Laboratoire Charles Fabry, EDF R&D

PhD manuscript

Quantum simulation for strongly interacting fermions with neutral atoms array
: towards the simulation of materials of interest

Antoine Michel

_Supervisors:_ Antoine Browaeys et Christophe Domain
_Co-supervisor:_ Thierry Lahaye

List of acronyms
AQS : Analog Quantum Simulation
AT : Adiabatic Theorem
BP : Barren Plateau
CDMFT : Cluster Dynamical Mean-Field Theory
CMFT : Cluster Mean Field Theory
DFT : Density Functional Theory
DFPT : Density Functional Perturbation Theory
DMFT : Dynamical Mean-Field Theory
DQS : Digital Quantum Simulation
EDF : Electricité de France
GGA : Generalized Gradient Approximation
HF : Hartree Fock
HPC : High Performance Computing
IASCC : Irradiated Assisted Stress Corrosion Cracking
LDA : Local Density Approximation
LTO : Long Term Operator
MF : Mean-Field
MIS : Maximum Independent Set
MOT : Magneto-Optical Trap
NISQ : Noisy Intermediate Scale Quantum
NPP : Nuclear Power Plant
PKA : Primary Knocked-on Atoms
PWR : Pressurized Water Reactor
PQC : Parametrized Quantum Circuit
QA : Quantum Annealing
QAOA : Quantum Approximate Optimization Algorithm
QC : Quantum Computing
QEC : Quantum Error Correction
QFT : Quantum Fourier Transform
QMC : Quantum Monte Carlo
QPU : Quantum Processor Unit
QuAltOA : Quantum Alternating Operator Ansatz
QUBO : Quadratic Unconstrained Binary Optimization
RQP : Rydberg Quantum Processor
SLM : Spatial Light Modulator
SSMF : Slave Spin Mean Field
STO : Slater Type Orbital
VQA : Variational Quantum Algorithm
VQE : Variational Quantum Eigensolver

## Remerciements

C’est peut-être la partie la plus difficile à écrire pour moi, car je sais que
c’est celle qui va être lue en premier, mais toute cette aventure n’aurait été
possible sans la participation de beaucoup de personnes que je souhaite, par
conséquent, remercier ici. Je souhaite tout d’abord remercier mon jury de
thèse, Silke Biermann, Benoît Vermersch, Guido Pupillo, David Clément, Bruno
Senjean et Thomas Ayral. Leur lecture approfondie de mon manuscrit ainsi que
leurs retours et leurs commentaires sur mon travail m’ont permis de mettre en
perspective ma recherche et sont des outils précieux pour la poursuivre.

Cette thèse étant CIFRE, je tiens à remercier la direction du laboratoire
Charles Fabry pour son accueil, et notamment Patrick Georges, mais aussi EDF
R&D et plus particulièrement Stéphane Taunier, Julien Stodolna et Marion Gorce
pour m’avoir donné la chance d’évoluer au sein du groupe T27 du département
MMC. Je tiens à remercier également Valérie Kervargant pour son aide très
précieuse sur l’organisation de mes voyages en France et à l’étranger.

Je souhaite aussi remercier grandement Antoine Browaeys et Thierry Lahaye, mes
deux directeurs de thèse, de m’avoir fait confiance pour ce projet ambitieux
et d’avoir relevé le défi d’encadrer pour la première fois une thèse CIFRE,
mais aussi une thèse théorique. Le mode d’encadrement était plus distendu qu’à
l’accoutumé et la pandémie n’a pas aidé, mais vous avez toujours fait l’effort
de garder le contact et un œil attentif sur mon travail. Vous avez toujours su
être présent dans les moments charnières de la thèse et vos conseils et
remarques ainsi que votre immense culture et expérience de la recherche m’ont
été très utiles pour mener à bien mon travail et me suivront, j’en suis sûr,
jusqu’à la fin de ma carrière. Je tiens aussi à remercier chaleureusement
Christophe Domain, mon co-superviseur, avec qui j’ai passé la plupart de mon
temps durant ces trois ans. Malgré la nouveauté du sujet, tu t’es investi avec
moi pour essayer de comprendre et de relier les enjeux d’EDF sur le
vieillissement des matériaux et la simulation quantique. Ta bienveillance et
ta patience sont des qualités rares, surtout pour un doctorant à qui l’on doit
expliquer encore au bout de trois ans comment fonctionne (et vieillit) une
centrale nucléaire. Ta culture du numérique (je suis toujours friand de tes
anecdotes sur le début du calcul HPC !) et de la physique m’a permis de
prendre énormément de recul sur mon sujet et de comprendre les enjeux du
calcul quantique. Merci de m’avoir introduit de la meilleure des manières à ce
tout nouveau monde pour moi et je suis ravi de continuer avec toi sur ce sujet
passionnant !

Cette thèse n’aurait tout simplement pas vu le jour sans le travail de Marc
Porcheron (que je n’ai pas eu la chance de côtoyer longtemps), Stéphane Tanguy
et Etienne Décossin qui ont lancé le programme quantique à EDF. Merci à vous
d’avoir été aussi visionnaire et de croire en la physique ! Je tiens aussi à
remercier tout particulièrement Joseph Mikael. Tu m’as tout de suite accueilli
à bras ouverts dans le groupe même après un an de COVID et je m’y suis senti
immédiatement dans mon élément. Je ne pense pas qu’il existe beaucoup de
personnes aussi enthousiastes pour le monde du calcul quantique, et l’énergie
que tu déploies pour faire vivre le projet m’impressionne un peu plus chaque
jour. Je souhaite aussi remercier toute l’équipe qui gravite autour du
quantique : Paulin, Pascale, Cyril, Cyril $\times 2$, Quentin, Rodolphe,
Mohamed, Ulysse, Naomi (et j’en oublie certainement), nos échanges et
diversités de compétences initiales furent un moteur pour ma thèse.

Durant toute la durée de ma thèse, j’ai pu compter sur le soutien de PASQAL et
tout particulièrement de Loïc Henriet qui a suivi et participé à mon travail,
merci pour la confiance !

Je souhaite aussi remercier grandement Sebastian Grijalva et Thomas Ayral.
Vous avez officieusement participé activement à l’encadrement de ma thèse et
avez pris énormément de votre temps pour travailler avec un jeune doctorant
que vous ne connaissiez pas et pour cela, je vous en serai éternellement
reconnaissant.

Je tiens aussi à remercier Julien Villmejane pour m’avoir fait confiance en me
confiant des TP et des TD d’électronique en première année de l’IOGS malgré
mes souvenirs qui étaient lointains.

Pendant les trois années de thèse (en fait les deux dernières), j’ai pu
partager mes galères avec Clément, mon coloc’ de bureau, ainsi qu’avec Clément
bis, Lucie, Maxime et je les remercie pour cela. j’ai aussi fait de super
rencontres à l’école des Houches que ce soit Andréas, Marion, Julien (j’espère
que tu bats tout ton labo au ping maintenant !), Alexandre, Gabriel et
d’autres que j’oublie !

En-dehors du monde quantique, le soutien de mes amis a été particulièrement
important et je souhaite tout d’abord remercier Alexis, Sylvain et Anis qui
ont fait le déplacement pour ma soutenance. C’est une amitié qui commence (la
sauce) à bien durer dans le temps et j’espère que cela perdurera jusqu’à nos
80 ans quand on ouvrira une bonne bouteille… Mais j’ai aussi pu compter sur
Laury, le chaleureux Antoine, Ambre, Antho, Thomas, Seddik (qui est le seul à
s’être engagé dans la même galère que moi, merci pour la balade jusqu’au lac
où on ne peut pas se baigner et pour les memes échangés), Alexandre et Amélie.
Je voudrais aussi remercier Kamil pour m’avoir carry sur Ash (contrairement à
Anis et Sylvain).

Parce qu’il est important de vider sa tête des codes Python et des équations,
je souhaite remercier le club de tennis de table de Villemomble et tout
particulièrement Nico, Jérèm, Loris, Lad, Cédric, Vibol, Yves et tout ceux que
j’oublie pour m’avoir pris dans leurs équipes ! Promis les gars, on remonte en
R3 bientôt.

Un grand merci aussi à ma famille (et ma belle-famille !) pour le soutien sans
faille et surtout mes parents.

Enfin, je souhaite finir par toi qui me suis depuis maintenant 7 ans. De ma
déprime d’école d’ingénieur à aujourd’hui, tu ne m’as jamais lâché. Ces trois
années ont été très intenses avec de précieux moments de joie, mais aussi des
moments difficiles. Néanmoins, nous sommes toujours restés soudés et pour rien
au monde, je n’aurais souhaité vivre ces années avec quelqu’un d’autre. Tu es
le dénominateur commun de toutes mes réussites, que tu le veuilles ou non.
J’espère que l’on regardera The Big Bang Theory encore ensemble quand on sera
vieux.

###### Contents

  1. 1 General introduction
    1. 1.1 Industrial material aging context
    2. 1.2 Quantum computing context
    3. 1.3 EDF projects
      1. 1.3.1 Multiscale modelling
      2. 1.3.2 The quantum project
    4. 1.4 Contributions of this work
    5. 1.5 Manuscript organization
  2. 2 Simulating correlated matter: from classical to quantum
    1. 2.1 Forewords
    2. 2.2 Ab initio or first principles methods
      1. 2.2.1 Hartree-Fock method
      2. 2.2.2 Fundamentals of density functional theory
      3. 2.2.3 The Local Density Approximation (LDA)
      4. 2.2.4 The generalized-gradient approximation (GGA)
      5. 2.2.5 The LDA + U method
      6. 2.2.6 Hybrid/metaGGA approach
      7. 2.2.7 Beyond DFT
      8. 2.2.8 Successes and limitations
    3. 2.3 Quantum approaches
      1. 2.3.1 The Hubbard model
      2. 2.3.2 The Mott physics: starting point of highly correlated electrons
      3. 2.3.3 The Dynamical Mean-Field Theory (DMFT)
      4. 2.3.4 Extensions of these methods
      5. 2.3.5 Quantum Monte Carlo
      6. 2.3.6 Conclusion
    4. 2.4 Overview of quantum computing
      1. 2.4.1 Basis
      2. 2.4.2 Many qubits system and entangled states
      3. 2.4.3 Density matrix and mixed states
      4. 2.4.4 Close system and quantum logic operators
      5. 2.4.5 Digital approach
      6. 2.4.6 Analog approach
      7. 2.4.7 Conclusion
    5. 2.5 Methods
      1. 2.5.1 The quantum phase algorithm
      2. 2.5.2 The Variational Quantum Algorithm (VQA)
      3. 2.5.3 Quantum Annealing
    6. 2.6 Quantum computing for chemistry and many-body physics
      1. 2.6.1 From qubits to fermions
      2. 2.6.2 Unitary coupled cluster ansatz
    7. 2.7 Conclusion
  3. 3 Quantum simulation with Rydberg atoms
    1. 3.1 Forewords
    2. 3.2 Neutral atom arrays
      1. 3.2.1 Initial state preparation
      2. 3.2.2 Generating spin-spin interactions
      3. 3.2.3 Register readout
    3. 3.3 Quantum simulation
      1. 3.3.1 Digital quantum simulation
      2. 3.3.2 Analog quantum simulation
    4. 3.4 Conclusion
  4. 4 Digital-analog variational quantum eigensolver for chemistry
    1. 4.1 Forewords
    2. 4.2 Introduction
    3. 4.3 Analog Variational Quantum Eigensolver with Rydberg atoms
      1. 4.3.1 Hamiltonians from Quantum Computational Chemistry
      2. 4.3.2 Rydberg Atom Quantum Processor
      3. 4.3.3 Variational Algorithms on a Rydberg atoms device
    4. 4.4 Description of the Protocols
      1. 4.4.1 UCC ansatz on an analog quantum processor: application on H2
      2. 4.4.2 Alternating pulses
      3. 4.4.3 Optimized Register and Iteratively Parameterized Pulses
    5. 4.5 Numerical Results
      1. 4.5.1 Application on LiH and BeH2 molecules
      2. 4.5.2 Roadmap for more complex molecules
    6. 4.6 Discussion
  5. 5 Using Rydberg platform to simulate strong fermionic correlations in the 2D-Hubbard model
    1. 5.1 Forewords
    2. 5.2 Introduction
    3. 5.3 Problem reduction through the slave-spin method.
      1. 5.3.1 Details of the main equations
      2. 5.3.2 Fulfillment of the constraint
      3. 5.3.3 Variants: Towards a multiorbital case
    4. 5.4 Solution of the two coupled subproblems
      1. 5.4.1 Solving the fermionic Hamiltonian $H_{\mathrm{f}}$ for $J$: Bogoliubov method
      2. 5.4.2 Solving the spin Hamiltonian via a cluster mean-field approach
      3. 5.4.3 Convergence of the self-consistent loop
    5. 5.5 Quantum algorithm for the spin Hamiltonian.
    6. 5.6 Solving the spin model with a Rydberg platform: details
      1. 5.6.1 Optimization of the geometry
      2. 5.6.2 Details of the annealing schedule
      3. 5.6.3 Experimental imperfections
    7. 5.7 Results at equilibrium
    8. 5.8 Dynamics of the Hubbard model with the slave-spin method
      1. 5.8.1 Dynamics in slave-spin theory
      2. 5.8.2 Frequency dependency on eigenenergies
      3. 5.8.3 Dynamics and constraint fulfillment
      4. 5.8.4 Results out of equilibrium
    9. 5.9 Conclusion
  6. 6 Summary and outlook
  7. A LiH and BeH${}_{\text{2}}$ Hamiltonians
  8. B Slave-spin theory for the Mott transition in the triangular lattice
    1. B.1 Forewords
    2. B.2 Solving the spin Hamiltonian $H_{\mathrm{S}}$ for $Q$: cluster mean-field
    3. B.3 Self-consistent outer cycle
      1. B.3.1 Solving the embedded model
    4. B.4 Discussion on units
    5. B.5 Results
      1. B.5.1 Exact diagonalization method
      2. B.5.2 Impact of experimental parameters
    6. B.6 Discussion
      1. B.6.1 Convergence of the method
      2. B.6.2 Experimental feasibility
  9. C Discussion on the correlation length of the system
    1. C.1 Forewords
    2. C.2 Correlation length with the slave-spin method implemented in a RQP
  10. D  Quasi-particle weight oscillations
    1. D.1 Forewords
    2. D.2 Data of the signal
    3. D.3 Fourier transform
    4. D.4 Effect of noise
  11. E Résumé en français

##  Chapter 1 General introduction

###  1.1 Industrial material aging context

The energy industry faces major challenges today with the threat of global
warming, growth of raw material prices and political instabilities. Regarding
this context, it is crucial for industries like Electricité de France EDF to
deliver a sustainable source of electricity. In France, this electric power is
mainly provided by nuclear energy (63% of the total production in 2022 111RTE
bilan électrique 2022 https://assets.rte-france.com/prod/public/2023-02/Bilan-
electrique-2022-synthese.pdf). This proportion dropped in 2022 due to the
discovery of stress corrosion in Nuclear Power Plant (NPP). Global warming is
expected to have a huge impact on our society [382]; therefore, reaching a
zero carbon emission energy industry in 2050 is a major goal. From this
standpoint, extending the lifetime of the nuclear power plant over 60 years
(Long Term Operation-LTO) has been raised to be an important step 222RTE
futures energétiques 2050 https://assets.rte-
france.com/prod/public/2021-12/Futurs-Energetiques-2050-principaux-
resultats.pdf. All currently operating NPPs in France use the pressurized
water reactor technology (PWR). One reactor is made of a pressure vessel
containing the core internals supporting the fuel assemblies in which the
fissile materials are, Fig. 1.1.

Figure 1.1: View of a vessel in PWR.

When operating, internal structures undergo harsh stresses: temperature
locally up to 350°C, mechanical constraints, intense neutron irradiation up to
120 dpa (displacement per atom), corrosion. The combination of these factors
induces creep, a potential risk of swelling (for which VVER or austenitic
materials such as fuel cladding are concerned) but no evidence of this
phenomena have been found in PWR reactors today, and thus changes the
properties of materials inside the vessels. It has been shown that irradiation
increases the formation of cracks in PWR components such as bolts [310]. These
phenomena are known as the Irradiation Assisted Stress Corrosion Cracking
(IASCC). To anticipate materials aging of NPPs, it is needed to understand
these phenomena at the atomic scale, where the formation of defects appears.
Several works have been done to study defects evolution in austenitic steels
[435] with ab initio method such as the Density Functional Theory (DFT) [441]
or Monte-Carlo algorithms (kinetic Monte-Carlo or rate theory) [345] with
encouraging results. Nevertheless, simulating complex phenomena where quantum
correlations are important, such as para-magnetism, is still difficult with
these tools. Austenitic steels are in the para-magnetic phase in power plants
conditions of temperature and pressure, and it is well known that magnetic
phases play an important role in energies of mitigation and defects [332,
260]. Therefore, being able to simulate accurately strong quantum correlations
could help, in fine, to increase NPP lifetime. Indeed, DFT is in a framework
multiscale modelling approach and output data of atomic scale simulation can
be used as input data of macroscopic scale simulations.

In addition, the aging of lithium-ion batteries is also a major issue to renew
the car fleet or for static energy storage, which is crucial to reduce
greenhouse gas emissions. Oxides are very difficult to simulate because most
of these materials have strong-correlated electrons that state-of-the-art
methods struggle to design [290]. We are surrounded by batteries in our
everyday life, and understanding how batteries age is decisive if an all
electrical world is considered. Another field where simulations are important
is solar cells. The difficulty is to reliably simulate excited states to
understand the efficiency degradation. Ab initio methods reach their limits
when it comes to study excited states of a material. The energy industry
therefore needs new methods of simulation to understand how materials of
interest age and to be able to avoid enormous costs like changing vessel
interns after a cracking or an explosion of a battery following a great loss
of the capacity. First principles computations have encountered great
successes [325, 508, 441] but they are limited in great significance area and
need to be improved or replaced because of quantum correlations being too
important.

Figure 1.2: Overview of quantum computers builders in the world with the three
main qubit carriers: the atoms, the electrons and the photons (figure taken
from [339]).

###  1.2 Quantum computing context

In 1982, the physicist Richard Feynman proposed to simulate quantum phenomena
with a quantum computer [342]. This is the birth of quantum computing. The
principle is to manipulate qubits, instead of classical bits, which lie in
vectorial space. Thanks to the superposition of quantum states, it is possible
to manipulate the coefficients $a$ and $b$ in a state $a\ket{0}+b\ket{1}$
where $|a|^{2}+|b|^{2}=1$. We are not dealing only with the bit $0$ or $1$ but
with both at the same time. In 1994, Peter Shor proposed an algorithm based on
quantum computing and Quantum Fourier Transform (QFT) to factor any integer
$N$ in a polynomial time [473], threatening to break public-key cryptography
schemes. In other words, if a quantum computer with enough noiseless qubits is
built, it could break most of the internet security and the world internet
network would be endangered. Thenceforth, a race for the universal quantum
computer has begun and is still ongoing today. The number of applications is
great: cryptography, optimization, differential partial equations,
combinatorial problems and of course materials and chemistry. Several
technologies are being tested with their own pros and cons all over the world
(examples in Fig. 1.2). Researchers also focus on designing original quantum
algorithms because it is impossible to convert a classical algorithm into a
quantum algorithm [420]. Nevertheless, quantum computing is still a promise as
no real quantum advantage has been shown up to date. Nowadays, we are in the
Noisy Intermediate Scale Quantum Computer (NISQ) era, [444] where noisy
quantum simulators with a few qubits are available. Therefore, original
algorithms considering noise have to be designed and a great amount of effort
is put in the emulation of quantum computing. This means that High Performance
Computing (HPCs) are used to numerically simulate the behavior of a quantum
computer, considering noise and limitations. This very important step in the
creation of an algorithm allows testing it with a few qubits to anticipate and
to mitigate the effect of noise. Moreover, quantum algorithms are often in two
parts:

  * •

A quantum part which is supposed to be solved by a real quantum computer.

  * •

A classical part that is tackled by a classical quantum computer.

The two parts are self-correlated and aim at minimizing an energy and finding
the groundstate energy of a system through the variational principle, for
instance.

The work presented in this manuscript lies between the energy industry
challenges described above and the breakthroughs in quantum computing. We show
how we can study strong-correlated electrons on realistic quantum simulators,
having in mind the issue of aging materials for energy industry.

###  1.3 EDF projects

My PhD is part of a quantum project and contributing to a material modelling
project:

  * •

a multiscale modelling approach to understand and anticipate defects and
develop aging models of materials in PWR (ULTIMATE project);

  * •

a quantum project (SI quantique) which aims at studying potential applications
of quantum computing for EDF.

####  1.3.1 Multiscale modelling

Ab initio atomic simulations are a part of a wide modelling program at EDF,
starting from simulating neutron irradiation to finite elements method. The
idea of this multiscale modelling is to use data and results of atomic scale
simulation to help modelling higher orders of magnitude problems, and so on
until the macroscopic scale is reached. This encompasses simulating the energy
distribution of the Primary Knocked-on Atoms (PKA) from the neutron spectrum,
ab initio methods such as DFT, Molecular Dynamics to characterize elementary
mechanisms at the atomic scale and also to simulate displacement cascades and
dislocation-defect interactions, Monte-Carlo and rate theory to simulate
microstructure evolution under irradiation, Dislocation Dynamics for
plasticity modelling and so on [257, 285].

A schematic representation of multiscale modelling is shown in Fig. 1.3.

Figure 1.3: Multiscale modelling applied in the PERFECT project to the
pressure vessel steels. Taken from [285].

Therefore, quantum computing could help to improve results of all the loop of
multiscale modelling by enhancing DFT materials modelling.

####  1.3.2 The quantum project

The Research and Development section of EDF is the biggest in Europe. Several
crucial issues are studied there to deliver sustainable source of electricity
in France (and other countries) and at the same time, keeping a balance in
economy. Quantum computing holds the promise to solve numbers of relatable
problems. The advent of QC could change everything in computing and
simulation, and there is no mapping between classical simulations and the
quantum world: one has to rewrite all algorithms to benefit QC. That was the
starting point of the quantum project at EDF in 2018: being active in this
field and starting to design our own algorithms for our own use case. Several
sector and direct applications have been identified such as partial
differential equation (with the HHL algorithm see Sec. 2.5.1), post-quantum
cryptography (Shor algorithm Sec. 2.5.1) the smart charging optimization
problem [317] and, of course, material simulation. All use case can be seen in
Tab. 1.1. The quantum group at EDF collaborated with a wide variety of actors
from French start-ups to IBM in the United States of America.

R&D sectors | Direct application
---|---
Production | Materials simulation (internals, batteries)
Probabilistic Safety Assessment studies
Partial differential equations
Quantum metrology networks
Network | Optimal power flow
Energy management | Optimisation/smart charging
Machine learning
Information Technologies | Post-quantum cryptography
Table 1.1: Sectors and direct applications of quantum computing for EDF

###  1.4 Contributions of this work

The contributions of this work are the following:

First, I present the first hybrid digital-analog variational algorithm on the
Rydberg Quantum Processor (RQP), a NISQ device, which aims at finding the
groundstate of molecules. This algorithm encompasses all specificities from
the device. I show a numerical implementation of this algorithm and prove that
it can be run on an actual device in a reasonable amount of time. The
efficiency of this algorithm is shown on molecules H${}_{\text{2}}$, LiH and
BeH${}_{\text{2}}$. All steps of the algorithm are discussed. The
transformation from a molecular algorithm to a qubits algorithm, the state
preparation, the measurement process with the derandomization method are
studied [418]. We propose a roadmap for bigger molecules and an experimental
implementation. This work results from a strong collaboration with the start-
up PASQAL.

Secondly, I introduce a new hybrid algorithm to observe the Mott transition on
2D-Hubbard model on a RQP. The slave-spin method is used to transfer the
complexity of the strongly-correlated electrons of Hubbard model on Ising-like
Hamiltonian coupled to a free electron’s system via a mean-field mapping. We
study the feasibility of this algorithm on an experimental device by exploring
the impact of all sources of error known up to date. I show that the Mott
transition can be observed on a RQP. I also demonstrate that we can observe
the effect of a quantum quench on the quasi-particle weight of the system on
RQP. This algorithm is the first step toward simulation of bulk materials on a
noisy quantum simulator. This work results from a strong collaboration with
the EVIDEN/ATOS company.

###  1.5 Manuscript organization

This work is structured as follows:

In chapter 2, I give an overview of state-of-the-art methods to simulate
electrons and magnetism "classically". Successes and limitations of these
tools are presented. Then, after an introduction of theoretical quantum
principles, I present some of the most recent NISQ quantum algorithms and
their applications with a focus on methods to simulate fermionic systems.

In chapter 3, I present the Rydberg platform, the physics behind it and how to
perform quantum computing with it. The digital and the analog approaches are
presented.

In chapter 4, I give the method and results of a digital-analog quantum
eigensolver designed for the RQP. After a presentation of the molecular to
qubits Hamiltonian mapping, we present an analog simulation implementation to
find the groundstate of the H${}_{\text{2}}$ molecule, which is inspired from
the Unitary coupled-cluster ansatz. Finally, I present the protocol for bigger
molecules and the results we obtain numerically.

In chapter 5, I introduce the slave-spin method and I show how it can be
employed to predict the Mott metallic-insulator transition and the out of
equilibrium behavior in the 2D-Hubbard model on a RQP. All the protocol and
theory behind the algorithm are described. The numerical simulation of a
realistic implementation on a noisy quantum simulator is shown and discussed.

Chapter 6 presents the conclusion and outlook of this work.

##  Chapter 2 Simulating correlated matter: from classical to quantum

###  2.1 Forewords

The famous citation of Richard Feynman "Nature isn’t classical, dammit, and if
you want to make a simulation of nature, you’d better make it quantum
mechanical, and by golly it’s a wonderful problem, because it doesn’t look so
easy" reminds us that simulating nature with a quantum computer seems to be a
good idea but most importantly, it is not straightforward. Indeed, there are
$\approx 10^{23}$ interacting electrons in all pieces ($\approx 1$ cm3) of all
the matter surrounding us. Even if only one-half spin degree of freedom is
considered, it means that $\approx 2^{10^{23}}$ states are allowed for a small
piece of matter, which is more than the number of particles in the universe.
Therefore, in front of this unreachable complexity, humans had to be creative.
The first step is attributable to Erwin Schrödinger and Werner Heisenberg who
first described the behavior of non-relativist quantum particles with the
Schrödinger equation (and matrix calculation):

$i\hbar\frac{\partial\psi(t,\vec{r})}{\partial
t}=-\frac{\hbar^{2}}{2m}\Delta\psi(t,\vec{r})+V(t,\vec{r})\psi(t,\vec{r})$
(2.1)

with $\hbar$ the Planck constant divided by $2\pi$, $\psi(t,\vec{r})$ the
wavefunction of the particle, $m$ its mass, and $V(t,\vec{r})$ all external
potentials the particle undergoes. This equation combined with periodic
boundaries condition (PBC) has led to the first quantum revolution with the
band theory [394] which is at the heart of all today’s electronic devices.
Nevertheless, approximations can not stand when considering certain class of
materials. For instance, complex alloys can not be described with this theory.
With the democratization of computers, the idea of solving the Schrödinger
equation with an algorithm to describe and understand complex phenomena has
been more and more studied. Indeed, as Richard Feynman said, the world is
quantum and if we want to improve our lives and our society, the quantum world
has to reveal all its secrets. Since particles are in interaction and the
complexity of materials is huge, approximations have to be made to hope for an
advancement. In this section, I review some of the most ubiquitous methods to
simulate strongly correlated electrons and spins. They all consider
approximations, but they have allowed to understand a large part of complex
materials. These methods are used today at EDF for instance to improve and
anticipate material aging. I discuss successes but also limitations of these
methods. These limitations can be really bounding in some fields where quantum
correlations are very strong [256, 375]. Quantum computing (QC) holds the
promise to solve really hard materials’ problem unreachable today. In the
second part of this chapter, I give an overview of the theoretical bases of QC
and how to simulate interacting electrons on a quantum computer. The last part
of this bibliography is dedicated to modern methods to perform quantum
computing on NISQ device and how to simulate fermionic problems on them.

###  2.2 Ab initio or first principles methods

The challenge of simulating materials relies on solving the Schrödinger
equation (Eq. 2.1). It is only possible in specific cases (the harmonic
oscillator for instance with
$\hat{H}=\frac{\hat{P}^{2}}{2m}+\frac{1}{2}k\hat{X}^{2}$). In general,
approximations are necessary to reach observable values. In this section, I
describe the Hartree-Fock method and the Density Functional Theory (DFT),
electronic structure-calculation methods widely used all around the world and
state-of-the-art methods in a large part of chemistry and material simulation
at the atomic level.

##### The Born-Oppenheimer approximation

Let us consider a periodic material. It is composed of electrons and nuclei in
interactions. The Hamiltonian of such a system can be written:

$H_{\text{mat}}=\sum_{k=1}^{M}\nabla_{\hat{\vec{R}}_{\text{k}}}^{2}-\sum_{i=1}^{N}\nabla_{\vec{\text{r}}_{\text{i}}}^{2}+\frac{1}{2}\sum_{i_{1}\neq
i_{2}=1}^{N}\frac{e^{2}}{|\vec{r}_{i_{1}}-\vec{r}_{i_{2}}|}+\frac{1}{2}\sum_{k_{1}\neq
k_{2}=1}^{M}\frac{Z_{k_{1}}Z_{k_{2}}}{|\vec{R}_{k_{1}}-\vec{R}_{k_{2}}|}-\sum_{i}^{N}\sum_{k}^{M}\frac{Z_{k}e}{|\vec{r}_{i}-\vec{R}_{k}|}$
(2.2)

with $M$ and $N$ being the number of nucleus and electrons respectively,
$\vec{R}_{k}$ and $\vec{r}_{i}$ the coordinates in 3D-space of nucleus and
electrons respectively. $e$ is the electric charge and $Z_{k}$ the number of
protons inside the nucleus. The first two terms are the kinetic energy of
protons and electrons, and the other terms are the Coulomb interaction energy.
The first approximation to simplify this equation was proposed by Max Born and
Robert Oppenheimer in 1927. It relies on the ratio of mass between nucleus and
electrons being equal to $\approx 2000$. It is therefore possible to consider
that electrons see the nucleus moving adiabatically. The kinetic energy of the
nucleus is then neglected, and the Coulomb interaction between protons is a
constant. From electron’s point of view, they undergo the following
Hamiltonian:

$H_{\text{e}}=E+U+V$ (2.3)

where $E$ is the kinetic energy of electrons, $U$ is the interaction between
electrons and $V$ the interaction of electrons with the stationary external
potential (protons interaction). The purpose now is to find the groundstate of
such a Hamiltonian for a well-defined $V$. It can actually be solved for small
systems in small basis set but obviously, trying to solve this equation with
the vectorstate $\ket{\psi_{0}}$ for a material is out of reach for classical
computer and will always be.

####  2.2.1 Hartree-Fock method

The Hartree-Fock (HF) method [330] is based on finding a good approximation of
the wave-function of the system to compute the ground state energy. The method
often assumes that the N-body wave-function describing the ground state of a
system is what we called a Slater determinant:

$\Psi(\textbf{r}_{1},\textbf{r}_{2},\dots,\textbf{r}_{N})=\frac{1}{\sqrt{N!}}\begin{vmatrix}\psi_{1}(\textbf{r}_{1})&\psi_{2}(\textbf{r}_{1})&\dots&\psi_{N}(\textbf{r}_{1})\\\
\psi_{1}(\textbf{r}_{2})&\psi_{2}(\textbf{r}_{2})&\dots&\psi_{N}(\textbf{r}_{2})\\\
\vdots&\vdots&\ddots&\vdots\\\
\psi_{1}(\textbf{r}_{N})&\psi_{2}(\textbf{r}_{N})&\dots&\psi_{N}(\textbf{r}_{N})\end{vmatrix}$
(2.4)

with orthogonality relations:

$\delta_{i,j}=\int\psi^{\dagger}_{i}(\textbf{x})\psi_{j}(\textbf{x})d\textbf{x}$
(2.5)

where $\textbf{x}=(\textbf{r},\sigma)$. The next approximation is to find how
to express these spatial-spin wave-functions. In general, linear combinations
of Gaussian functions (STO-nG) are chosen as basis set. This leads to good
approximations without too much complexity [480]. The variational principle is
then used to solve the HF equations.

This principle states that all approximations of ground state will lead to an
energy greater or equal to the exact ground state energy. The Rayleigh Ritz
wave-function variational principle can be expressed as:

$E_{0}=\underset{\ket{\psi}}{\text{min}}\bra{\Psi}{H_{\text{e}}}\ket{\Psi}$
(2.6)

where $\bra{\psi}\ket{\psi}=1$.

The HF method is at the center of many quantum chemistry numerical simulations
today [370].

####  2.2.2 Fundamentals of density functional theory

Finding a good wave-function for the system can be complicated and can lead to
huge approximations. Another approach is to focus on the one-particle density
of the system. Indeed, this observable gives a lot of information of the
system and can be easier to compute than the wave-function. This is the very
first step of the Density Functional Theory (DFT).

The variational theorem can be reformulated in terms of one-electron density
defined as:

$n(\textbf{r})=N\int\dots\int|\ket{\psi(\textbf{x}_{0}\textbf{x}_{1}\dots\textbf{x}_{N})}|^{2}d\textbf{x}_{0}d\textbf{x}_{1}\dots
d\textbf{x}_{N}.$ (2.7)

Indeed, the Hohenberg-Kohn theorem states that:

  * •

the external potential is a unique functional of the electronic density up to
a constant; 111Proof: let’s consider two external potentials
$V_{1}(\textbf{r})$ and $V_{2}(\textbf{r})$ such as $V_{1}(\textbf{r})\neq
V_{2}(\textbf{r})+\mathrm{cst}$. The two corresponding Hamiltonians
$H_{1}(V_{1})$ and $H_{2}(V_{2})$ have therefore different groundstates
$\ket{\psi_{1}}$ and $\ket{\psi_{2}}$ and groundstate energies $E_{1}$ and
$E_{2}$ but let’s suppose the same one-electron density n(r). We have
$E_{1}=\bra{\psi_{1}}H_{1}\ket{\psi_{1}}=\bra{\psi_{1}}H_{2}+V_{1}-V_{2}\ket{\psi_{1}}>E_{2}+\bra{\psi_{1}}V_{1}-V_{2}\ket{\psi_{1}}=E_{2}+\int(V_{1}(r)-V_{2}(r))n(r)dr$
and
$E_{2}=\bra{\psi_{2}}H_{2}\ket{\psi_{2}}=\bra{\psi_{2}}H_{1}+V_{2}-V_{1}\ket{\psi_{2}}>E_{1}+\int(V_{2}(r)-V_{1}(r))n(r)dr$.
This leads to $E_{1}+E_{2}>E_{1}+E_{2}$ which is absurd. So, the one-electron
density has to be different for the two Hamiltonians.

  * •

the functional that gives the energy of the system gives the ground state
energy if and only if the density is the ground-state density.

This is the foundation of the density functional theory. If we define the
universal density functional,

$F[n]=\bra{\psi[n]}{E}+{U}\ket{\psi[n]},$ (2.8)

The variational principle can be written:

$E_{0}=\underset{\text{n}}{\text{min}}\Big{(}F[n]+\int
V(\textbf{r})n(\textbf{r})d\textbf{r}\Big{)}.$ (2.9)

The final step is to map this interacting functional of energy onto $N$ non-
interacting one-electron systems with an effective external potential
$V_{\text{eff}}(\textbf{r})$ generating the same density n(r) as the real
system. The state of such a system can be described as a Slater determinant of
one-body wave functions $\phi_{i}(\textbf{r})$. Therefore, for the free-
electrons (or Kohn-Sham) system, the density is
$n(\textbf{r})=\sum_{i}|\phi_{i}(\textbf{r})|^{2}$ where $i$ browses all
occupied orbitals and the ground-state energy is
$E_{0}=\underset{\phi}{\mathrm{min}}\Big{(}\bra{\phi}T_{0}\ket{\phi}+\bra{\phi}V_{\text{eff}}\ket{\phi}\Big{)}$
With $T_{0}$ the kinetic energy of the free-electrons system. By decomposing
$F[n]$ as $\bra{\phi}T_{0}\ket{\phi}+E_{\text{Hxc}}[n]$ with
$E_{\text{Hxc}}[n]=\frac{1}{2}\int\int\frac{n(\textbf{r}_{1})n(\textbf{r}_{2})}{|\textbf{r}_{1}-\textbf{r}_{2}|}d\textbf{r}_{1}\textbf{r}_{2}+E_{\text{xc}}[n]$,
one can obtain:

$E[n]=T_{0}[n]+\frac{1}{2}\int\int\frac{n(\textbf{r}_{1})n(\textbf{r}_{2})}{|\textbf{r}_{1}-\textbf{r}_{2}|}d\textbf{r}_{1}\textbf{r}_{2}+E_{\text{xc}}[n]+\int
V(\textbf{r})n(\textbf{r})d\textbf{r}$ (2.10)

with
$T_{0}[n]=\sum_{i}\int\phi_{i}^{*}(\textbf{r})(\frac{-\nabla^{2}}{2})\phi_{i}(\textbf{r})d\textbf{r}$.
As a result,
$V_{\text{eff}}(\textbf{r})=V(\textbf{r})+\int\frac{n(\textbf{r'})}{|\textbf{r'}-\textbf{r}|}d\textbf{r}^{\prime}+\frac{\delta
E_{\text{xc}}[n]}{\delta n(\textbf{r})}$.

We obtain the Kohn-Sham equations:

$\Big{(}-\frac{1}{2}\nabla^{2}+V(\textbf{r})+\int\frac{n(\textbf{r'})}{|\textbf{r'}-\textbf{r}|}d\textbf{r}^{\prime}+\frac{\delta
E_{\text{xc}}[n]}{\delta
n(\textbf{r})}\Big{)}\phi_{i}(\textbf{r)}=\epsilon_{i}\phi_{i}(\mathbf{r}).$
(2.11)

This equation can be solved by a self-consistent loop described in Fig. 2.1.
One starts from a first guess for the density $n^{0}(\textbf{r})$. From this
density, one can calculate $V_{\text{eff}}(\textbf{r})$ and solve the Kohn-
Sham equation, Eq. 2.11. We then recover $n^{k}(\textbf{r})$ from wave-
functions $\phi_{i}(\textbf{r})$ and we compare it to the previous density
$n^{k-1}(\textbf{r})$. The loop goes on until self-consistency is reached.
Everything in the DFT is exact except that $E_{xc}$ and $\frac{\delta
E_{xc}[n]}{\delta n(\textbf{r})}$ are unknown and must be approximated. I will
give a few examples of common approximations used in this theory.

Figure 2.1: Self-consistency loop to solve Kohn-Sham equations.

####  2.2.3 The Local Density Approximation (LDA)

A strong but simple approximation is to consider that $E_{xc}$ depends on the
density the same way as an electron gas:

$E_{\text{xc}}[n]=\int\epsilon_{\textbf{xc}}(n(\textbf{r}))n(\textbf{r})d\textbf{r}$
(2.12)

and

$\frac{\delta E_{\text{xc}}[n]}{\delta
n(\textbf{r})}=\epsilon_{\text{xc}}+n(\textbf{r})\frac{\delta\epsilon^{\text{LDA}}_{\text{xc}}(n(\textbf{r}))}{\delta
n(\textbf{r})}$ (2.13)

where $\epsilon_{xc}$ is the exchange-correlation term of a uniform electron
gas of density $n(\textbf{r})$. This approximation relies on empirical results
and on powerful transferability and universality properties of interaction,
leading to successful results during the last decades. It is easy to consider
spinfull model with the Local Spin Density Approximations where densities of
spins up $n_{\uparrow}(\textbf{r})$ and spins down
$n_{\downarrow}(\textbf{r})$ are added in the term:

$E_{\text{xc}}^{\text{LSDA}}[n_{\uparrow},n_{\downarrow}]=\int\varepsilon_{\text{XC}}(n_{\uparrow},n_{\downarrow})n(\mathbf{r})\,\mathrm{d}\mathbf{r}$
(2.14)

####  2.2.4 The generalized-gradient approximation (GGA)

Another approach consists in considering gradients $\nabla n(\textbf{r})$,
$\nabla^{2}n(\textbf{r})$ and so on, of the density. whereas results can be
worse than LDA in some cases (like describing gas-phase reaction barriers), it
is used to describe metals and molecule–metal surface reactions [401, 353]. In
certain cases, LDA can have better results than GGA [282].

####  2.2.5 The LDA + U method

The LDA method is blind to the orbital dependency of the Coulomb and exchange
interaction. In [266], the authors proposed to add an interaction term, for
localized electrons, driven by an interaction $U$ just as proposed by P.W
Anderson [263]. This method gives a correct description of the Mott insulators
(see Sec. 2.3.2) and oxides such as NiO [266, 286] and U$\mathrm{O}_{2}$ [507,
472, 326] which are materials of interest for EDF.

####  2.2.6 Hybrid/metaGGA approach

metaGGA methods includes a part in the exchange-correlation function which
depends on the kinetic energy, and higher order of gradients of the density.
In addition, one, can mix a Hartree-Fock (HF) and a Kohn-Sham exchange to
obtain better results [486].

####  2.2.7 Beyond DFT

In many DFT works, paramagnetic phases are modeled as non-magnetic which can
lead to erroneous conclusions [256]. A realistic treatment of magnetism in
materials can be crucial to perform a predictive description of some
properties, like defect’s energies in austenitic steels. Two orthogonal
pictures were first proposed to simulate magnetism:

  * •

the itinerant model,

  * •

the local magnetic moment model.

The first one is based on the band theory of electrons whereas the latter
assumes localized electrons on atoms which carry therefore a local moment. The
answer lies in mixing these two approaches [256]. For instance, the spin
dynamics is based on the equivalence between a strongly interacted system with
onsite interactions and an electron living in a system with fluctuating charge
and spin fields. For instance, the Disordered Local Moment (DLM) model
considers that the full electrons model does not cover its phase space in
time. Indeed, each of the spins can flip after a time denotes the spin-flip
time and points in more or less random direction. An example of magnetic
interactions for this type of model is the longitudinal spin-fluctuation
Hamiltonian $H_{\mathrm{lsf}}$:

$H_{\mathrm{lsf}}=NJ^{(0)}(\overline{M})+\sum_{i}J^{(1)}(\overline{M},M_{i})-\sum_{i,j\neq
i}J_{i,j}(\overline{M},M_{i},M_{j}))\mathbf{M}_{i}\mathbf{M}_{j}$ (2.15)

with $\mathbf{M}_{i}$ being the magnetization vector and $M_{i}$ its norm,
$J^{(0)}(\overline{M})$ is the energy of the magnetically disordered system
where the mean value of the local magnetic moment is
$\overline{M}=1/N\sum_{i}M_{i}$. $J^{(1)}(\overline{M},M_{i})$ is the energy
needed to alter the value of the spin $i$ into $M_{i}$ from the imposed
average value $\overline{M}$. Finally, $J_{i,j}$ is the interaction energy
between two spins where all other local moments are disordered.

Many other approaches exist, [256], and choosing the good magnetic model
strongly depends on the problem.

####  2.2.8 Successes and limitations

The potency of the DFT relies on its simplicity and its polynomial complexity.
When it first came out, the method exceeded all previous methods in terms of
efficiency and duration of computation. Despite being quite simple, the LDA
approximations have shown his efficiency and has confirmed a lot of
experimental results [299]. Several improvements have been done such as
Density Functional Perturbation Theory (DFPT) to study phonons in crystals and
periodic materials [279]. However, several complex systems remain unreachable
for DFT.

  * •

In practice, computing excited energies is still difficult with this method as
it relies on variational principle which is efficient to find the ground
state. Therefore, computing gap in semiconductor or photovoltaic materials are
still out of reach [299].

  * •

In batteries, complex phenomena can happen, especially for oxides (for
instance Li-ion batteries or NiO materials [449]) and ab initio simulations
struggle to recover good results corresponding to experimental data [290,
411]. Simple and empirical approximations of the exchange-correlation terms
are not enough to encompass all phenomena observed experimentally, especially
when electrons tend to be delocalized [299].

  * •

In magnetic materials, spins have a quantum behavior and the simulation of a
paramagnetic material suffers from too many approximations [256]. An example
is the austenitic steel in nuclear power plant: it has been shown [441] that
simulating paramagnetism with DFT (and molecular dynamics as DFT works at $0$
K) leads to find the antiferromagnetic phase the most stable one to study
defects, whereas it is known the paramagnetic phase is the most stable. In
that specific case, one must go further to properly study magnetic phases in
materials.

###  2.3 Quantum approaches

####  2.3.1 The Hubbard model

Starting from first principles methods is not the only way to describe the
quantum world. Another approach to study strongly correlated electrons system
is to describe it with quantum mechanics formalism. By considering second
quantization [312], one can express states and observables in the Fock state,
which is much more convenient when the number of particle is big (such as in
materials). Electrons are described by quantum mechanics and then tend to be
delocalized to minimize their energy (wave behavior) but as charged particles,
they repel one another and thus try to be as localized as possible in order to
avoid paying this potential energy price. The minimal way of describing this
strive is the famous example of the toy model called “the Hubbard model”. The
system is simple: we consider fermions (or bosons) on a lattice where
particles have a probability to jump on nearest-neighbors sites (tight-
binding) and interact locally when they see each other on the same site [269]:

$\displaystyle\begin{split}H_{\text{Hubbard}}=&\sum_{i,j,\sigma}t_{i,j}d_{i\sigma}^{\dagger}d_{j\sigma}+U\sum_{i}n_{i,\uparrow}n_{i,\downarrow}\\\
&+\mu\sum_{i}(n_{i,\uparrow}+n_{i,\downarrow})\end{split}$ (2.16)

where $t_{i,j}$ is the hopping term between sites $i$ and $j$,
$d_{i,\sigma}^{\dagger}$ ($d_{i,\sigma}$) the fermionic creation
(annihilation) operator following anticommutation rule
$\\{d_{i,\sigma}^{\dagger},d_{j,\sigma^{\prime}}\\}=\delta_{i,j,\sigma,\sigma^{\prime}}$,
$n_{i,\sigma}=d_{i,\sigma}^{\dagger}d_{i,\sigma}$, $U$ the onsite interaction
(describing the Coulomb interaction) and $\mu$ the chemical potential. This is
the single-band Hubbard model (only one orbital per site). This model in 2
dimensions is conjectured to be describing (reasonably) high-temperature
superconductors [264]. The doping and several orbital cases are still active
fields of research in 2D and 3D. Solving this model could lead to major
discoveries and breakthroughs in fields of experimental and theoretical
physics [381].

The ratio $U/t$ controls the importance of interaction energy over kinetic
energy. If $U/t\rightarrow 0$, this means that we are dealing with free
electrons and the Hamiltonian can be diagonalized in the Fourier space by
setting $c_{k,\sigma}^{\dagger}=\sum_{i}e^{i.k.R_{i}}c_{i,\sigma}^{\dagger}$
such that $H_{\mathrm{Hubbard}}=\sum_{k}\epsilon_{k}c_{k}^{\dagger}c_{k}$. The
many-body ground state is a Slater determinant made up of these modes of
energy $<\epsilon_{f}$ (Fermi energy) propagating with the dispersion relation
$\epsilon(k)$. On the other hand, if $U/t\rightarrow\infty$, the Hamiltonian
is diagonal in the real space and eigenstates are of the form
$\ket{n_{(0,\sigma)},\dots,n_{(i,\sigma)},\dots,n_{(N,\sigma)}}$ where
$n_{i,\sigma}=0$ or $1$. In the case of half-filling, each site is populated
with one particle and the first excited state (so one site double-occupied)
costs an energy of $U$. We can therefore observe two regimes: one where
electrons can move freely such as in a metal and the other where the kinetic
energy is negligible, and the system is an insulator. This describes what we
call the Mott physics.

####  2.3.2 The Mott physics: starting point of highly correlated electrons

Back in the early 1900s, band’s theory was one of the greatest breakthroughs
discovered thanks to the quantum theory. It is the starting point of
transistors discovery which has led to the huge establishment of computers in
our lives (for instance the one on which I am typing these lines) and the
Moore’s law [422]. Still, while band theory provided a good classification for
many solids at the time, Verwey and de Boer [488] discovered in 1936 that some
materials such a nickel oxide (NiO), an oxide with 3D valence electrons, were
very poor conductors. From this postulate, Mott and Peierls [423] proposed in
1937 to hypothesize that in these materials, “the electrostatic interaction
between the electrons prevents them from moving at all”, explaining their
insulating behavior. Mott’s paper was the opening remark of the field of
strongly correlated materials.

####  2.3.3 The Dynamical Mean-Field Theory (DMFT)

###### The mean-field approximation

Let’s consider the toy model of statistical physics, the Ising model:

$H_{\mathrm{Ising}}=J\sum_{i,j}\sigma_{i}\sigma_{j}+h\sum_{j}\sigma_{j}$
(2.17)

with $\sigma=\\{-1;+1\\}$ and $z$ the coordination number of the lattice. This
model is exactly solvable in 1D and 2D [432]. Nevertheless, a first approach
to solve such a model is the single-site Mean-Field (MF) method [497]. This
means that we only consider one site interacting with a MF obtained by a self-
consistent loop. Mathematically,
$\sigma_{i}\sigma_{j}\approx\langle\sigma_{i}\rangle\sigma_{j}+\langle\sigma_{j}\rangle\sigma_{i}-\langle\sigma_{i}\rangle\langle\sigma_{j}\rangle$.
We can define $m=\langle\sigma\rangle$ and as a result, Eq. 2.17 can be
written (neglecting constant terms):

$H_{\mathrm{Ising}}\approx(mJz+h)\sum_{i}\sigma_{i}.$ (2.18)

$h_{\mathrm{eff}}=mJz+h$ is called the Weiss field and must be found by the
self-consistency condition $m=\mathrm{tanh}(\beta mJz+h)$. $m$ is the relevant
degree of freedom, as known as the order parameter. The dynamical mean-field
theory is based on the same reasoning. In this section, I describe the basis
of the DMFT following [350].

###### The Green function

Treating interacting particles is very hard, and the complexity of such a
system grows exponentially with the number of particle (or size of the system
in the Fock space). We therefore need some general way of examining the change
of the system in response to external effects (potential, temperature…) even
though we can not diagonalize the Hamiltonian. One of the mathematical tools
often used in many-body physics is the Green’s function. It is defined as:

$\mathcal{G}(\textbf{r},\textbf{r}^{\prime},t-t^{\prime})=-i\bra{\phi}Td_{i,\sigma}(\textbf{r},t)d_{j,\sigma^{\prime}}^{\dagger}(\textbf{r}^{\prime},t^{\prime})\ket{\phi}$
(2.19)

where $\ket{\phi}$ is the many-body groundstate and r, $t$ are space
coordinates, time coordinates respectively. $T$ is the ordering operator, such
as:

$\displaystyle\begin{split}Td_{i,\sigma}(\textbf{r},t)d_{j,\sigma^{\prime}}^{\dagger}(\textbf{r}^{\prime},t^{\prime})&=d_{i,\sigma}(\textbf{r},t)d_{j,\sigma^{\prime}}^{\dagger}(\textbf{r}^{\prime},t^{\prime})\,(t>t^{\prime})\\\
Td_{i,\sigma}(\textbf{r},t)d_{j,\sigma^{\prime}}^{\dagger}(\textbf{r}^{\prime},t^{\prime})&=\pm
d_{j,\sigma^{\prime}}^{\dagger}(\textbf{r}^{\prime},t^{\prime})d_{i,\sigma}(\textbf{r},t)\,(t<t^{\prime}).\end{split}$
(2.20)

In the case of periodic system, $\mathcal{G}$ only depends on
$(\textbf{r}-\textbf{r}^{\prime})$. In the Hubbard model, fermionic operators
only depends on time and site number. We can define the local Green’s function
at a given lattice site:

$\mathcal{G}^{\sigma}(\tau-\tau^{\prime})=-\langle
Td_{i,\sigma}(\tau)d_{i,\sigma^{\prime}}(\tau^{\prime})\ket{\phi}$ (2.21)

where $\tau$ is the imaginary time defined as $\tau=-it$. In the classical
mean-field theory, the local magnetization $m_{i}$ is represented as a single
spin on site $i$ coupled to an effective Weiss field $h_{\mathrm{eff}}$. The
reasoning is rigorously analog in the DMFT [350]. Let’s consider the
Hamiltonian of an Anderson impurity model,
$H_{\mathrm{And}}=H_{\mathrm{atom}}+H_{\mathrm{bath}}+H_{\mathrm{coupling}}$
in which:

$\displaystyle\begin{split}H_{\mathrm{atom}}&=Un_{\uparrow}^{c}n_{\downarrow}^{c}-\mu(n_{\uparrow}^{c}+n_{\downarrow}^{c})\\\
H_{\mathrm{bath}}&=\sum_{l,\sigma}\epsilon_{l}a_{l,\sigma}^{\dagger}a_{l,\sigma}\\\
H_{\mathrm{coupling}}&=\sum_{l,\sigma}V_{l}(a_{l,\sigma}^{\dagger}c_{\sigma}+c_{\sigma}^{\dagger}a_{l,\sigma}).\end{split}$
(2.22)

with $\mu$ the chemical potential of the impurity/atom. This Hamiltonian is
describing free fermions (a bath described by the $a_{l}^{\dagger}$’s) coupled
(via the interaction $V_{l}$) to a single-site (the “impurity”). The idea is
to consider that the Green’s function of this model coincides with the local
Green’s function of the lattice Hubbard model under consideration. The
parameters $V_{l}$ and $\epsilon_{l}$ are only taken into account in the
hybridization function

$\Delta(i\omega_{n})=\sum_{l}\frac{|V_{l}|^{2}}{i\omega_{n}-\epsilon_{l}}.$
(2.23)

The Weiss dynamical field (as known as the self-energy of the impurity model)
of this method is then defined as:

$\displaystyle\begin{split}\hat{\sum}_{imp}(i\omega_{n})&=\hat{\mathcal{G}}_{0,imp}^{-1}-\hat{G}_{imp}^{-1}(i\omega_{n})\\\
&=i\omega_{n}+\mu-\Delta(i\omega_{n})-\hat{G}_{imp}^{-1}(i\omega_{n})\end{split}$
(2.24)

with $\mathcal{G}_{0}$ being the Green’s function of the impurity model when
$U=0$ and $G$ the interacting Green’s function. Let’s now consider the Green’s
function of the original lattice model (the Hubbard model):

$\hat{G}(\textbf{k},i\omega_{n})=\frac{1}{i\omega_{n}+\mu-\epsilon_{\textbf{k}}-\hat{\sum}(\textbf{k},i\omega_{n})}.$
(2.25)

The approximation relies now on considering that the lattice self-energy
coincides with the impurity. This means that:

$\hat{\sum}_{i,i}\approx\hat{\sum}_{imp}\,\,\,,\,\hat{\sum}_{i\neq j}\approx
0.$ (2.26)

As a result, we obtain the self-consistency relation:

$\sum_{\textbf{k}}\frac{1}{\Delta(i\omega_{n})+\hat{G}(i\omega_{n})^{-1}-\epsilon_{\textbf{k}}}=\hat{G}(i\omega_{n}).$
(2.27)

Figure 2.2: The DMFT loop (inspire from [272])

A schematic representation of DMFT loops is shown in Fig. 2.2. The DMFT is
exact in the limit of infinite coordination lattices. This is also true for
the mean-field in classical mechanics. It is also true in the extreme cases
$U=0$ and $t_{i,j}=0$. The DMFT has encountered many successes in the
calculation of Mott transition (see [351] for more details).

####  2.3.4 Extensions of these methods

The major weakness of the DMFT is the lack of correlations due to considering
a single site. The Cluster DMFT (CDMFT) has been developed to consider a
cluster of sites coupled to a bath resulting in a better comprehension of Mott
transitions[436, 305]. In addition, the DFT and DMFT can be combined [289,
366, 437, 439, 403, 477] which enables an efficient calculation of the total
energy of realistic correlated electron systems (see Fig. 2.3).

Figure 2.3: Double loop in the DFT+DMFT method. Taken from [403].

####  2.3.5 Quantum Monte Carlo

To end this section, I will talk a bit about Quantum Monte Carlo (QMC)
methods. This approach aims at solving complex many-body problems by
estimating explicitly the groundstate wave-function of the system. To this
aim, a stochastic numerical integration is performed with the famous
Metropolis algorithm [499, 417]. These algorithms have a polynomial complexity
($O(N^{3})-O(N^{4})$) and are widely used to solve and simulate spins model
[454, 479]. Some works also study materials [491, 336]. A mix between Ab
initio approaches and QMC is possible [492]. However, the famous sign problem
arises for fermionic systems [434] but also for frustrated spin systems [372,
259]. QMC approaches are methods used for decades with a lot of successes but
some strong limitations exist.

####  2.3.6 Conclusion

In this section, I have briefly described first principles methods, Green
function methods and quantum Monte-Carlo approaches to many-body systems.
These approaches are at the center of modern numerical methods to understand
properties of materials at the atomic scale. During the last decades, these
methods have contributed to understand and design complex phenomena in
materials in academic and industry research. Investigations on improving them
are still ongoing. Nevertheless, describing correlations in strong correlated
systems seems to be out of reach without approximations. A spark in the dark
has appeared with the promise of quantum computing.

###  2.4 Overview of quantum computing

Whether quantum theory is exact or holds hidden variables has been a thorny
debate among physicists in the early 1900s. Albert Einstein said that “gods
don’t play dice” and there was room for doubt. Nevertheless, the quantum
theory was confirmed with many experiments during all the century until the
final hit, the Aspect’s experiment that has violated Bell’s inequalities in
1982 [271]. Since this major breakthrough awarded by the Nobel Prize in 2022,
researchers and industrials invest a huge amount of money and energy to build
what we call a quantum computer. In this section, I describe the foundation of
quantum computing principles and some advanced methods to use it. I also
describe the impact of this research and the promise of quantum computing for
materials and chemistry, which could change our society.

In "classical" computing, the smallest unit of information is stored in Bits
"0" or "1". Thanks to transistors and their reduced size ($\approx$ few nm),
it is possible to apply millions of operations per second in modern computers,
such as the ’AND’ or the ’XOR’ operators. In the quantum world, the
information is stored in vector states. The combination of such states allows
performing this kind of operation to bits ’0’ and ’1’ at the same time. We are
not dealing with bits anymore, but with qubits. These two states are
distinguishable, and we will use the Dirac notation in the following
$\ket{0}=\begin{pmatrix}1\\\ 0\end{pmatrix}$ and $\ket{1}=\begin{pmatrix}0\\\
1\end{pmatrix}$ (equivalent to $\ket{\uparrow}$ and $\ket{\downarrow}$ or
$\ket{g}$ and $\ket{r}$ in our notations). These states can be created via
photons, excited states in atoms or superconducting devices. One of the main
properties of these states is their dynamical behavior, being driven by the
Schrödinger equation. It is then possible to apply unitary operators to evolve
the system through time. We have therefore access to a new algebraic set of
"quantum gates" replacing the "classical gates" of qubit. In the following,
$\hbar=1$. Most of this section is inspired from [428].

####  2.4.1 Basis

Let’s start with a pure state. One can write the state of a qubit as:

$\ket{\psi}=a\ket{0}+b\ket{1}$ (2.28)

where $a$ and $b$ are complex coefficients and $|a|^{2}+|b|^{2}=1$. A qubit is
always written in the $z$ basis. A qubit can evolve under the application of
operators. A very important set of such operators are the Pauli matrices,

$\mathcal{I}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},\quad
S^{x}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},\quad
S^{y}=\begin{pmatrix}0&-i\\\ i&0\end{pmatrix},\quad
S^{z}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$ (2.29)

It is a basis for the $2\times 2$ Hermitian matrices real vector. This means
that any hermitian operator $U$ ($U^{\dagger}=U$) is a linear combination of
these 4 matrices. They are also unitary ($U^{\dagger}U=\mathcal{I}$). Usually,
a qubit is represented on a Bloch sphere, Fig. 2.4. Indeed, in the $z$ basis,
any state can be written
$\cos(\frac{\theta}{2})\ket{0}_{z}+e^{i\phi}\sin(\frac{\theta}{2})\ket{1}_{z}$
with $\phi\in[0,2\pi]$ and $\theta\in[0,\pi]$ up to a global phase. As an
example, $\ket{0}_{x}$ (the eigenstate of $S^{x}$ with the corresponding
eigenvalue $1$) can be read on the Bloch sphere

$\cos(\frac{\pi}{4})\ket{0}+e^{0}\sin(\frac{\pi}{4})\ket{1}=\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})=\ket{0}_{x}$
(2.30)

but also $\ket{1}_{y}$ (the eigenstate of $S^{y}$ with the corresponding
eigenvalue $-1$)

$\cos(\frac{\pi}{4})\ket{0}+e^{-i\frac{\pi}{2}}\sin(\frac{\pi}{4})\ket{1}=\frac{1}{\sqrt{2}}(\ket{0}-i\ket{1})=\ket{1}_{y}.$
(2.31)

Figure 2.4: The Bloch sphere is a graphical tool to represent a qubit. A state
$\psi$ lies on the surface of the sphere and can entirely be described by the
two angles $\theta$ and $\phi$.

####  2.4.2 Many qubits system and entangled states

Considering only one qubit is not very useful but it is possible to deal with
several qubits (let’s note $N$ qubits) by considering the tensor product of
states. For instance, if we suppose that a system $A$ is in the state
$\psi_{A}=a_{A}\ket{0}+b_{A}\ket{1}$ and a system $B$ is in the state
$\psi_{B}=a_{B}\ket{0}+b_{B}\ket{1}$, we can describe the full system $AB$ by
the separable state

$\ket{\psi_{A}}\otimes\ket{\psi_{B}}=a_{A}a_{B}\ket{00}+a_{A}b_{B}\ket{01}+b_{A}a_{B}\ket{10}+b_{A}b_{B}\ket{11}.$
(2.32)

Separable means that the state of the whole system can be written as a tensor
product of the state of the two sub-systems. One can generalize with
$\ket{\Psi}_{\mathrm{tot}}=\bigotimes_{i=1}^{N}\ket{\psi}_{i}$ but also for
any operator $\hat{O}_{\mathrm{tot}}=\bigotimes_{i=1}^{N}\hat{O_{i}}$. What
makes quantum physics distinct from other physics fields is the strange
property of entanglement. If we go back to the two qubits system, it can be,
for instance, in what are called Bell states:

$\frac{\ket{00}+\ket{11}}{\sqrt{2}}\,\,\,\,\text{or}\,\,\,\,\frac{\ket{01}+\ket{10}}{\sqrt{2}}.$
(2.33)

These states can not be written as a tensor product
$\ket{\psi_{A}}\bigotimes\ket{\psi_{B}}$. In addition, performing a
measurement on one system imposes the state of the other system instantly.
This counter-intuitive concept is described by Albert Einstein as “a spooky
action at a distance” and is at the center of the promises of quantum
computing. One can remark that the amount of information to store to be able
to fully describe a quantum state grows exponentially ($2^{N}$) with the
number of qubit.

####  2.4.3 Density matrix and mixed states

The most general way of representing the state of a quantum system is with the
density matrix representation:

$\rho=\sum_{i}p_{i}\ket{\psi}_{i}\bra{\psi}_{i}.$ (2.34)

In this representation, we consider a statistical ensemble of $N_{s}$ system,
each one being in a pure state $\ket{\psi}_{i}$. $p_{i}$ is the probability of
finding the quantum system in a pure state $\ket{\psi_{i}}$ and
$\sum_{i}p_{i}=1$. This state is called a mixed state. Using density matrices
and mixed states is useful when the state prepared is not fully known. For
instance, if we consider two ensembles of one qubit with one ensemble in the
state $\ket{0}$ and the other in the state $\ket{1}$, the corresponding
density matrix will be:

$\rho=\begin{pmatrix}\frac{1}{2}&0\\\ 0&\frac{1}{2}\end{pmatrix}.$ (2.35)

The state described here is a fully mixed state. Non-zero off-diagonal
elements in the density matrix identify the presence of quantum coherence in a
system. One interesting value to compute is the purity of the system, i.e
$P=Tr(\rho^{2})$. If $P=1$ the system is in a pure state and, on the other
hand, if $P=\frac{1}{2^{N}}$, we have a fully mixed state. The density matrix
representation is very useful to describe open system, as it is very difficult
to know the state of the whole system:
$(\mathrm{system}\bigotimes\mathrm{environment})$.

####  2.4.4 Close system and quantum logic operators

For a closed system, the evolution of a quantum state is driven by the
Schrödinger equation, leading to:

$i\hbar\frac{\partial\ket{\psi(\tau)}}{\partial\tau}=H(\tau)\ket{\psi(\tau)}$
(2.36)

where $H(\tau)$ is the time dependent Hamiltonian of the system. A direct
consequence is the unitary nature of any evolution operator acting on a close
system (a unitary operator means $U^{\dagger}U=\mathcal{I}$). Therefore, only
unitary operators can be used to evolve a quantum state. This can be
interpreted as the norm preservation of a quantum state. To know the state
after the time-evolution, one must measure the state (so the operator $S^{z}$
in the $z$ basis) several times. Indeed, if the system is in the state
$\frac{\ket{0}+\ket{1}}{\sqrt{2}}$, performing on measurement of $S^{z}$ one
time will give $-1$ with probability $|a|^{2}=1/2$ and $1$ with probability
$|b|^{2}=1/2$. One has to perform many measurement repetitions to obtain these
probabilities. The mean value of $S^{z}$ is obtained by probability of mean
value $-\frac{N_{0}}{N_{\mathrm{m}}}+\frac{N_{1}}{N_{\mathrm{m}}}$ and a
standard-deviation proportional to $1/\sqrt{N_{\mathrm{m}}}$ where $N_{0}$ is
the number of $1$ obtained after measurement, $N_{1}$ is the number of $-1$
obtained after measurement and $N_{\mathrm{m}}=N_{0}+N_{1}$ is the total
number of measurements. This is a direct consequence of the quantum projection
principle (or collapse of the wave-function), and it is called the quantum
projection noise as it brings errors to the computation of an observable (this
will be called shot noise in the following).

The quantum computing is based on applying unitary operators or quantum
logical gates (abbreviated as gates) on the system, just as in classical
computing. The procedure to perform quantum computing is the following:

  * •

start with a register of $N$ qubits in a well-known initial state;

  * •

a quantum circuit which carries all time-ordered gates to apply to the
register is defined;

  * •

at the end of the quantum circuit, a measurement is performed to obtain a set
of $N$ classical bits (due to the quantum projection noise).

[font=] $\ket{0}$&U_01
$\ket{0}$ [2]U_1,2
$\ket{0}$

Figure 2.5: Example of a quantum circuit with three qubits. The initial state
is $\ket{000}$

An example of a quantum circuit is shown in Fig. 2.5. A quantum circuit allows
manipulating $2^{N}$ coefficients, whereas only $N$ bits are controlled at a
time in classical computing. Single qubit operators
$U=\begin{pmatrix}U_{11}&U_{12}\\\ U_{21}&U_{22}\end{pmatrix}$ are then
$2\times 2$ unitary matrices. It changes coefficient $a$ and $b$ in
$\ket{\psi}=a\ket{0}+b\ket{1}$ into $a^{\prime}$ and $b^{\prime}$ such as:

$U\ket{\psi}=\begin{pmatrix}U_{11}&U_{12}\\\
U_{21}&U_{22}\end{pmatrix}\begin{pmatrix}a\\\
b\end{pmatrix}=\begin{pmatrix}a^{\prime}\\\
b^{\prime}\end{pmatrix},\,\,\,\,|a^{\prime}|^{2}+|b^{\prime}|^{2}=1$ (2.37)

This means that any quantum evolution on a quantum circuit is reversible
(contrary to classical circuit). More generally, applying a unitary matrix to
a qubit is like moving the quantum state on the surface of the Bloch sphere.
One performs rotations in all directions while conserving the norm. Thus, any
arbitrary single qubit unitary operator can be written in the form:

$U=e^{i\alpha}R_{\textbf{n}}(\theta)$ (2.38)

with
$R_{\textbf{n}}(\theta)=exp(-i\theta(\textbf{n}\cdot\boldsymbol{S})/2)=\cos{\frac{\theta}{2}}\mathcal{I}-i\sin{\frac{\theta}{2}}(n_{x}S^{x}+n_{y}S^{y}+n_{z}S^{z})$
and $\alpha$ a real number $\in[0,2\pi]$. Here,
$\textbf{n}=(n_{x},n_{y},n_{z})$ is a real unit vector in three dimensions and
$\boldsymbol{S}=(S^{x},S^{y},S^{z})$. Some very useful gates can be pointed
out, such as the Hadamard gates (denoted $H$) or the phase shift gate (denoted
$R_{\phi}$):

$H=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\
1&-1\end{pmatrix}=ie^{-i\frac{\pi}{2}(S^{x}-S^{z})/\sqrt{2}},\,\,\,\,\
R_{\phi}=\begin{pmatrix}1&0\\\
0&e^{i\phi}\end{pmatrix}=e^{i\frac{\phi}{2}}e^{-i\phi S^{z}}$ (2.39)

Still, considering only one-qubit operators is not very useful as we can not
create entangled states. Two-qubit gates can do the job. The idea is to make
two qubits interact. For instance, the CNOT gate considers two input qubits:
one is the control qubit and the other is the target qubit such as if the
control qubit is in the state $\ket{1}$, the other qubit "flips"
$\ket{0}\rightarrow\ket{1}$ and $\ket{1}\rightarrow\ket{0}$, otherwise it
remains the same. The matrix representation of the CNOT gate is

$\mathrm{CNOT}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{pmatrix}$ (2.40)

and it is graphically represented in Fig. 2.6.

[font=] &1

Figure 2.6: Graphical, a representation of a CNOT gate in a quantum circuit

Combining 1-qubit and 2-qubit gates can bring entanglement in the system. For
instance, if the target qubit is in state $\ket{0}$ and the control qubit is
in state $\ket{1}$ at the start of the circuit, applying a Hadamard gate to
the control qubit and then a CNOT creates a Bell state for the target qubit
$\frac{\ket{00}-\ket{11}}{\sqrt{2}}$. Some other 2-qubit gates are widely
used, such as the SWAP gate

$\textit{SWAP}=\begin{pmatrix}1&0&0&0\\\ 0&0&1&0\\\ 0&1&0&0\\\
0&0&0&1\end{pmatrix}.$ (2.41)

We can also think of N-qubit gates, but they are very difficult to implement
in practice. Hopefully, universal quantum gates sets exist, which means that
only with specific finite size sets of 1-qubit and 2-qubit gates, one can
approximately reproduce any unitary operation. The error on the approximation
reduces to $0$ when the number of gates is infinite. For example, the rotation
operator $R_{\textbf{n}}(\theta)$, the phase shift gate $R_{\phi}$ and the
CNOT gate are commonly used to form a universal quantum gate set.

The quantum computing is then based on the evolution through time of a state
of a closed system thanks to 1-qubit and 2-qubits unitary operators. In
practice, two very different approaches exist [352]: the digital and the
analog approaches.

####  2.4.5 Digital approach

Following the fact that any unitary operator can be (approximately) decomposed
in product of 1-qubit and 2-qubit gates, it is possible to emulate the
evolution of any Hamiltonian with a quantum circuit. This is the Digital
Quantum Simulation (DQS). In the following, this Hamiltonian is called the
target Hamiltonian. In quantum computing, the ansatz is a way to prepare a
specified quantum state with a circuit. It encompasses the initial state
preparation, but also the circuit itself. The purpose of a good ansatz is to
reach the desire quantum state with the fewer gates or measurements as
possible.

Preparing the initial state of an ansatz can be very difficult if some
specific properties are needed. In the case of fermionic computation, it can
be very useful to start with a state that respects antisymmetries in the
quantum state. Usually, the state $\ket{0}^{\bigotimes N}$ is the easiest to
prepare as it is the native state of the device, and it has been proven that
one can prepare a fermionic state with all possible permutations with
polynomial resources [255]. Ancilla qubit can also be useful to prepare the
initial state [494]. An ancilla qubit is a qubit acting in the circuit but
which is not a qubit describing the original problem. It is added to the
quantum circuit only to help the algorithm to run faster. It does not contain
properties of the wanted state, but it helps in the quantum computation.

DQS is not easy because sometimes the number of gates needed grows
exponentially with the size of the system but for most Hamiltonians with local
terms, this number grows in a polynomial way [428]. Let us consider that the
target Hamiltonian is a sum of constant local interaction terms
$H=\sum_{l}H_{l}$. Therefore, mimicking the evolution of $H$ in time means
that the circuit must be as close as possible of
$U(t)=\operatorname{e}^{-i\sum_{l}H_{l}t}$. Two cases are possible:

  * •

Either, for all indices $l$ and $l^{\prime}$, $[H_{l},H_{l^{\prime}}]=0$ and
$U=\prod_{l}e^{-iH_{l}t}$ and the decomposition in gates is direct.

  * •

Either, and that is the vast majority of cases, $[H_{l},H_{l^{\prime}}]\neq 0$
and one need to use the Trotter decomposition.

The idea is to cut the total time of evolution in infinitesimal slices, such
as $U(t)\approx(e^{-i\sum_{l}H_{l}\Delta t})^{\frac{t}{\Delta t}}$. Therefore,
we obtain [428]

$U(t)\approx\prod_{N_{t}=\frac{t}{\Delta t}}(\prod_{l}e^{-iH_{l}\Delta
t})^{N_{t}}+O((\Delta t)^{2})$ (2.42)

and when $\Delta t\rightarrow 0$ we recover a product of local terms.
Similarly, one can decompose $e^{i(A+B)\Delta t}$, where $[A,B]\neq 0$, into:

$e^{i(A+B)\Delta t}=e^{iA\Delta t/2}e^{iB\Delta t}e^{iA\Delta t/2}+O((\Delta
t)^{3}).$ (2.43)

In some cases, the goal is to reach a specific state with specific properties
instead of mimicking the evolution of a specific Hamiltonian. The Shor
algorithm [473] is a perfect example of this. Details about this algorithm
will be provided in Sec.2.5. For this type of algorithm, the QFT is very
useful. The idea is the same as for the classical discrete Fourier transform,
where we take a vector of complex numbers $(x_{0},x_{1},\dots,x_{N-1})$ as
inputs, and it outputs the transformed data as a vector of complex numbers
$(y_{0},y_{1},\dots,y_{N-1})$ defined as:

$y_{k}=\frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}x_{j}e^{2i\pi jk/N}$ (2.44)

The QFT is exactly the same transformation applying on an orthonormal basis.
We define states $\ket{0},\dots,\ket{n-1}$ with $\ket{0}=\ket{0}^{\bigotimes
N},\ket{1}=\ket{0}^{\bigotimes(N-1)}\bigotimes\ket{1},\dots,\ket{n-1}=\ket{1}^{\bigotimes
N}$ and $n=2^{N}$. Considering a state
$\ket{\psi}=\sum_{j=0}^{n-1}\alpha_{j}\ket{j}$ The QFT is the defined as:

$\displaystyle\begin{split}\ket{\psi}&\rightarrow\sum_{k=0}^{n-1}\beta_{k}\ket{k}\\\
\beta_{k}&=\frac{1}{\sqrt{n}}\sum_{j=0}^{n-1}e^{2i\pi
jk/n}\alpha_{j}\ket{j}.\end{split}$ (2.45)

The digital approach is based on the fact that "anything" can be simulated
with only a universal set of gates. The gates are applied with in an ordered-
time sequence. The last step is the measurement of the state at the output of
the circuit.

As described in previous sections (Sec. 2.4.4), measuring a quantum state
means getting the eigenvalue $\lambda_{k}$ of a hermitian operator $\hat{O}$
(measurement tool) corresponding to the eigenstate $\ket{k}$ with the
probability $|\bra{\psi}\ket{k}|^{2}$ where $\ket{\psi}$ is the state just
before the measurement. If $\ket{\psi}$ is not an eigenstate of $\hat{O}$ but
a linear combination of its eigenstates, $N_{m}$ measurements are needed to
reconstruct statistically the value of $\langle\hat{O}\rangle$ with an error
of $\approx 1/\sqrt{N_{m}}$. In general, Quantum State Tomography (QST) can be
used to characterize a quantum state but it requires resources that grow
exponentially with the size of the system, making it inefficient for large
quantum systems [319]. The measurement can also be performed in the quantum
circuit and not only at the end, that the case for protective measurements
[309], to perform Quantum Error Correction (QEC). The QEC for digital quantum
computing is out of the scope of this thesis but it is a crucial part of
digital quantum computating today [448].

The measurement procedure is therefore a major step of quantum computing, and
a good measurement procedure is needed to reduce the cost of computation.

In practice, we do not have access to perfect qubits but to what is called
physical qubits i.e. qubits with errors. In classical computing, the system is
repeated a great amount of time in parallel to avoid errors but in the quantum
world, the no-cloning theorem [502] prevents this method. Two types of error
can appear in qubits:

  * •

the bit-flip error $\ket{0}\rightarrow\ket{1}$ and $\ket{1}\rightarrow\ket{0}$

  * •

the phase-flip error
$\frac{\ket{0}+\ket{1}}{\sqrt{2}}\rightarrow\frac{\ket{0}-\ket{1}}{\sqrt{2}}$
and
$\frac{\ket{0}-\ket{1}}{\sqrt{2}}\rightarrow\frac{\ket{0}+\ket{1}}{\sqrt{2}}$.

A contrario, logical qubits are noiseless qubits and can be directly used for
computation. QEC allows to count the number of effective logical qubits that
are available for an algorithm from the number of physical qubits available in
the device. Another source of error is the gate fidelity and most importantly
the fidelity of entangling gates. The fidelity $\mathcal{F}$ of a gate is a
number between $0$ and $1$ which measures the capacity of an experimental gate
to reproduce the theoretical expected state. If the fidelity is
$\mathcal{F}=0$, it always reproduces an orthogonal state whereas if
$\mathcal{F}=1$ (impossible in practice), it reproduces exactly the target
state at each application.

####  2.4.6 Analog approach

That being said, a "perfect" quantum computer or simulator (without noise or
with very efficient QEC) does not exist today and the number of qubits or
gates is limited. As an example, the Shor algorithm would need several
thousands logical qubits which means millions of physical qubits… which is far
from being reached. John Preskill [444] has defined this era as the Noisy
Intermediate Scale Quantum (NISQ) era where only few number of noisy qubits
are available and, unfortunately, DQS boils down to the number of logical
qubits and the fidelity of gates. Nevertheless, some quantum devices can
simulate quantum mechanics by quantum means. This means that the intrinsic
physics of the device follows a quantum behavior and, therefore, can mimic
another system of interest. The Hamiltonian of the system is called the
resource Hamiltonian and the Hamiltonian to mimic is still the target
Hamiltonian. Here, the challenge is to map the resource Hamiltonian onto the
target Hamiltonian. It is not always possible, and in most cases, only a
partial mapping can be done with, for instance, symmetries in common. An
important advantage of Analog Quantum Simulation (AQS) is the fidelity of the
simulation of the resource Hamiltonian. Usually, this type of device can
control more qubits with a better fidelity than for digital quantum computing.
In addition, even in the presence of noise, one can recover physical
properties of interest. For instance, when looking at a quantum phase
transition, the qualitative behavior with not all quantitative quantities can
be enough. At first glance, AQS seems simpler than DQS, but it comes with
major inconveniences. First, the mapping between the resource and the target
Hamiltonian is not always direct. For instance, qubits are not fermions and
therefore looking at properties of a fermionic system with a driven qubits
Hamiltonian needs several steps. Second, AQS is way less flexible than DQS and
not anything can be simulated. In most cases, states reached by AQS are not
known and problems which are hard to describe with a physical Hamiltonian
(such as the Shor algorithm) can not be solved with analog simulators. An
example of the success of AQS has been done on trapped-ions simulator [395].
They have investigated the physics of the Schwinger model — a toy problem for
lattice quantum electrodynamics by leveraging the similarity between the
symmetries of a 20-ions quantum simulator and those of the Schwinger model.
The resource Hamiltonian is the ’XY’ Hamiltonian

$H_{\mathrm{R}}=\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}J_{i,j}(S_{i}^{+}S_{j}^{-}+S_{i}^{-}S_{j}^{+})+B\sum_{i=1}^{N}S_{i}^{z}$
(2.46)

with $J_{i,j}$ the long-range antiferromagnetic interaction term. This term is
controllable and $J_{i,j}\approx J_{0}/|i-j|^{\alpha}$ with $0<\alpha<3$. $B$
is an effective, uniform magnetic field. $S_{i}^{+}=(S_{i}^{x}+iS_{i}^{y})/2$
and $S_{i}^{-}=(S_{i}^{+})^{\dagger}=(S_{i}^{x}-iS_{i}^{y})/2$. On the other
hand, the target Hamiltonian (Schwinger model) is

$H_{\mathrm{T}}=w\sum_{j=1}^{N-1}(S_{j}^{+}S_{j+1}^{-}+S_{j+1}^{-}S_{j}^{+})+\frac{m}{2}\sum_{j=1}^{N}(-1)^{j}S_{j}^{z}+\overline{g}\sum_{j=1}^{N}L_{j}^{2}$
(2.47)

where $L_{j}$ depends on $S^{z}$. The crucial point, here, is the shared
symmetries of the two Hamiltonians: in both cases,
$[H,S^{z}_{\mathrm{tot}}]=0$. This helps to reduce the complexity of the
"quantum circuit" and the number of parameters in the quantum computation (see
Sec. 4.4.1). The result is the observation of a quantum phase transition up to
20 qubits despite all the noise and errors in the system (Fig. 2.7) with a
divergent derivative for one observable and a divergence for the other.
Because of the finite size system, these divergences do not go to infinity.
Nevertheless, the behavior of the target Hamiltonian is reproduced at least
qualitatively.

Figure 2.7: Quantum phase transition observed on trapped-ions device with
Analog Quantum Simulation. (a) is the order parameter of the phase transition
and (b) is the second-order Rényi entanglement entropy. Experimental data are
represented by circles, and solid lines are theoretical results from exact
diagonalization. The error bars are standard errors of the mean
$\langle\hat{\mathcal{O}}\rangle$ and $S^{(2)}_{\text{AB}}=$
tr($\rho^{2}_{\text{A}})$ (with the A–B bipartition shown in the inset),
respectively. Taken from [395]

####  2.4.7 Conclusion

Quantum computing can be performed in two different ways:

  * •

A digital approach with the possibility to simulate any Hamiltonian at a cost
of a high number of gates and qubits needed to reach an advantage.

  * •

An analog approach leading to simulate with a better precision specific
Hamiltonian feasible in practice.

The promises could be world changing or at least improve drastically the
search of solutions for certain classes of problems and, hence, overcome
classical limitations. Despite many public and private investments, it is far
from a forgone conclusion that quantum computing will be inevitable one day
because of noise and errors. A new field has therefore emerged: the hybrid
algorithms. The quantum algorithms are divided in two parts: one quantum part
where the quantum simulator or computer solves a very specific quantum problem
and one classical part which helps the quantum part to reach his target. More
globally, many hybrid and fully quantum algorithms are developed today to
mitigate errors and being implemented on today’s devices. In the next section,
I will present some of these methods with their success and also the
improvements needed to reach the famous "quantum supremacy" or, more
realistically, "quantum advantage".

###  2.5 Methods

In this section, I describe some of state-of-the-art methods of quantum
computing. I also depict the slave-spin theory and how to implement it on a
Rydberg Quantum Processor (RQP) leading to an hybrid quantum-classical
algorithm.

####  2.5.1 The quantum phase algorithm

In many quantum problems, the goal is to find the eigenstates of an operator
$U$. When $N$ is too big, classical computer struggles to diagonalize a
$2^{N}\times 2^{N}$ matrix. The quantum phase algorithm proposes to overcome
this. The first thing to notice is that an eigenvector of a unitary operator
is only defined by its phase as its modulus square is $1$. If $\lambda$ is an
eigenvalue corresponding to the eigenstate $\ket{\psi}$ of a unitary operator,
$\bra{\psi}U^{\dagger}U\ket{\psi}=\bra{\psi}\ket{\psi}\lambda^{*}\lambda=|\lambda|^{2}$
and because $U^{\dagger}U=\mathcal{I}$, $|\lambda|^{2}=1$. Therefore, any
eigenstate is of the form $\lambda=e^{2i\pi\phi}$ with $\phi\in[0,1]$. As
$\phi\in[0,1]$, one can write it in binary notation
$\phi=0.\phi_{1}\phi_{2}\dots\phi_{N}=\phi_{1}\times 2^{-1}+\phi_{2}\times
2^{-2}+\dots\phi_{N}\times 2^{-N}$. Let’s consider the circuit, Fig. 2.8.
After the Hadamard gate, the state is
$(\ket{0}\ket{\psi}+\ket{1}\ket{\psi})/\sqrt{2}$. Then the $C_{U^{2^{j}}}$
gate is applied: this means that if the control qubit is $\ket{1}$, the gate
${U^{2^{j}}}$, with $j$ an integer, is applied to $\ket{\psi}$, otherwise it
is not. The power of $2^{j}$ will be explained later. Here, $j=0$ and hence,
the state at the end of the circuit is
$(\ket{0}\ket{\psi}+e^{2i\pi\phi}\ket{1}\ket{\psi})/\sqrt{2}$.

[font=] $\ket{0}$&H1
$\ket{\psi}$ U^2^0

Figure 2.8: Quantum circuit for the quantum phase estimation at first order.

Let’s now consider Fig. 2.10. It is straightforward to see that
$U^{2}\ket{\psi}=e^{2\pi(2\phi)}$ and more generally,
$U^{2^{j}}\ket{\psi}=e^{2\pi(2^{j}\phi)}$. In addition,
$\phi=0.\phi_{1}\phi_{2}=\phi_{1}\times 2^{-1}+\phi_{2}\times 2^{-2}$,
$2\phi=\phi_{1}.\phi_{2}$ and therefore,
$e^{i2\pi(2\phi)}=e^{i2\pi(\phi_{1})}e^{i2\pi(0.\phi_{2})}=e^{i2\pi(0.\phi_{2})}$
because $\phi_{1}$ is an integer. More generally,
$e^{i2\pi(2^{j}\phi)}=e^{i2\pi 0.\phi_{j+1}\phi_{j+2}\dots}$. Going back to
the case $j=1$, the output state is $\Big{(}(\ket{0}+e^{2i\pi
0.\phi_{1}\phi_{2}}\ket{1}(\ket{0}+e^{2i\pi
0.\phi_{2}}\ket{1})\ket{\psi}\Big{)}/2$ if $\phi=0.\phi_{1}\phi_{2}$ is
assumed. For an arbitrary $n$, the output state is

$\frac{1}{\sqrt{2^{n}}}\sum_{k\in\\{0,1\\}^{n}}e^{2i\pi(\phi\times
k)}\ket{k}\ket{\psi}$ (2.48)

with $k$ a binary number. The final step is to consider the inverse Fourier
transform ($\mathrm{QFT^{-1}}$) in the canonical basis
$\ket{x},\,\,\,\,x\in\\{0,1\\}^{n}$. The $\mathrm{QFT}$ is defined as

$\ket{x}\rightarrow\ket{x_{\mathrm{QTF}}}=\frac{1}{\sqrt{2^{n}}}\sum_{k\in\\{0,1\\}^{n}}e^{2i\pi(x\times
k)/2^{n}}\ket{k}$ (2.49)

and looks very much like Eq. 2.48. Finally, applying the inverse Fourier
transform leads to the state
$\ket{\phi}=\ket{\phi_{n}}\ket{\phi_{n-1}}\dots\ket{\phi_{1}}$ which can be
directly measured (ignoring the state $\ket{\psi}$ which remains the same).
Thanks to this method, one can estimate the $n$ first digits of the phase and
therefore the eigenvalue of any unitary operator. A strong asset of this
method is the state $\ket{\psi}$ can be unknown. Indeed, one can decompose the
state $\ket{\psi}$ in the basis of eigenvalue of $U$ and thus recover the
result described above for all eigenstates.

[font=] $\ket{0}$&H2
$\ket{0}$H 1
$\ket{\psi}$ U^2^0U^2^1

Figure 2.9: Quantum circuit for the quantum phase estimation at second order.

[font=] $\ket{0}$&H4 … [4]QFT^-1
$\ket{0}$H3 …
⋮ ⋮ ⋮ ⋮ ⋮ ⋮
$\ket{0}$ H … 1
$\ket{\psi}$ U^2^0U^2^1 … U^2^j

Figure 2.10: Complete quantum circuit for the quantum phase estimation.

This algorithm and the QFT are at the heart of several algorithms which are
candidate to show quantum supremacy : the Shor algorithm to decompose any
number into prime numbers [473] with polynomial resources, the HHL (stands for
Harrow, Hassidim and Lloyd, the authors of the original article) [361]
algorithm which solves linear equations with a complexity of
$O(\mathrm{log}(N))$ where $N$ is the number of variables in the system and
the Grover algorithm [358] for unstructured search that finds with high
probability the unique input to a black box function that produces a
particular output value. For instance, consider a function
$f\\{0,1,\dots,N-1\\}\rightarrow\\{0,1\\}$ and you want to find out the
relation between inputs and outputs of the function. Classically, the fastest
way to do so is to test all possibilities, which means $O(N)$ measurement. The
Groover algorithm proposes to find the solution with a probability higher than
$1/2$ with only $O(\sqrt{N})$ measurements.

Despite being very promising, these algorithms need a lot of qubits and high
fidelity gates that are not accessible today, and we do not know when they
will be. The challenge is very hard and it is a long journey before being able
to show a quantum advantage over classical algorithms but the reward would be
to crack most communication encryption of our modern world, so maybe it is
worth it. But it remains a bet. Only the number $15$ was decomposed into prime
numbers thanks to the Shor algorithm [421]. Because no one wants to wait
several decades, some other methods have been explored.

####  2.5.2 The Variational Quantum Algorithm (VQA)

As described in Sec. 2.2, computing the groundstate of strongly-correlated
many-body Hamiltonians would help researchers but also companies in energy,
chemistry, health, drugs and therefore change the society. The good news is
that, it often relies on finding the groundstate of such a Hamiltonian. In all
minds, quantum computing aims at exceeding classical computing. However,
classical algorithms benefit from several decades of development, and they are
far from being obsolete. The idea to combine both quantum and classical worlds
was first proposed by Alberto Peruzzo et al. in [440]. Such an algorithm which
combines quantum and classical realms is called a hybrid quantum-classical
algorithm. More specifically, one specific algorithm is well suited to find
the groundstate of an operator: the Variational Quantum Algorithm (VQA). The
principle is to prepare a state with a gate-parametrized quantum circuit. This
means that parameters can be manually adjusted in gates of the circuit (such
as angle $\theta$ in rotation matrix $R_{\textbf{n}}(\theta)$) to change the
output as a function of parameters. The aim is to be able to span the most of
Hilbert space possible with the different parameters. One starts with an input
state and a unitary parametrized gate (called Parametrized Quantum Circuit
PQC) $U(\mathbf{\theta})$ is applied to it. The state is measured at the end
of the circuit, and the target operator (often the target Hamiltonian, noted
as $H_{\mathrm{T}}$ in the following) is measured in this state by statistical
repetition of the measurement. When the goal of VQA is to find the lower eigen
value of an operator, it is often called the Variatonal Quantum Eigensolver
(VQE). The key point is that now a classical optimizer looks for the
parameters to minimize the measured value. Then the computation is done again
with the new parameters and the loop goes on until a criterion is reached. At
the end, the energy measured in the state obtained should be close to the real
ground state energy of the target operator. A graphical example of the VQA
algorithm is shown in Fig. 2.11. In the following, I describe each step of
this algorithm and how to leverage them to increase the performance of VQA.

Figure 2.11: The Variational Quantum Algorithms loop. An initial state is
prepared as an input. Then time-ordered parametrized (the parameters are
regrouped in a vector $\mathbf{\theta}$) gates are applied to this state. The
output is a vector state depending on parameters. This state is measure to
obtain a bitstring and the target Hamiltonian is measured in this bitstring
state. This procedure is repeated $N$ times to estimate the mean value
$E(\mathbf{\theta}_{\mathrm{tot}})=\langle H_{\mathrm{T}}\rangle$. Then a
classical optimizer looks for new parameters to minimize this mean value. The
procedure is done again with the new parameters. These loops are performed
until a criteria is reached on the value of $\langle
H_{\mathrm{target}}\rangle$ is reached or a fixed number of loops is exceeded.
The final value of $\langle H_{\mathrm{T}}\rangle$ is expected to be very
close to the real ground-state energy.

One of the main advantages of VQA methods is that they provide a general
framework that can be used to solve a wide array of problems. Indeed, the
basic elements are always the same, whereas the structure can be different
from a problem to another. VQA algorithm is an active field of research [413,
301, 287, 303, 415] and have been performed on a lot of architectures [440,
392, 431, 425, 313]. It is at the center of quantum computing today and the
most promising way to show a quantum advantage in the next few years.

The first step of VQA is to encode our problem as a cost function to minimize
$C$. This cost function is usually the mean value of a Hamiltonian:

$\bra{\psi_{\mathrm{init}}}U(\mathbf{\theta})^{\dagger}H_{\mathrm{T}}U(\mathbf{\theta})\ket{\psi_{\mathrm{init}}}=\langle
H_{\mathrm{T}}\rangle_{\mathbf{\theta}}$ (2.50)

where $\ket{\psi_{\mathrm{init}}}$ is the initial state and
$U(\mathbf{\theta})$ the PQC describing the circuit. Other cost functions can
be chosen, such as the Gibbs objective function [406]

$G=-\mathrm{ln}\langle e^{-\eta H_{\mathrm{T}}}\rangle$ (2.51)

where the parameter $\eta$ has to be tuned. But in the end, when $\eta$ is
small, it reduces to the problem Eq. 2.50. Another target of VQA can be a
specific state, $\ket{\Psi}$ and then this algorithm is used to test the
fidelity of a circuit. The cost function is then

$\bra{\psi_{\mathrm{init}}}U(\mathbf{\theta})^{\dagger}\ket{\Psi}\bra{\Psi}U(\mathbf{\theta})\ket{\psi_{\mathrm{init}}}=|\bra{\Psi}U(\mathbf{\theta})\ket{\psi_{\mathrm{init}}}|^{2}.$
(2.52)

That is the case in [304, 367, 277] for specific problems. However, one needs
to know the target state to use this cost function, and it is often not the
case in materials where the groundstate is obviously not known. The main goal
of VQA is to find the set of parameters $\theta$ to reach:

$\mathrm{min}_{\theta}(C(\mathbf{\theta}))$ (2.53)

and therefore the ground-state energy of the target Hamiltonian through the
variational principle (see 2.6) in the specific case of a VQE algorithm for
instance.

One has to ensure that the target Hamiltonian can be measured in the
computational basis, i.e. it is a sum of Pauli strings

$H_{\mathrm{T}}=\sum_{s=1}^{N_{S}}{\bf
c}_{s}\bigg{(}\bigotimes_{j=1}^{N}S_{j}^{(s)}\bigg{)}$ (2.54)

with $N_{S}$ the number of Pauli strings and $N$ the number of qubits. $S_{j}$
are the Pauli matrices. If $H_{\mathrm{T}}$ is expressed this way, the
measurement in the computational basis is straightforward. Unfortunately, the
original problem is not always described with qubits and therefore, one must
find a way to map the initial Hilbert space into the qubit Hilbert space.
Depending on the problem considered, this step can be easy or can need a
mapping (see Sec. 4.3.1). Then an expectation value of $H_{\mathrm{T}}$ is the
sum of expectation values of Pauli operators

$\langle H_{\mathrm{T}}\rangle_{\mathbf{\theta}}=\sum_{s=1}^{N_{S}}{\bf
c}_{s}\bigg{(}\langle\bigotimes_{j=1}^{N}S_{j}^{(s)}\rangle_{\mathbf{\theta}}\bigg{)}.$
(2.55)

The next essential building block is of course the quantum circuit or the
ansatz. The common definition of all VQA algorithm is the state obtained at
the end of the circuit:

$\ket{\psi(\mathbf{\theta})}=U(\mathbf{\theta})\ket{\psi_{\mathrm{init}}}$
(2.56)

where $\ket{\psi_{\mathrm{init}}}$ is the initial state of the circuit. The
initial state preparation strongly relies on the problem and the circuit. A
good start offers better performance with for instance an initial state that
follows problem symmetries or is easy to prepare on the quantum hardware. The
choice of $U(\mathbf{\theta})$ will greatly impact the success of the VQA.
From the perspective of the problem, the ansatz influences both the
convergence speed and the closeness of the final state to a state that
optimally solves the problem. On the other hand, one can have to take into
account the hardware on which the VQA is performed: gates, parameters, errors,
and even the method (digital or analog) can differ a lot between
architectures. For instance, some entangling gates such as CNOT can have a
great fidelity in some hardwares, but it would be better to choose the
$\mathrm{SWAP}$ gate on another ones. In practice, most of the ansatz
developed can be classified as problem-inspired or hardware efficient,
depending on their structure and application.

###### Problem-inspired ansatz

The idea here is to draw from the target Hamiltonian to get the circuit. One
example is the Trotter decomposition, Eq. 2.42. Chemists have developed their
own ansatz way before the advent of the VQA. It is called the Unitary coupled
cluster ansatz [283] and will be developer deeper in Sec. 2.6 and Sec. 4. In a
simpler manner, the variational Hamiltonian ansatz aims at reducing the number
of parameters and accelerate the convergence [415, 496] by considering terms
of the fermionic target Hamiltonian itself. For this purpose, we consider that
$H_{\mathrm{T}}=\sum_{l}H_{l}$ and the ansatz is then

$U=\prod_{l=1}e^{i\theta_{l}H_{l}}.$ (2.57)

A very promising algorithm of the NISQ era is the Quantum Approximate
Optimization Algorithm (QAOA) [340]. It aims at approximate solutions to
combinatorial optimization problems. The cost function is designed to encode a
combinatorial problem by means of bit strings that form the computational
basis. A key point of this algorithm is a theoretical guarantee of convergence
when the depth of the quantum circuit increases. It consists of applying $p$
times two layers of non commutating operators $C$ (the operator of the cost
function) and $M$ the mixing operator. Mathematically, the prepared wave
function is of the form (taking notations from [317]):

$\ket{Z_{\mathbf{\gamma},\mathbf{\beta}}}=e^{-i\beta_{p}M}e^{-i\gamma_{p}C}\dots
e^{-i\beta_{0}M}e^{-i\gamma_{0}C}\ket{\psi_{\mathrm{init}}}.$ (2.58)

$(\mathbf{\gamma},\mathbf{\beta})$ are sets of parameters to optimize and $p$
is called the QAOA level or depth. The performance of the QAOA algorithm
improves with this value in a perfect case (in the absence of noise). For
satisfying constraints of the original problem or experimental device
limitations, penalties can be added to the cost function. Nevertheless, it is
often not sufficient and thinking on how to encode the constraints directly in
the ansatz seems to be the most promising way [427]. This approach was
implemented on a quantum emulator considering Rydberg atoms device on Maximum
Independent Set (MIS) and max-k-cut problems with promising results [317]. The
two problems are drawn from the rapidly growing sector of smart-charging of
electrical vehicles, in which EDF is strongly involved. The QAOA method can be
compared with a Trotterisation of the annealing method (see 2.5.3). One can
generalize this method with layers of two unitary operators $U(\gamma)$
(phase-separation) and $V(\beta)$ (mixing) which do not necessarily emerge
from the time evolution of a specific Hamiltonian. This is called the Quantum
Alternating Operator Ansatz [360] (also QAOA in the literature). Following
[287], we will abbreviate this ansatz as QuAltOA. In my work, I have tested
this ansatz for chemistry problems with Rydberg atoms device and compared it
with a new hardware-efficient digital-analog ansatz and I have shown that in
this case, hardware-ansatz is more efficient.

###### Hardware-efficient ansatz

Problem-inspired ansatz seems great theoretically with good symmetries between
the problem and the unitary operator, reduced parameters number, better
efficiency… But when it comes to implement it on a real quantum computer,
reality is catching up to us. Indeed, each architecture can implement gates
with more or less efficiency (some gates are impossible to implement on some
devices) and errors, decoherence, limited fidelity impose to use hardware-
efficient circuits. That is hardware-efficient ansatz. Even if the circuit is
not well suited for the original problem, choosing gates that can be well
implemented on a specific device can lead to better results than with a
problem-inspired device. The first example of this is in [392] where IBM chose
to build a parametrized circuit with specific gates that can be well
implemented on a superconductor device. Back then, it was one of the most
impressive results of quantum computing for chemistry and spin system. Often,
hardware efficient ansatz relies on only one or two entangling gates and just
a few single-qubit gates. For the specific case of superconducting qubit, the
lack of connectivity between qubits leads to choose specific gates that are
not linked with the original problem. All the difficulty of studying VQA in
The NISQ era is to balance the two ansatz. As explained in the previous
section, it is possible to map a circuit with specific gates into another
circuit with another gates, with the same results. Most of the studies take
the problem-inspired ansatz and implement it with a hardware efficient circuit
[425, 371]. In addition, one can choose a specific architecture to solve a
problem as symmetries of implementable gates match with symmetries of the
problem [395]. In fact, it seems that choosing the architecture with respect
to problem symmetries to reduce the complexity and length of the circuit is
one of the major challenges of the NISQ era (and maybe beyond).

###### Classical optimization

The quantum part of VQA is very important and challenging, but the
optimization of parameters is also an active field of research. Thus far, the
effect of noise and errors have only been considered for the quantum part, but
the impact on the optimizer is huge. Even natural laws of physics themselves
limit the efficiency of the optimization: one have to perform many
measurements to obtain the mean value of an observable with a great precision.
Thus, a good optimizer should try to minimize the number of measurements or
function evaluations. Last but not least, it should be resilient to noisy
values coming from limited fidelity, quantum decoherence and so on. In other
words, the perfect optimizer should converge toward the minimum value of an
observable with only few noisy sets of data which is not an easy task.
Therefore, choosing a good optimizer is crucial [400]. The first idea is to
use gradient descent algorithm, which is a local based research algorithm. It
is based on the postulate that one can have access to the derivative of the
cost function $\frac{\partial C(\mathbf{\theta})}{\partial\theta_{i}}$. This
value indicates the direction in which the objective function shows the
greatest change [442]. More sophisticated or free-gradient algorithms have
been tried such as the genetic algorithm [493, 418] which is a global
optimizer (it aims at finding a global minimum) at a cost of more function
evaluations. Finding the good optimizer for different problems is still an
active field of research [348, 295] that I will not develop here, but useful
insights can be found in [496, 287, 303].

###### Measurements

Once the state is obtained at the output of the circuit, one must measure the
state in the most efficient way to facilitate the work of the quantum
optimizer and reduced the number of total measurements. The most direct
approach is to decompose the output state in the basis of the target operator
$O$. To proceed, a unitary operation can be applied to project the state into
the diagonal basis of the observable. Let’s consider an example: we have a
state $\ket{\psi(\mathbf{\theta})}_{z}$ of $N$ qubits at the output of the
circuit, diagonal in the $z$ basis, and we want to know the mean value of the
operator $\bigotimes^{N}Y$. One needs to perform a rotation at the end of the
circuit to project the state in the $y$ basis:

$Y=R^{\dagger}_{x}(\pi/2)ZR_{x}(\pi/2)$ (2.59)

and

$\langle
Y\rangle=\bra{\psi(\mathbf{\theta})}R^{\dagger}_{x}(\pi/2)ZR_{x}(\pi/2)\ket{\psi(\mathbf{\theta})}.$
(2.60)

Therefore, the rotation $R_{x}(\pi/2)$ must be applied to all qubits at the
end of the circuit. This example is quite simple, but let’s now consider a
general target Hamiltonian $H_{\mathrm{T}}=\sum_{s=1}^{N_{S}}{\bf
c}_{s}\bigg{(}\bigotimes_{j=1}^{N}S_{j}^{(s)}\bigg{)}$ where $S$ can be either
$S^{z}$, $S^{x}$ or $S^{y}$. For each Pauli string, a specific tensor product
of rotations is needed. This means that the number of gates to apply to each
qubit is proportional to the number of different Pauli strings in the
Hamiltonian. In many-body and chemistry target Hamiltonian, the number of
terms can grow rapidly and therefore the number of measurement can become
considerable (keeping in mind that for each Pauli strings, $N_{s}$
measurements are needed to have an error of $1/\sqrt{N_{s}}$ on the mean
value). For one Pauli string, the number of measurement samples needed to
estimate $\bigotimes^{N}_{j}S_{j}$ with an additive error of at most
$\epsilon$ with a failure probability of at most $\delta$ is bounded by
Hoeffding’s inequality [378]

$N_{s}\geq\frac{2}{\epsilon^{2}}\mathrm{log}(\frac{2}{\delta}).$ (2.61)

The first idea is to group Pauli strings that can be measured simultaneously
to minimize the number of measurements [392, 415] but this can be a NP-hard
problem.

One of the most powerful methods is the classical shadow method with
randomized [379] and derandomized [380] measurement (details about this method
is shown in 4.4.3).

One advantage of VQA algorithm is the possibility to implement it with AQS
[395, 418] and DQS [392, 425].

###### Limitations and challenges

VQA algorithm offers a good alternative to full quantum algorithm and many
promising results have yet been shown theoretically and experimentally in this
paradigm. Yet, theoretical limitations (so not considering device limitations
and errors) have been demonstrated recently. The first one is the Barren
Plateau (BP) [414]. In this article, it has been shown that the expectation
value of the gradient of the cost function decays exponentially to zero as a
function of the number of qubits. This is true for Randomized Parameterized
Quantum Circuit (RPQC) but also for a wide class of reasonable parameterized
quantum circuits. As a result, the parameters landscape is essentially flat.
Hence, in a BP, one needs an exponentially large precision to resolve against
finite sampling noise and determine a cost-minimizing direction. This
phenomenon is true for free-gradient methods [270] and of course gradient-
based methods. Even noise has been shown to be a source of BP regardless of
the ansatz employed [495]. This could annihilate any hope on the much
anticipated quantum advantage, and methods to overcome this issue have to be
found.

Other challenges remain, such as the expressibility of the ansatz (the
capability of the parametrized quantum gates to explore all vector states of
the Hilbert space) or the reachability (whether it is easy to find a quantum
state that minimize the cost function). Even the effects of noise are still
studied today [397, 349, 346].

VQA algorithms pave the way to reach a quantum advantage. Combining High-
performance computing (HPC) and Quantum Processor Unit (QPU) is a hot topic,
and this could help to overcome difficulties described above. One can also
think of parallelization of quantum device, just as the start-up Welinq
proposes [498].

Many challenges remain but in the NISQ era, variational algorithms seem to be
the most promising way to show that quantum computing can surpass most
advanced classical methods in specific problems.

####  2.5.3 Quantum Annealing

Another possible approach with NISQ is the Quantum Annealing (QA) [343, 389,
267]. The idea is to use quantum fluctuation to reach a specific state of a
Hamiltonian. In practice, the Adiabatic Theorem (AT) is often used [258].
Starting from a well-known eigenstate of a realizable Hamiltonian $H_{0}$, one
can slowly tune the parameters of the device to reach a more complicated
Hamiltonian (the target Hamiltonian $H_{\mathrm{T}}$). The adiabatic theorem
stipulates that the state of the system at the end of annealing is
(approximately) the corresponding eigenstate of the target Hamiltonian (the
eigenstate with the same level of excited energy). For instance, if the system
is in the groundstate of $H_{0}$ at the beginning, it will be in the
groundstate of $H_{\mathrm{T}}$ at the end. If it starts in the first excited
state of $H_{0}$, it will end in the first excited state of $H_{\mathrm{T}}$
and so on for all eigenstates to the most excited state. Mathematically, a
time-dependent parameter $s(t)$ is slowly tuned from $0$ to $1$

$H(s)=s(t)H_{\mathrm{T}}+(1-s(t))H_{0}\,\,\,\,$ (2.62)

where $s(t=0)=0$ and $s(t=T_{\mathrm{final}})=1$ and $T_{\mathrm{final}}$ is
the final time of the procedure. QA has been performed with DQS on the D-Wave
machine [390]. It attempts to solve problems in a particular form called
Quadratic Unconstrained Binary Optimization (QUBO) with the Ising Hamiltonian
[409]. However, no quantum advantage or speed-up over classical simulations
have yet been found [308, 261]. AT is well suited for AQS if the problem can
be translated into finding the groundstate of the resource Hamiltonian. A
recent success of quantum annealing is with Rydberg Quantum Processor where an
anti-ferromagnetic has been built up to 196 spins (or qubits) [464] by
dynamically tuning the parameters of the Hamiltonian quasi-adiabatically
whereas numerical simulations struggle at calculating ground state of a
quantum Ising Hamiltonian for more than 32 spins.

The adiabatic theorem is not really flexible in terms of problems that can be
tackled, but if a solvable problem with this method is found, it can provide
good results and a potential path towards a quantum advantage (see Sec. 5).

###### Conclusion on methods

We have seen the limitations of full-quantum algorithm such as QPE on today’s
device. Reaching a quantum advantage with the Shor algorithm is only
conceivable in a distant future, hence, many other tools have been developed
to run algorithms on NISQ computers. One of them are hybrid algorithms, which
combine a quantum part with few gates and qubits and a classical part helping
the quantum part to reach its target. Despite having theoretical challenges
that remain, hybrid algorithms are the most promising way to solve real-world
problems. Materials and chemistry simulation could help to find room
temperature superconductor, new drugs, new materials are to anticipate
materials aging. In the next section, I present modern methods to apply
quantum computing to solve many-body physics problems.

###  2.6 Quantum computing for chemistry and many-body physics

####  2.6.1 From qubits to fermions

In atoms, electrons are assumed to be in a fixed potential created by nucleus
(Born-Oppenheimer approximation). Therefore, we can describe them with a
kinetic potential and the Coulomb interaction between electrons. In the second
quantization, the electronic Hamiltonian is used to rewrite Eq. 2.2 as:

$H=\sum_{p,q}h_{pq}a_{p}^{\dagger}a_{q}+\frac{1}{2}\sum_{p,q,r,s}h_{pqrs}a_{p}^{\dagger}a_{q}^{\dagger}a_{r}a_{s}.$
(2.63)

where the coefficients $h_{pq}$ and $h_{pqrs}$ encode the spatial and spin
configuration of each of the electrons and depend on the inter-nuclear and
inter-electron distances $\mathbf{R},\mathbf{r}$:

$\displaystyle h_{pq}$ $\displaystyle=\int
d\mathbf{x}\phi_{p}^{*}(\mathbf{x})\left(-\frac{\nabla^{2}}{2}-\sum_{i}\frac{\mathcal{Z}_{i}}{|\mathbf{R}_{i}-\mathbf{r}|}\right)\phi_{q}(\mathbf{x})$
(2.64) $\displaystyle h_{pqrs}$ $\displaystyle=\int
d\mathbf{x}_{1}d\mathbf{x}_{2}\frac{\phi_{p}^{*}(\mathbf{x}_{1})\phi_{q}^{*}(\mathbf{x}_{2})\phi_{r}(\mathbf{x}_{1})\phi_{s}(\mathbf{x}_{2})}{|\mathbf{r}_{1}-\mathbf{r}_{2}|}.$

Next, we map the fermionic $a^{\dagger}$ operators acting on Fock states of
$n$ orbitals to a Hilbert space of operators acting on spin states of $N$
qubits. This corresponds to the quantum processors’ effective interaction
Hamiltonians, quantum gates and measurement basis. In my work, I have studied
VQA algorithms applied to electrons system and so on, fermions. As it is well-
known, fermions follow the Pauli principle and fermionic wave-functions have
to be antisymmetrized to respect this fundamental principle. Qubits (or spin)
do not have the same statistics, and the corresponding states are not
naturally antisymmetrized. Hopefully, several methods have been developed to
map qubits to fermions and vice versa. Here, I describe the ones I have used
during my PhD: the Jordan-Wigner transform [387] and the Bravyi-Kitaev
transform [296]. Let’s consider a many-body electrons system. The annihilation
$c$ and creation $c^{\dagger}$ operators follow an anticommutation rule
$\\{c_{i},c_{j}^{\dagger}\\}=\delta_{i,j}$. The system’s Hamiltonian is
written with this type of operator. For Pauli operators
$\\{S^{i},S^{j}\\}=2\delta_{i,j}\mathcal{I}$ where $i$ and $j$
$\in\\{0,x,y,z\\}$ and, therefore, it is not easy to describe the target
Hamiltonian in terms of Pauli operators. To alleviate this issue, Pascual
Jordan and Eugene Wigner proposed the following transformation [468]

$\displaystyle\begin{split}c_{1}^{\dagger}&=(S^{-})\otimes I\otimes
I\dots\otimes I\\\ c_{2}^{\dagger}&=\sigma^{z}\otimes(S^{-})\otimes I\otimes
I\dots\otimes I\\\ \vdots\\\
c_{n}^{\dagger}&=\sigma^{z}\otimes\sigma^{z}\otimes\dots\otimes\sigma^{z}\otimes(S^{-}).\\\
\end{split}$ (2.65)

where $S^{-}=\frac{S^{x}+iS^{y}}{2}$ and
${S^{-}}^{\dagger}=S^{+}=\frac{S^{x}-iS^{y}}{2}$. Therefore, the relations

$\\{c_{i},c_{j}^{\dagger}\\}=\delta_{i,j},\quad\\{c_{i},c_{j}\\}=0,\quad\\{c_{i}^{\dagger},c_{j}^{\dagger}\\}=0$
(2.66)

are well fulfilled. The main advantage of this mapping is its simplicity, but
as a result, any fermionic operator becomes a Pauli string of $N$ Pauli
matrices, and we lost locality properties of the original Hamiltonian. It is
therefore really hard to emulate such a Hamiltonian as
$H_{\mathrm{T}}\neq\sum_{l}H_{l}$. In addition, this transformation needs
$O(N)$ operations on qubits to perform one fermionic operation. In JW
transform, the occupation stays local but parity of the wave-function is
delocalized.

The parity mapping is the exact inverse [468]. In this mapping, the parity is
stored locally. The idea is to consider the occupation number basis state
(fermionic state) $\ket{f_{n},f_{n-1},\dots,f_{0}}$ (in the following, all
sums of binary variables are taken modulo 2) and apply the transformation:

$p_{i}=\sum_{j}[\pi_{n}]_{i,j}f_{j}$ (2.67)

where $n$ is the number of orbitals and $\pi_{n}$ is upper triangular matrix:

$[\pi_{n}]_{i,j}=1\,\,\,\mathrm{if}\,\,\,i<j\,\,\,,\,\,\,[\pi_{n}]_{i,j}=0\,\,\,\mathrm{otherwise}.$
(2.68)

Unlike the JW transform, one can not represent the creation and annihilation
operators of a particle in an orbital $j$ simply by applying $S^{\pm}_{j}$
because one qubit does not store the occupation of an orbital. It is needed to
look at qubit $(j-1)$. If the qubit $(j-1)$ is in the state $\ket{0}$, then it
represents accurately the occupation orbital $j$ and then one needs to apply
$S^{+}$ to the parity basis to simulate $a^{\dagger}$ one the fermionic basis.
On the other hand, if the qubit $(j-1)$ is in the state $\ket{1}$, the qubit
$j$ inverts parity and the operator $S^{+}$ needs to be applied. The same
reasoning applied for the annihilation operator. The equivalent of $S^{\pm}$
in the parity basis is:

$P_{j}^{\pm}=S_{j}^{\pm}\otimes\ket{0}\bra{0}_{j-1}-S_{j}^{\mp}\otimes\ket{1}\bra{1}_{j-1}=\frac{1}{2}(S^{x}_{j}\otimes
Z_{j-1}\mp iS^{y}_{j})$ (2.69)

Furthermore, creating or annihilating a particle in orbital $j$ implies a
change in the parity data and we must update the cumulative sums for all $k>j$
by applying the operator $S^{x}$. It corresponds to add $1$ to all qubits
$k>j$ modulo 2.

At the end of the day, the creation and annihilation operators in the parity
mapping are written:

$\displaystyle\begin{split}a^{\dagger}&=S^{x}_{n}\otimes
S^{x}_{n-1}\otimes\dots\otimes S^{x}_{j+1}\otimes P_{j}^{+}\\\
a&=S^{x}_{n}\otimes S^{x}_{n-1}\otimes\dots\otimes S^{x}_{j+1}\otimes
P_{j}^{-}.\end{split}$ (2.70)

As it is clearly seen, we just replaced the loss of locality with matrices $Z$
in the JW transform with the loss of locality with matrices $X$ in the parity
basis. Thus, the total operation to apply on each qubit is still $O(N)$.

In the middle of both transforms, there is the Bravyi-Kitaev mapping (BK)
[296]. The idea is to store the parity and occupation non-locally. A qubit $j$
stores the occupation only if $j$ is even. If $j$ is odd, the corresponding
qubit holds a partial sum of the occupation set of orbitals less than index
$j$. The transformation is quite complicated and I refer to [468] for more
details. What is important is the BK transform only needs
$O(\mathrm{log}_{2}(N))$ transformation to go from fermions to qubits and
therefore is the simplest transform from a digital circuit complexity point of
view.

####  2.6.2 Unitary coupled cluster ansatz

The simplest approximation for a complete basis of Fock space for fermions is
the set of Slater Determinants (Eq. 2.4). In other words, we approximate the
eigenstates of the target Hamiltonian to be of the form:

$\ket{\psi}=\prod_{j}(a^{\dagger}_{j})^{\gamma_{j}}\ket{00\dots 0}$ (2.71)

where $\gamma_{j}=0,1$ and $a^{\dagger}_{j}$ creates a particle in the orbital
$j$. The corresponding state with the JW transform is

$\ket{\psi}_{\mathrm{JW}}=\prod_{j}(S^{x}_{j})^{\gamma_{j}}\ket{00\dots 0}$
(2.72)

Usually, the method to generate Slater determinants is the Thouless algorithm
[484]. Starting from a Slater determinant $\ket{\psi_{0}}$ one can generate an
ensemble of new determinants given by [356]:

$\ket{\psi(Z)}=e^{i\sum_{i,j}Z_{i,j}a^{\dagger}_{i}a_{j}}\ket{\psi_{0}}$
(2.73)

where Z is hermitian. In VQA algorithms [356], this matrix is optimized with a
PQC to minimize $\bra{\psi_{Z}}H_{\mathrm{T}}\ket{\psi_{Z}}$.

Thus far, we did not take into account correlations which are at the heart of
the goal of quantum computing. Starting from the Thouless algorithm, one can
extend it to gradually insert excitation into the system. This is the Unitary
Coupled Cluster (UCC) ansatz.

It is constructed from a parametrized cluster operator $T(\theta)$ which is a
sum of particle-hole excitations of different orders:

$\displaystyle\begin{split}T(\theta)&=T^{1}(\mathbf{\theta})+T^{2}(\mathbf{\theta})+\dots\\\
T^{1}(\theta)&=\sum_{i,j}\theta_{i,j}a^{\dagger}_{i}a_{j}\\\
T^{2}(\theta)&=\sum_{i,jkl}\theta_{i,jkl}a^{\dagger}_{i}a^{\dagger}_{j}a_{k}a_{l}.\end{split}$
(2.74)

The sum is truncated due to the decrease impact of higher order terms.

The anstaz of the PQC is then:

$\ket{\psi(\theta)}=e^{(T(\mathbf{\theta})-T^{\dagger}(\mathbf{\theta}))}\ket{\psi_{\mathrm{init}}}.$
(2.75)

UCCSD refers to the truncation after the double excitation and UCCSDT to the
truncation after the triple excitation. The next step is transforming the
operators $T$ into qubits operators with methods describe in Sec. 2.6.1.
Finally, the Trotter decomposition can be used to decompose
$e^{T(\theta)-T^{\dagger}(\theta)}$ into a sequence of gates. This method is
very well suited for DQS.

This method is quite old and has been proposed decades ago [369] and has been
consequently widely applied. A good summary of implementations of the UCC
ansatz can be found in [262]. These problems often concern small molecules, as
this method is still benchmarked on NISQ computers.

Thus far, we only discussed the quantum part of the algorithm and how to
leverage it for molecular and fermionic many-body problems but the problem can
be directly simplified with chemistry and theoretical simplifications. For
instance, molecular wave-functions in Eq. 2.64 have to be chosen. They often
constructed from linear combinations of atomic orbitals, themselves considered
as linear combination of Gaussian functions (STO-NG) basis. The more common
ones are the STO-3G (for three gaussians) or STO-6G (for six gaussians) [368].
Therefore, choosing 3 or 6 functions can reduce or increase the complexity of
the target Hamiltonian and thus, the efficiency of the VQA algorithm. But in
the other hand, choosing a more accurate representation of orbitals leads to a
better description of correlations in molecules or materials. The difficulty
lies in finding the balance between chemical accuracy and VQA efficiency. In
addition, some low energy orbitals can be considered as "frozen" whereas high
energy orbitals can be considered as unoccupied just as in [371] where they
only focused on a subspace of the orbitals in the lithium molecule. With all
these approximations, they reduced the target Hamiltonian from 12 qubits and
200 Pauli strings to an effective Hamiltonian of only 3 qubits and 13 Pauli
strings.

In conclusion, efforts have to be done in optimizing the quantum algorithm but
also in the chemistry side where approximations are really important to
simplify the problem.

###  2.7 Conclusion

Simulating matters at atomic scale is at the center of much research since the
advent of computing. It enables to anticipate the behavior of complex system
and even creating new materials. Several methods (DFT, DMFT, QMC) have
encountered great successes and are used all around the world for fundamental
physics or applications. Recently, a new approach has emerged to help or even
surpass these "classical" approaches: the quantum computing. Many methods
theoretically exist to provide a quantum advantage on a quantum computer. Yet,
noiseless computers with thousands of logical qubits and high fidelity gates
do not exist and it is not even sure they will one day. What do exist today
are NISQ device, with experimental proofs of their utility. Therefore, hybrid
methods such as VQA have been developed to show a quantum advantage on these
devices. There have been some successes but also failures with the discovery
of Barren Plateau for instance. Today, a real quantum advantage on important
problems for the society has not been found. The field of quantum computing is
an experimental and theoretical exploration where industrial giants are
involved with the promise to revolutionize the world and many architectures
have their pros and cons. Among this emulation, one technology has proven a
quantum advantage in theoretical physics problems: the neutral atoms (or
Rydberg Quantum Processor RQP) [297]. Groundbreaking results have emerged from
this technology, considering digital [338, 391, 314] or analog [464, 306, 462]
quantum computing. RQP is one of the best candidates to simulate many-body
physics. Regarding the goal of my PhD to perform quantum simulation of
fermionic many-body physics, it has appeared to me that RQP is an excellent
candidate to get closer to a quantum advantage in this field and, most
importantly, being able to implement my algorithms on a real device soon.

##  Chapter 3 Quantum simulation with Rydberg atoms

###  3.1 Forewords

In this chapter, I give an overview of the experimental apparatus of the
Rydberg Quantum Processor (RQP) (first section) and the protocol to perform
digital and analog quantum computing (second section). Most of the contents of
are adapted from [297, 374, 463].

###  3.2 Neutral atom arrays

Figure 3.1: (a) Schematic representation of the main hardware components
constituting the RQP. In red, the trapping laser is modulated by the Spatial
Light Modulator (SLM) to produce microtraps i.e. optical tweezers with the
geometry desired (see inset). In purple, the moving tweezers allow
reorganizing the atoms after the first fluorescence picture. They are
controlled by a 2D acousto-optic laser beam deflector (AOD). In green, the
fluorescence light is captured by a camera to obtain a bitstring. (b) Picture
of the heart of experimental apparatus. The register is prepared at the center
of the photography. Taken from [374].

####  3.2.1 Initial state preparation

Figure 3.2: The Rydberg blockade and computation cycle (a) The two chosen
electronic states of atoms are coupled via the Rabi frequency $\Omega$. (b)
Two atoms separated by a distance $R<R_{b}$ are only coupled to the state
$\ket{+}=(\ket{rg}+\ket{gr})/\sqrt{2}$ and not $\ket{rr}$. If the distance is
greater than $R_{b}$, the state $\ket{rr}$ is accessible. (c) Graphical
representation of Rydberg blockade. A regular array with spacing $a$ is
created with the SLM, each of the atoms prevent the atoms in the circle of
radius $R_{b}$ from its center to be in the state $\ket{r}$ with it. (d)
Temporal sequence of one computational cycle. The register is initialized with
atoms trapped in a tweezer with a probability $1/2$. The atoms are then
rearranged in the desired geometry. The quantum processing can now start and
the cycle end with a picture of atoms: the register readout. All the process
has to be done again for each measurement. The three first figures are
inspired from Fig. B1 of [297] and the last figure is inspired from Fig. 5B of
[374].

Quantum computing is by definition the manipulation of individual quantum
objects (qubits) by experimental techniques. Quantum objects can take various
forms such as gazes, dots, ions, photons, Josephson junctions and atoms. Until
the end of the 1900s, controlling qubits with enough precision to entangle and
apply logic operations on them was still a theoretical idea. Today, many
platforms are able to control up to several hundreds of qubit at the same time
and thus, give a hope to simulate quantum systems better than classical
methods. RQP is based on ensembles of individual atoms trapped in optical
lattices. In this section, I give an overview of breakthrough methods which
have led to being able to manipulates hundreds of neutral atoms individually
and make them interact.

Figure 3.3: Examples of geometry in 2D and 3D experimentally feasible. Taken
from [281].

RQP is fully controlled by light, from the state preparation and the set-up of
the register to the quantum processing part and finally to readout the
register. Each step needs different type of lights with different wavelengths
and power. At the same time, electronic components and software stack monitor
steps like rearranging atom tweezers or tune the magnitude of lasers achieving
desired Hamiltonian. The first step to generate arrays of neutral atoms is to
prepare a diluted atomic vapor inside an ultra-high vacuum system (to isolate
atoms from the environment). To do so, one gram of solid rubidium (Rb) is
heated up in an oven to transform it into a vapor. Contrary to other
architectures such as superconductors, the device operates at room-
temperature, leveraging the complications of a cooling system. At the output
of the oven, atoms are slowed down to reach $\approx 10$ m/s. The atoms of gas
are then trapped inside a 3D magneto-optical trap (3D MOT). These atoms are
the reservoir to a second trapping laser system which isolates each individual
atoms, the so-called optical tweezers [458]. Two very useful characteristics
of optical tweezer can be pointed out: at first, each tweezer contains at most
1 atom. This is ensured by the very fast light-assisted collision which
happens if two atoms are in the trap. The number of tweezers and their
arrangement in the 2D or 3D space is controlled by a Spatial Light Modulator
(SLM) [429] which is a "mask" that modulates the phase of the light into a
desired intensity pattern (see Fig. 3.3). A strong asset of RQP is to fully
control the geometry of the array and therefore connectivity between atoms
(more details in the next section, Sec. 3.2.2). Nevertheless, the occupation
of a trap is not deterministic: this means that a trap is filled with one atom
with a probability of $p\sim 0.5$. To palliate this issue, a picture of atoms
is taken by looking at their fluorescence and tweezers are moved one by one to
arrange atoms in the desired way. This procedure has a success rate of $99\%$
and takes few tens of milliseconds. This is enabled by an algorithm
calculating the optimal moves for each tweezer from the first picture. After
all of this, the register is ready for the quantum processing. A schematic
representation of the experimental apparatus is shown in Fig. 3.1 with also a
picture of the heart of the device.

####  3.2.2 Generating spin-spin interactions

In this subsection, I describe the physics behind the QRP and the tunable
Hamiltonians that can be simulated thanks to this physics. In order to perform
quantum computing, atoms have to interact. To this aim, two electronic states
are chosen to be the qubit $\ket{0}$ and $\ket{1}$: the groundstate and a high
excited state of the atoms. A key asset of Rydberg atoms for quantum computing
is the Rydberg blockade [453].The strong van der Waals interactions between
two atoms in the Rydberg state can be used to prevent simultaneous excitations
of two atoms. In an isolated atom, a laser field is coupling the ground state
and the Rydberg state of an atom with a Rabi frequency, $\Omega$ Fig. 3.2(a).
In the case of two atoms separated by a distance $R$, they undergo a van der
Waals interaction for the state $\ket{rr}$ and therefore there is a shift in
energy by the quantity $C_{6}/R^{6}$ to reach this state. Thus, one can define
the blockade radius defined by:

$R_{b}=(C_{6}/\hbar\Omega)^{1/6}.$ (3.1)

If the blockade condition is fulfilled $\hbar\Omega\ll C_{6}/R^{6}$ i.e. $R\ll
R_{b}$, the state $\ket{rr}$ is non-resonant and the two atoms evolve from the