of next-generation energy storage systems. Progress in this field
relies not only on the discovery of novel materials but also on the refinement
of simulation methods to accurately characterize key properties of lithium-ion
batteries. The spectrum of properties influencing their performance is diverse
and encompasses mechanical and electrochemical attributes, thermal stability
of the cathode, the electrochemical windows of the electrolyte, formation of
the solid-electrolyte interphase, and ionic mobility, among others. This
chapter is based on publication [1].

Today, the limiting factor in the deployment of electric vehicles is their
range, that is, the number of kilometers they are able to travel without
recharging the battery. Although this measure depends on numerous factors,
such as aerodynamics, vehicle weight, driving style, tires, etc., the factor
with the greatest impact is the capacity of the electric batteries. These
batteries have been increasing their capacity to have autonomies that range
from 400 to 800 km. However, it is expected that this range will be much
greater with the development of technology.

To increase the capacity of an electric battery, it must be possible to
simulate the chemical processes that occur inside the battery. Understanding
the optimum structure of the anode and cathode, how to achieve the best
storage and discharge rate, or how to avoid battery degradation are
fundamental processes for extending the autonomy and life of these batteries.

All these processes have in common the need to use a GSP and GSEE algorithm.
As seen above, the classical algorithms that do these tasks have low accuracy
due to their limited scalability. Therefore, quantum algorithms have become
the most promising solution in the medium term. There are numerous approaches
to accelerate the development of new algorithms, TFermion proposes to study
their cost and compare the techniques that implement each of them to obtain
certain conclusions.

Following this motivation, this paper reports the collaboration with the
start-up Xanadu and the company Volkswagen for the application of TFermion to
the study of quantum algorithms that allow to know and design with detail and
precision electric batteries.

####  9.2 Methodology

The task of simulating all the processes that occur in an electric battery is
too complex, not to mention that there are numerous technologies for creating
electric batteries. In this case, the work is focused on the simulation of a
lithium-ion battery, and the study focused on the cathode.

The unit cell material, Li2FeSiO4, is studied because it is a common candidate
material for the cathode of batteries. Other properties such as the ionic
mobility inside the cathode, and the thermal stability of the material, can
also be predicted from the ground state energy of the different phases of the
material. In summary, this work is focused on the simulation of cathode
materials, which is crucial for predicting important properties of a battery
cell.

The battery cell comprises a positive electrode (cathode) and a negative
electrode (anode), separated by a porous membrane (separator) and immersed in
an ion-conducting material (electrolyte). Chemical reactions occurring at the
electrode-electrolyte interface drive the conversion of chemical energy into
electrical energy. During discharge, an oxidation reaction at the anode
generates electrons and lithium ions. The electrons flow through an external
circuit, while the lithium ions migrate through the electrolyte and
intercalate into the cathode material. Conversely, during charging, an
external voltage is applied to reverse this process. The lithium ions are
extracted from the cathode, transported back through the electrolyte, and
intercalated into the anode material.

In this chapter, three fundamental aspects of electric batteries are
investigated: the equilibrium cell voltage, the ionic mobility, and the
thermal stability of the cathode material. These aspects are critical for
enhancing the capacity of lithium-ion batteries and are instrumental in
studying the electronic structure of cathode materials.

The equilibrium voltage plays a crucial role in defining the energy storage
capacity of a battery relative to its volume (energy density) and weight
(specific energy). It represents the average voltage (V) of an electrochemical
cell, which converts chemical energy into electrical energy. The Nernst
equation governs the equilibrium voltage, taking into account factors such as
the number of charges transferred, the Faraday constant, and the change in
free energy associated with the cell reaction [131].

The chemical diffusivity (D) serves as a key parameter characterizing the
mobility of lithium ions within a material. In scenarios where diffusion
mechanisms remain independent of temperature, a microscopic model can
effectively illustrate the process of lithium ion hopping from its initial
position to an adjacent vacant site within the crystal structure of the host
material.

Numerous processes contribute to the degradation of battery performance over
time, encompassing the formation of the solid electrolyte interphase,
degradation of cathode active materials, lithium plating on the anode, and the
growth of lithium dendrites, among others [132]. Simulating these phenomena
poses a significant challenge, as it requires a bottom-up approach spanning
from the atomic level to the macroscopic scale [133].

Anticipating the thermal stability of cathode materials is crucial for
enhancing the safety of lithium-ion batteries, particularly as they can become
unstable in their charged state. The removal of more lithium ions from oxide-
based cathode materials can lead to their degradation into other phases of the
material [134].

At present, first-principles calculations of the electronic structure of
cathode materials are largely performed using density functional theory (DFT)
methods [135], explained in chapter 8. This chapter is dedicated to exploring
methods suitable for calculating the ground-state energies of cathode
materials, a crucial parameter for simulating battery properties, specifically
quantum phase estimation algorithm. A quantum phase estimation (QPE) algorithm
has been used with simulation of the Hamiltonian by qubitization, plane waves
to represent the quantum state, and first quantization to encode the system.

The quantum algorithm begins by taking the Hamiltonian, which describes the
interactions among electrons within the material’s unit cell, as input. It
then generates an approximation of the smallest eigenvalue of this
Hamiltonian, which corresponds to the ground-state energy. To facilitate this
process, a method for representing and constructing Hamiltonians is necessary,
tailored specifically to the quantum algorithm.

Studying the electronic Hamiltonian is crucial to understand why employing a
first-quantization approach in a plane-wave basis is well-suited for
simulating battery materials [126]. This approach allows for an effective
representation of the electronic structure and interactions within the
material, making it particularly suitable for quantum simulations.

However, preparing an approximate ground state to serve as input for the
quantum phase estimation algorithm presents challenges, especially in the case
of periodic materials and within the context of first quantization. Careful
consideration is required to identify suitable methods for achieving this task
effectively.

Once the ground state is prepared, the qubitization formalism comes into play,
enabling the encoding of the Hamiltonian into a suitable unitary
transformation [23]. This step is essential for implementing the quantum phase
estimation algorithm.

To analyze the overall complexity of the algorithm and ensure compatibility
with fault-tolerant architectures, TFermion is employed. TFermion facilitates
the compilation of all necessary operations into a universal set of quantum
gates, ensuring the algorithm’s feasibility and scalability on quantum
hardware.

Plane waves offer a well-suited framework for studying periodic systems,
providing concise representations of Hamiltonians. However, achieving high
accuracy often demands a significant number of plane waves, resulting in an
impractically large number of qubits in second quantization. To address this
challenge, first-quantization techniques are favored for materials simulation.
This approach leverages qubitization-based quantum phase estimation
algorithms, specifically designed for first-quantized Hamiltonians represented
in a plane-wave basis.

The atomic configuration of a cathode material is characterized by its unit
cell, comprising a collection of atoms that can be spatially translated to
cover the entire crystal. To determine the material’s electronic structure,
the Schrödinger equation is solved within the unit cell, subject to periodic
boundary conditions. While any comprehensive set of basis functions can
represent the first-quantized Hamiltonian $H$, for periodic systems, employing
plane waves with the lattice’s periodicity emerges as a fitting selection.

In the quantum phase estimation algorithm, achieving an input state with
substantial overlap with the actual ground state is crucial. In many quantum
chemistry algorithms, including quantum phase estimation, this is accomplished
by generating a state of non-interacting electrons characterized by single-
particle wave functions, or orbitals, which are optimized using the Hartree-
Fock method.

The situation is more complicated when studying periodic materials in first
quantization using a plane-wave basis. Here it is necessary to apply the
Hartree-Fock method to extended materials and provide an algorithm to prepare
the resulting Hartree-Fock state in a plane-wave basis, which must be
explicitly antisymmetrized.

Studying periodic materials in first quantization using a plane-wave basis
presents a more intricate scenario. In this context, applying the Hartree-Fock
method to extended materials becomes necessary. Moreover, an algorithm is
required to prepare the resulting Hartree-Fock state in a plane-wave basis,
with explicit antisymmetrization.

For the Quantum Phase Estimation (QPE) algorithm is necessary to select a
method to encode the Hamiltonian into an appropriate unitary operator. The
qubitization method stands out due to its ability to achieve exact
implementation without any approximation. Quantum phase estimation, executed
on the qubitization operator, relies on operators of preparation, $PREP_{H}$,
and selection, $SEL_{H}$. The compilation strategy for implementing these
operators is derived from [136].

Unlike the methods analyzed in TFermion, for this particular use case, the
cost of preparing the ground state approach has been taken into account. This
cost has been divided into cost in T-gates, Toffoli gates and an estimate of
the algorithm execution time in quantum hardware.

To carry out this study, it has been necessary to make slight modifications to
the TFermion library. First, the combination of techniques to be used for the
simulation was included in a concrete way. In addition, new performance
metrics, Toffoli gates, and execution time have been included. Finally, the
code has been optimized to include a calculation of the cost of the algorithm
for a molecule of a higher complexity than those analyzed in the TFermion
work.

Figure 31 illustrates the relationship between the overall cost of the
algorithm, quantified by the number of Toffoli gates, and varying parameters
such as the number of plane waves (N) and the error in estimating the ground-
state energy. Given the discrete nature of qubit requirements for representing
the quantum state, which directly influences the Toffoli gate cost (set at
100), it is advisable to select a number of plane waves (N) corresponding to
integer values of np.

The insights from figure 32 shed light on the computational demands of
executing a quantum algorithm, particularly in relation to precision
requirements and clock frequency. At a modest clock rate of 10 kHz, even with
a low precision of np = 4, the estimated runtime extends over several years.
Conversely, with an optimistic but feasible clock rate of 100 MHz, the runtime
for np = 4 is reduced to less than a day. However, for np = 9, the estimated
runtime extends to approximately a year. It may be prudent to limit the number
of plane waves to N = $10^{6}$, corresponding to np = 7, as this adequately
captures the system’s accuracy even in large basis sets like cc-pV5Z. With
this choice and a 100 MHz clock rate, the runtime is projected to be
approximately a few weeks.

Figure 31:  Non-Clifford gate cost for initial state preparation and quantum
phase estimation. (a) The non-Clifford gate cost due to Givens rotations used
in the circuit for initial state preparation. (b) Toffoli gate cost of the
quantum phase estimation algorithm. All calculations are done for the unit
cell of Li2FeSiO4 with 156 electrons. The total number of qubits is 2,375 for
np = 4 and 6,652 for np = 9. In the right figure it only depicts Toffoli gate
count, as the number of T gates is much smaller (¡ 3 × 105). The total error
includes contributions from different approximations throughout the algorithm,
but it does not take into account the error derived from a finite basis set.
The slope of the Toffoli gate cost for fixed target precision is a consequence
of the leading cost term. These calculations were performed with the T-Fermion
library. Figure 32: Estimation of the time required to run the algorithm. This
figure illustrates total runtime for synthesizing all the Toffoli gates
indicated in figure 31. All calculations are done for the unit cell of
Li2FeSiO4 with 156 electrons, and it is assumed that the number of plane waves
used in the state preparation and quantum phase estimation are the same. The
total number of qubits is 2,375 for np = 4 and 6,652 for np = 9. The
distillation time is computed as the product of the number of Toffoli gates,
the surface code distance d, and the clock frequency, all divided by a small
np factor that parallelize the CSWAPs and arithmetic computations. It is
emphasized that these are rough estimates whose main purpose is to provide a
method to interpret the gate cost.

####  9.3 Results

  * ✓

Application of TFermion to a highly complex industrial problem, with results
that help to learn more about the needs of quantum devices.

  * ✓

Extension of the TFermion library to obtain two new metrics, Toffoli gates and
execution time. In addition, the analysis of ground state preparation cost has
been included.

  * ✓

Analysis of how quantum chemistry algorithms should be improved to approach
the capabilities of quantum hardware.

  * ✓

Demonstration that the resources required to run Quantum Phase Estimation
algorithms on quantum hardware and gain practical quantum advantage, while far
from the capabilities of current devices, are in line with medium-term device
growth estimates.

### Conclusions

T

his thesis has focused on the study of algorithms that follow the hybrid
classical-quantum paradigm. As has been explained, this paradigm represents a
middle way that tries to take advantage of the widely studied classical
algorithms and the new possibilities offered by quantum algorithms. Although
it may not be a definitive solution, it seems more logical to move towards a
scenario in which the algorithms are either classical or quantum, it is an
intermediate step that facilitates the execution of quantum algorithms and
possible scenarios in which existing algorithms are accelerated in highly
complex problems.

Hybrid algorithms encompass a heterogeneous group of techniques ranging from
classical algorithms with a quantum module in charge of performing a complex
but bounded task, algorithms that only include a classical preprocessing
module to reduce the volume of data to be encoded in the classical circuit, to
architectures in which the quantum circuit is optimized by a classical
algorithm. There are even other formulations in which a classical algorithm
analyzes a quantum algorithm to optimize its cost and analyze its properties.

The first problems that quantum computing turned its attention to were
simulation problems. Later, it was realized that the best scenario for quantum
simulation with practical applications was quantum chemistry. In the next
generation of quantum computing development and with more algorithms proposed,
optimization problems also began to be thought of as good candidates to be
accelerated by quantum algorithms. Therefore, in this work, these two families
of problems, optimization and quantum chemistry, have been chosen to apply the
developed algorithms. The motivation behind this choice was to create
algorithms that can have a high impact on the industrial sector.

In relation to optimization problems, the study has focused on a specific
subcategory, namely search & sample algorithms. This subcategory identifies
algorithms capable of performing a search in a state space to find the
distribution of the data, or a minimum or a maximum. For this purpose, the
main classical algorithms belonging to this category have been analyzed, and
their limitations have been identified.

Different quantum proposals and their limitations were analyzed. Then, one has
chosen the one that has until now had aroused less interest in the world of
quantum optimization, the quantum walks. Using quantum walks as the central
algorithm, other authors have built on them a quantum version of the
Metropolis-Hastings algorithm. In this work, these versions of the M-H quantum
algorithm were taken as a reference and modified to implement it in a quantum
circuit. This implementation has resulted in a software tool called quantum
Metropolis Solver, QMS, which has been applied to different use cases.

QMS has been applied to the artificial intelligence use case because the
fundamental process of many AI algorithms is search, especially machine
learning algorithms. ML algorithms perform an abstraction and inference
process, in which they require a search process to find the best hypothesis to
explain the input examples. Since performing tests on these algorithms is too
complex due to the limited capabilities of quantum hardware, a reduced problem
that is classically used as a benchmark for these problems, the N-Queen
problem, has been used. The results show that there is a polynomial advantage
of the hybrid algorithm over the purely quantum algorithm.

Another field in which there are numerous problems that can be understood in
the context of optimization and where classical algorithms have strong
scalability limitations is the analysis of space exploration data. In this use
case, this work focused on the analysis of gravitational waves. For this
purpose, QMS was adapted to the gravitational wave parameter estimation
problem, observing quantum advantage over a classical algorithm used by the
LIGO collaboration to obtain its results.

The last use case in which QMS was applied is the protein folding problem. In
this case, there are many processes in nature that can only be simulated by
trying many combinations until the one that best fits the description of the
problem is found. In this case, QMS was applied to protein folding as an
assistant to a classical deep learning algorithm. Results were obtained that
validated the initial intuition that a quantum search algorithm was able to
improve the initial results offered by a deep learning algorithm.

In the quantum chemistry part II, a different methodology for hybrid
algorithms has been presented. In this case, the hybrid algorithm allows a
detailed analysis of the cost of quantum algorithms when executed on a quantum
device.

For this purpose, a software tool, TFermion, has been designed to calculate
the T-gate cost of a quantum algorithm applying Quantum Phase Estimation. The
results have made it possible to understand and compare the cost of state-of-
the-art quantum chemistry algorithms and the evolution required by quantum
devices to run these algorithms.

Finally, the application of TFermion on a real problem with industrial
interest, the design of electric batteries, has allowed to understand the
usefulness of this tool. The results of TFermion in this problem have shown
that the algorithms based on simulation of the Hamiltonian by qubitization,
plane waves to represent the quantum state and first quantization to encode
the system, obtain the best results for the design of the cathode.

This thesis aims to demonstrate that hybrid algorithms offer, at present, many
advantages for the main problems that are candidates for the early fault-
tolerant quantum advantage. In addition, this hybrid paradigm enables the
implementation of the algorithms in current and future devices in the near
term by reducing the size and depth of quantum circuits. Hybrid algorithms do
not seem to be the ultimate solution but they are a small step for algorithm
design but a big step for the quantum computing giant.

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