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CENTRE D’ETUDES DOCTORALES - SCIENCES ET TECHNOLOGIES

N° d’ordre: 3838

THÈSE

En vue de l’obtention du : DOCTORAT

Structure de Recherche : Physique des Hautes Énergies- Modélisation et
Simulation
Discipline : Physique
Spécialité : Physique mathématique et Information quantique

Présentée et soutenue le : 23/09/2023 par :

Hamid SAKHOUF

Quantum information processing with superconducting circuits:
realizing and characterizing quantum gates and algorithms in open quantum
systems

JURY

Mohammed Loulidi |  | PES,Université Mohammed v@, Faculté | Président/Rapporteur
---|---|---|---
|  | des sciences-Rabat. |
Lahoucine Bahmad |  | PES,Université Mohammed v@, Faculté |  Rapporteur/Examinateur
|  | des sciences-Rabat. |
Abderrahim El Allati |  | PH,Université Abdelmalek Essaadi, Faculté | Rapporteur/Examinateur
|  | des sciences et techniques- Al-Hoceima |
Mohammed DAOUD |  | PES,Université Ibn Tofail, Faculté |  Directeur de thèse
|  | des sciences-Kénitra. |
Rachid Ahl Laamara |  | PES,Université Mohammed v@, Faculté | Directeur de thèse
|  | des Sciences-Rabat. |

Année Universitaire : 2022/2023

⌧ Faculté des Sciences, avenue Ibn Battouta, BP. 1014 RP, Rabat –Maroc

☎ 00212(0) 37 77 18 76, ⍾Fax: 00212(0) 37 77 42 61 ; http://www.fsr.um5.ac.ma

## Dedication

First and formost, I want to express my deep appreciation to my family. I am
immensely grateful to my parents, siblings, and extended family for their
unwavering support, encouragement, and motivation throughout my academic
journey. Their belief in me and their constant presence in my life were the
driving forces behind the completion of this thesis.

In addition, I would also like to thank all members, past and present, of the
LPHE-MS, with whom I have had the pleasure to interact, for the many valuable
discussions and shared moments.

## Acknowledgment

During the preparation of this thesis, many things changed in my life, both
personally and scientifically. Before diving into the content of this thesis,
I would like to express my gratitude and thanks to those who have been present
in this period of my life and have contributed directly or indirectly to this
final work.

First, I would like to thank Mr. El Hassan SAIDI Professor at the Faculty of
Science of Rabat for giving me the opportunity to enroll in a Master’s degree
in mathematical physics, which in turn allowed me to enroll in the Ph.D.
program at the Laboratory of High Energy Physics, Modelisation, and Simulation
(LPHE-MS). I appreciate their advice and encouragement throughout this work.
My sincere thanks for their availability as well as the efforts they have
always made for the success of several activities of the LPHE-MS.

In addition to the director of LPHE-MS laboratory, I would like to warmly
thank my supervisor Mr. Rachid AHL LAAMARA, professor at the Faculty of
Sciences of Rabat for welcoming me to the LPHE-MS. His continuous
encouragement gave me the strength and courage to complete this thesis as well
as for his teaching of the group theory course and Lie algebra course.

Additionally, I would like to warmly thank my supervisor Mr. Mohammed DAOUD
Professor at faculty of science of Kenitra for his magnificent guidance
throughout my graduate studies. Without him, this thesis would never have been
possible. I appreciate his continuous support and guidance with formidable
motivation and support in my studies for my Master’s and Ph.D.

My regards also go to Mr. Mohammed LOULIDI, Professor at the Faculty of
Science of Rabat, who accepted the presidency of the jury also agreeing to be
a reporter of my modest work. I also acknowledge him for his help, discussion,
encouragement, and motivation.

Let me also thank the rapporteurs of the thesis : I will start with Mr.
Lahoucine BAHMAD, Professor at the Faculty of Science of Rabat. I would like
to thank him warmly for accepting to be a reporter of my modest work. Yours
sincerely.

My acknowledgment also goes to Mr. Abderrahim El ALLATI, PH, Faculty of
Sciences and Techniques Al-Hoceima, for accepting to be the reporter of my
thesis. I thank him for his interest and responsibility as an examiner of this
work and for his bearing the hardships of traveling from Al-Hoceima to Rabat
to be with the jury members.

## Résumé

Cette thèse se concentre sur le traitement de l’information quantique à l’aide
d’un dispositif supraconducteur, en particulier sur la réalisation de portes
quantiques et d’algorithmes dans des systèmes quantiques ouverts. Un tel
dispositif est construit par des qubits supraconducteurs de type transmon
couplés à un résonateur supraconducteur. Pour la réalisation des portes
quantiques et des algorithmes, une approche en une seule étape est utilisée.
Nous proposons des schémas plus rapides et plus efficaces pour réaliser des
portes de $X$-rotation et des portes d’enchevêtrement pour deux et trois
qubits. Au cours de ces opérations, le nombre de photons du résonateur est
annulé en raison du fort champ de micro-ondes ajouté. Elles ne nécessitent pas
que le résonateur soit initialement préparé dans l’état de vide et les schémas
est insensible à la désintégration du résonateur. En outre, la robustesse de
ces opérations est démontrée en incluant l’effet de la décohérence des
systèmes transmon et la désintégration du résonateur dans une équation
maîtresse, ce qui permet d’obtenir une grande fidélité dans la simulation
quantique. En outre, en utilisant les portes de $X$-rotation mises en œuvre
ainsi que les portes de phase, nous présentons une autre façon de mettre en
œuvre l’algorithme de Grover pour deux et trois qubits qui ne nécessite pas
une série de portes simples. En outre, nous démontrons en simulant
numériquement l’utilisation de la tomographie des processus quantiques pour
caractériser pleinement la performance d’une porte d’enchevêtrement à un seul
coup pour deux et trois qubits et nous obtenons des fidélités de processus
supérieures à 93$\%$. Ces portes sont utilisées pour créer des états intriqués
de Bell et de Greenberger-Horne-Zeilinger (GHZ).
Mots clés: Traitement de l’information et calcul quantique, Portes quantiques,
Algorithme de recherche de Grover, Portes d’intrication, Etats de Bell et GHZ,
circuits supraconducteurs, Haute fidélité.

## Abstract

This thesis focuses on quantum information processing using the
superconducting device, especially, on realizing quantum gates and algorithms
in open quantum systems. Such a device is constructed by transmon-type
superconducting qubits coupled to a superconducting resonator. For the
realization of quantum gates and algorithms, a one-step approach is used. We
suggest faster and more efficient schemes for realizing $X$-rotation and
entangling gates for two and three qubits. During these operations, the
resonator photon number is canceled owing to the strong microwave field added.
They do not require the resonator to be initially prepared in the vacuum state
and the scheme is insensitive to resonator decay. Furthermore, the robustness
of these operations is demonstrated by including the effect of the decoherence
of transmon systems and the resonator decay in a master equation, and as a
result high fidelity will be achieved in quantum simulation. In addition,
using the implemented x-rotation gates as well as the phase gates, we present
an alternative way for implementing Grover’s algorithm for two and three
qubits, which does not require a series of single gates. As well, we
demonstrate by a numerical simulation the use of quantum process tomography to
fully characterize the performance of a single-shot entangling gate for two
and three qubits and obtaining process fidelities greater than 93$\%$. These
gates are used to create Bell and Greenberger-Horne-Zeilinger (GHZ) entangled
states.

Keywords: Quantum information processing and computation, Quantum gates,
Grover’s search algorithm, Entangling gates, Bell and GHZ states, High
fidelity, superconducting circuits.

## Résumé détaillé

Cette thèse se concentre sur la réalisation et la caractérisation de portes
quantiques et d’algorithmes à l’aide d’une approche en une seule étape basée
sur un dispositif quantique supraconducteur. Un tel dispositif quantique est
construit en couplant des qubits supraconducteurs de type transmon à un
résonateur supraconducteur, piloté par un champ classique. À l’aide de ce
dispositif, nous proposons un schéma rapide pour préparer un registre
quantique comprenant de nombreux états de base en superposition quantique en
réalisant la porte de rotation X pour deux qubits ou plus, ne nécessitant
qu’une seule opération. De plus, en utilisant la même approche, nous réalisons
la porte d’intrication en une seule étape pour deux qubits ou plus. Ces portes
sont utilisées pour créer un état totalement intriqué tel qu’un état de Bell à
deux qubits et un état GHZ à trois qubits. En utilisant une simulation
strictement numérique basée sur des hypothèses réalistes concernant les
paramètres du système, nous étudions la dynamique quantique de ces portes et
démontrons que ces schémas peuvent être mis en œuvre avec une grande fidélité.

De plus, en utilisant les portes de rotation X mises en œuvre, nous présentons
une alternative pour mettre en œuvre l’algorithme de Grover à deux et trois
qubits, qui ne nécessite pas une série de portes individuelles. Notre
implémentation a montré que le schéma proposé permet de trouver efficacement
l’état correct avec une fidélité élevée d’environ 90$\%$. Ces schémas
présentent les avantages suivants : (i) ils ne nécessitent pas la combinaison
des portes de base, par exemple les portes individuelles et à deux qubits,
(ii) le résonateur n’a pas besoin d’être initialement préparé dans l’état de
vide, (iii) nos schémas sont insensibles à la dégradation de la cavité, ce qui
facilite les expériences pratiques. De plus, nous démontrons, en simulant
numériquement l’utilisation de la Tomographie du Processus Quantique (TPQ)
pour caractériser pleinement les performances d’une porte d’interaction en une
seule étape pour deux qubits ou plus, que nous obtenons des fidélités de
processus supérieures à 93$\%$.

Dans le premier chapitre, nous discuterons brièvement des bases du traitement
et du calcul de l’information quantique. Cela comprend les aspects
mathématiques et géométriques des bits quantiques, ainsi que la description de
la façon dont ils peuvent être manipulés et de la différence entre un
ordinateur quantique et son équivalent classique. Nous identifierons également
un ensemble universel de portes logiques quantiques, comprenant des portes
individuelles et multi- qubits, ainsi que leur contribution à la construction
d’algorithmes quantiques, tels que les algorithmes de Grover et Deutsch-Jozsa,
les protocoles d’intrication quantique et la préparation d’états de
superposition. Dans le deuxième chapitre, nous aborderons la mise en œuvre
physique des bits quantiques en utilisant des circuits électriques pour
réaliser des qubits supraconducteurs. Nous passerons en revue certains types
de qubits tels que la boîte à paires de Cooper (qubit de charge) et le qubit
transmon qui est utilisé dans cette thèse, ainsi que l’oscillateur LC
harmonique quantique en tant que composant essentiel dans électrodynamique
quantique en circuit(circuit QED). Nous décrirons également l’interaction
entre les qubits supraconducteurs et la ligne de transmission supraconductrice
ou l’oscillateur LC (le champ électromagnétique) à l’intérieur de la cavité de
la QED des circuits, qui est une version similaire de l’électrodynamique
quantique en cavité (cavité QED) classique qui sera également abordée. De
plus, nous discuterons du régime dispersif nécessaire pour le traitement de
l’information quantique. Enfin, nous terminerons ce chapitre en associant la
circuit QED au traitement de l’information quantique, où nous identifierons
les portes à qubit unique, ainsi que les portes C-Phase et iSWAP. Le chapitre
III de cette thèse vise à réaliser la porte de rotation X pour deux et trois
qubits, qui peut être utilisée pour préparer des états de superposition et
d’autres portes appelées portes d’intrication, utilisées pour générer des
états totalement intriqués tels que les états de Bell et GHZ. Leur mise en
œuvre est basée sur un dispositif quantique comprenant des qubits transmon
couplés capacitivement à un résonateur, piloté par un champ micro-onde, en
utilisant une approche en une seule étape. La robustesse de ces portes est
démontrée en prenant en compte l’effet de la décohérence des systèmes transmon
et de la dégradation du résonateur dans un formalisme d’équation maîtresse.
Pour des hypothèses réalistes concernant les paramètres du système transmon-
résonateur, une grande fidélité sera obtenue lors de la simulation quantique.
Au cours de ces opérations, le nombre de photonsdu résonateur est annulé en
raison du fort champ de micro-ondes ajouté. Elles ne nécessitent pas que le
résonateur soit initialement préparé dans l’état de vide et les schémas est
insensible à la désintégration du résonateur. Enfin, nous discuterons de la
faisabilité expérimentale des schémas proposés et les comparerons aux derniers
résultats expérimentaux dans le domaine des dispositifs supraconducteurs.

Mots clés: Traitement de l’information et calcul quantique, Portes quantiques,
Algorithme de recherche de Grover, Portes d’intrication, Etats de Bell et GHZ,
circuits supraconducteurs, Haute fidélité.

## List of publications

### This thesis is based in part on the following published articles:

 $\bullet$ Hamid Sakhouf, Mohammed Daoud, Rachid Ahl Laamara , “Quantum
process tomography of the single-shot entangling gate with superconducting
qubits,” Journal of Physics B: Atomic, Molecular and Optical Physics 56,
105501 (2023).

$\bullet$ Hamid Sakhouf, Mohammed Daoud, Rachid Ahl Laamara , “Simple scheme
for implementing the Grover search algorithm with superconducting qubits,”
Journal of Physics B: Atomic, Molecular and Optical Physics 54, 175501 (2021).

$\bullet$ Hamid Sakhouf, Mohammed Daoud, Rachid Ahl Laamara , “Implementation
of Grover’s Search Algorithm in the QED Circuit for Two Superconducting
Qubits,” International Journal of Theoretical Physics 59, 3436–3448 (2020).

###### List of Figures

  1. I.1 Bloch sphere representation of a qubit.
  2. I.2 Quantum circuit representation for CNOT gate. Here $x,y=0$ or $1$ and $\oplus$ denotes the addition modulo2, while the vertical line between the control and target qubit represents the circuit diagram of CNOT gate.
  3. I.3 Quantum circuit for creating the Bell state.
  4. I.4 Quantum circuit for creating the GHZ state.
  5. I.5 Quantum circuit for evaluating $f(x)$. ${U_{f}}$ is the quantum circuit which transforms input states ${\left|x\right\rangle_{n}}{\left|y\right\rangle_{m}}$ to ${\left|x\right\rangle_{n}}{\left|{y\oplus f\left(x\right)}\right\rangle_{m}}$.
  6. I.6 Evolution of amplitudes for each state during the Grover search algorithm.
  7. I.7 Geometric representation of Grover’s algorithm for a single Grover iteration, $G$.
  8. I.8 Quantum circuit implementing the general Deutsch-Jozsa algorithm.
  9. II.1 Schematic of a Josephson tunnel junction. It consists of two superconductors separated by an insulating barrier. The two sides of the junction are characterized by the number of Cooper pairs ${{n_{1}}}$, ${{n_{2}}}$ and the phases ${{\theta_{1}}}$, ${{\theta_{2}}}$. The charge ${n_{2}}-{n_{1}}$ and phase difference ${\theta_{2}}-{\theta_{1}}$ between the two metals are the essential parameters describing the behavior of the junction.
  10. II.2 Schematic of a Josephson tunnel junction. (a) The symbol of the Josephson junction (Cross in a box) .(b) The equivalent electrical circuit of the Josephson junction.
  11. II.3 Schematic of a SQUID. The two Josephson junctions form a loop, which is crossed by a flux magnetic field.
  12. II.4 a) Schematic of the Cooper pair box consisting of a Josephson junction (blue) which connects an island (green) with a reservoir (orange). b) The Josephson junction is replaced by a SQUID in which the qubit becomes more controllable by a magnetic flux.
  13. II.5 a) Circuit diagram corresponding to the Cooper pair box schematic in Fig. (II.4a). b) Circuit diagram corresponding to the Cooper pair box schematic in Fig. (II.4b).
  14. II.6 Energy spectrum of the charge qubit for the ratio as a function of gate charge $N_{g}$. The ${E\mathord{\left/{\vphantom{E{{E_{C}}}}}\right.\kern-1.2pt}{{E_{C}}}}$ is lower, the first two levels are isolated from the third for values of $N_{g}$ near the half-integer. a) the bold curve designates the energy spectrum of the Hamiltonian with $E_{C}$ = 0.2, the fine curve with ${E\mathord{\left/{\vphantom{E{{E_{C}}}}}\right.\kern-1.2pt}{{E_{C}}}}=0$. b) the energy spectrum for $0\leq{N_{g}}\leq 1$.
  15. II.7 The equivalent circuit of the Transmon qubit, an additional capacitance (Cs) has been added in parallel to the junction.
  16. II.8 Eigenenergies for three energy levels ($k=0,1,2$) of the Cooper pair box Hamiltonian (II.31) for different ratios ${{{E_{J}}}\mathord{\left/{\vphantom{{{E_{J}}}{{E_{C}}}}}\right.\kern-1.2pt}{{E_{C}}}}$. For very large ${{{E_{J}}}\mathord{\left/{\vphantom{{{E_{J}}}{{E_{C}}}}}\right.\kern-1.2pt}{{E_{C}}}}$, the spectrum practically becomes that of the harmonic oscillator. The system gets less sensitive to gate charge when the ${{{E_{J}}}\mathord{\left/{\vphantom{{{E_{J}}}{{E_{C}}}}}\right.\kern-1.2pt}{{E_{C}}}}$ is increased from the charging regime with ${{{E_{J}}}\mathord{\left/{\vphantom{{{E_{J}}}{{E_{C}}}}}\right.\kern-1.2pt}{{E_{C}}}}=0.5$ (a) to the transmon regime with ${{{E_{J}}}\mathord{\left/{\vphantom{{{E_{J}}}{{E_{C}}}}}\right.\kern-1.2pt}{{E_{C}}}}=50$ (c). The figure used from Ref.[66].
  17. II.9 QED cavity architecture includes an electromagnetic field mode of decay rate $\kappa$ coupled to a two-level system of decay rate $\gamma$ and the transition time ${t_{transit}}$ with a coupling force $g$.
  18. II.10 An electromagnetic cavity of length $L$, the electric field must therefore be zero at the extremities of the cavity.
  19. II.11 Two level atom.
  20. II.12 Energy diagram in the resonant case.
  21. II.13 Energy diagram in the dispersive regime.
  22. II.14 Energy diagram in the dispersive regime.
  23. II.15 Representation of the transmission line by a set of inductors in parallel with capacitors.
  24. II.16 Representation of an inductor in parallel with the capacitor.
  25. II.17 Circuit QED for the transmon qubit: (a) The transmon qubit is coupled to the $LC$ resonator through a gate capacitance $C_{d}$. The two Josephson junctions, with capacitance $C_{J}$ and Josephson energy $E_{J}$, are in parallel with an additional large capacitance $C_{B}$. (b) The transmon device consists of a transmon qubit coupled to a transmission line. The figure used from Ref.[43].
  26. III.1 Electrical circuit diagram of a three-qubit device[119].
  27. III.2 (Color online) (a) Fidelity of the two-qubit $X$-rotation gate as a function of the decay rate of resonator $\kappa$ (a) and as a function of the coherence time $T$ (b). The blue solid line means that our calculation was performed by using the Hamiltonian (III.7) for $j$=1,2, while the calculation referred to by the solid line was performed using the Hamiltonian (III.11) for $j$=1,2. The system parameters used in the numerical simulation are mentioned in the text.
  28. (a)
  29. III.3 (Color online) (a) Fidelity of the three-qubit $X$-rotation gate for different values of the cavity decay rate $\kappa$ (a) and initial cavity Fock states $\left|n\right\rangle$ (b). The blue solid line means that our calculation was performed by using the Hamiltonian (III.7) for $j$=1,2,3, while the calculation referred by the solid line was performed using the Hamiltonian (III.11) for $j$=1,2,3. The system parameters used in the numerical simulation are the same as a figure (III.2) while the others are mentioned in the text.
  30. (a)
  31. III.4 (Color online) Simulated measurements of the single-shot entangling gate (Eq.III.72) (a) The occupation probabilities versus the interaction time. The system parameters used in the numerical simulation are ${g\mathord{\left/{\vphantom{g{2\pi}}}\right.\kern-1.2pt}{2\pi}}=40$ MHz, $2{\Omega_{R}}=9g$, ${T_{c}}=20\mu s$, and ${\kappa\mathord{\left/{\vphantom{\kappa{2\pi=1.5}}}\right.\kern-1.2pt}{2\pi=1.5}}$ MHz. (b) Fidelity versus the number of the two-qubit entangling gates with environment decoherence under the Hamiltonian in equation (III.11) and without environment decoherence under the following Hamiltonian ${H_{2I}}=2{\Omega_{R}}{S_{x}}+2\lambda S_{x}^{2}$. The data parameters used are the same as in panel (b).
  32. (a)
  33. III.5 (Color online) (a) Quantum circuit for generating $GHZ$-state using the single-shot entangling gate(Eq.III.72) (b) The occupation probabilities as a function of the operation time ${T_{m}}$[132]. The system parameters used here are ${g\mathord{\left/{\vphantom{g{2\pi}}}\right.\kern-1.2pt}{2\pi}}=20$ MHz, $2{\Omega_{R}}=9g$, ${T_{c}}=20\mu s$, and ${\kappa\mathord{\left/{\vphantom{\kappa{2\pi=1.5}}}\right.\kern-1.2pt}{2\pi=1.5}}$ MHz.
  34. (a)
  35. III.6 (Color online) (a) Numerical simulation of the fidelity as a function of the number of entangling gates under both Hamiltonians in equation (III.11) and ${H_{3I}}$. (b) A number of entangling operations of the error are caused by decoherence. $\Delta F$ denotes the difference of the fidelity for generating $GHZ$ state with and without dissipation[132]. The data parameters used are the same as in figure (III.5a).
  36. (a)
  37. IV.1 Schematic circuit of a three-qubit Grover’s algorithm[128].
  38. IV.2 (Color online) The probability of finding the state after the first (a) and second (b) Grover’s iteration.
  39. (a)
  40. IV.3 Schematic circuit of Grover’s algorithm[128].
  41. IV.4 Schematic circuit of the two-qubit Grover algorithm[128].
  42. IV.5 Numerical results for the fidelities of two-qubit in Grover algorithm for the four possible oracles as the function $b=2\lambda t$. These plots are drawn for ${{{\Omega_{R}}}\mathord{\left/{\vphantom{{{\Omega_{R}}}\lambda}}\right.\kern-1.2pt}\lambda}=13.5$[128].
  43. IV.6 Schematic circuit of the three-qubit Grover algorithm[119].
  44. IV.7 Numerical results of the algorithm fidelity for the desired state $\left|{{1}{1}{1}}\right\rangle$ versus $b=2\lambda t$ for different values of $h={{{\Omega_{R}}}\mathord{\left/{\vphantom{{{\Omega_{R}}}\lambda}}\right.\kern-1.2pt}\lambda}$. Other parameters of the system are referred to in the text. (a) Fidelity ${F_{1}}$ for $O=c{P_{000}}$. (b) Fidelity ${F_{2}}$ for $O=c{P_{001}}$,$c{P_{010}}$,$c{P_{101}}$.(c) Fidelity ${F_{3}}$ for $c{P_{011}}$,$c{P_{100}}$,$c{P_{110}}$. (d) Fidelity ${F_{4}}$ for $O=c{P_{111}}$.
  45. IV.8 $QPT$ of the single-shot two-qubit entangling gate. Absolute values of the matrix elements of the ${\chi_{ideal}}$ (a) and the ${\chi_{sim}}$ (b). The matrix elements are shown in the operator basis $\left\\{{I\otimes I,I\otimes X,I\otimes Y,...,Z\otimes X,Z\otimes Y,Z\otimes Z}\right\\}$, where $I$ is identity matrix and $\left\\{{X,Y,Z}\right\\}$ are the Pauli operators $\left\\{{{\sigma_{x}},-i{\sigma_{y}},{\sigma_{z}}}\right\\}$.
  46. (a)
  47. IV.9 $QPT$ of the single-shot entangling gate for three qubits. Absolute values of the matrix elements of the ${\chi_{ideal}}$ (a) and the ${\chi_{sim}}$ (b). The matrix elements are shown in the operator basis $\left\\{{I\otimes I\otimes I,I\otimes I\otimes X,...,Z\otimes Z\otimes Y,Z\otimes Z\otimes Z}\right\\}$, where $I$ is identity matrix and $\left\\{{X,Y,Z}\right\\}$ are the Pauli operators $\left\\{{{\sigma_{x}},-i{\sigma_{y}},{\sigma_{z}}}\right\\}$.
  48. (a)
  49. IV.10 Performance of the single-shot entangling gate for three systems. Numerical simulation of the gate fidelity $F_{mean}$ versus the coherence times $T\geq 0.6\mu s$. Assuming that $T=T_{1}=T_{2}$, while the other system parameters are the same as in figure III.6.

Liste of figures

## List of Abbreviations

QIP | Quantum information processing
---|---
NMR | nuclear magnetic resonance
QED | quantum electrodynamics
CP or C-phase | conditional phase
GHZ | Greenberger-Horne-Zeilinger
CNOT | Controlled Not
QPT | Quantum process tomography
SQUID | superconducting quantum interference device
RWA | rotating wave approximation

## List of Symbols

iSWAP |  i-swap gate.
---|---
X, Y, Z |  single-qubit Pauli operator, also defned as ${\sigma_{x,y,z}}$.
$\sqrt{iSWAP}$ |  square-root of i-swap gate.
${R_{i}}\left(\theta\right)$ |  rotation around the axis $i$ by angle $\theta$.
$H$ |  Hadamard gate.
$Toffoli$ |  controlled-controlled-X gate.
$Fredkin$ |  controlled SWAP operation.
${I_{c}}$ |  the critical current of the junction
${\Phi_{0}}$ |  the magnetic flux quantum
$L$ |  the Lagrangian of a Josephson junction
${E_{C}}$ |  the charging energy of a Cooper pair
${E_{J}}$ |  the Josephson tunneling energy
$p$ |  the canonical momentum operator
${U_{T}}$ |  the potential energy of the SQUID
${C_{J}}$ |  capacitance of the Josephson junction
$\alpha$ |  anharmonicity of transmon qubit
${\omega_{q}}$ |  the qubit transition frequency
$g$ |  vacuum Rabi coupling frequency
${\omega_{r}}$ |  frequency of the resonator
${T_{1}}$ |  qubit relaxation time
${T_{2}}$ |  qubit dephasing time
${\Omega_{R}}$ |  Rabi frequency
${\omega_{d}}$ |  microwave feld frequency
$\bigotimes,\quad\bigoplus$ |  tensor product, direct sum

###### Contents

  1. Dedication
  2. Acknowledgment
  3. Résumé
  4. Abstract
  5. Résumé détaillé
  6. List of publications
  7. List of Abbreviations
  8. List of Symbols
  9. General Introduction
  10. I Fundamental concepts of quantum information processing and computation
    1. I.1 Quantum bit
      1. I.1.1 Definition
      2. I.1.2 Many qubits or quantum register
    2. I.2 Quantum gates and circuits
      1. I.2.1 Single-qubit gates
    3. I.2.2 Multi-qubit gates and universilaty
    4. I.2.3 Quantum circuit for two- and three-qubit entanglement
    5. I.3 Quantum algorithms
      1. I.3.1 Quantum parallelism needed in an algorithm
      2. I.3.2 Grover quantum algorithm
    6. I.3.3 Deutsch-Jozsa quantum algorithm
    7. I.4 Chapter summary
  11. II Quantum superconducting circuits
    1. II.1 Superconducting qubits
      1. II.1.1 The Josephson Junction
    2. II.1.2 SQUID
    3. II.1.3 Charge qubit( The Cooper pair box)
    4. II.1.4 Transmon qubit
    5. II.2 Cavity quantum electrodynamics
      1. II.2.1 Quantification of the field in a cavity
      2. II.2.2 Jaynes-Cummings model
      3. II.2.3 Approximation of the dipole moment and the rotating wave
      4. II.2.4 Resonance regime and dispersive regime
    6. II.3 Circuit quantum electrodynamics
      1. II.3.1 Quantum harmonic LC oscillator
      2. II.3.2 Transmission line (Coplanar waveguide)
      3. II.3.3 Coupling a transmon-type superconducting qubit to a coplanar waveguide resonator
    7. II.4 Quantum gates with superconducting device
      1. II.4.1 Single-qubit gates
      2. II.4.2 Two-qubit gates
    8. II.5 Chapter summary
  12. III Realizing two- and multi-qubit quantum gates in open quantum systems via QED crcuit for superconducting qubits
    1. III.1 Multi-qubit device description
      1. III.1.1 Multi-qubit device
      2. III.1.2 Hamiltonian description and the corresponding time evolution operator
    2. III.2 Open system dynamics
      1. III.2.1 Master equation
      2. III.2.2 Fidelity under master equation
    3. III.3 Realizing $X$-rotation gate for multi-qubit in the open quantum systems
      1. III.3.1 $X$-rotation gate for two transmon systems
        1. III.3.1.1 Two-qubit $X$-rotation gate generation
        2. III.3.1.2 Fidelity and discussion
      2. III.3.2 $X$-rotation gate for three transmon systems
        1. III.3.2.1 Three-qubit $X$-rotation gate generation
        2. III.3.2.2 Fidelity and discussion
    4. III.4 Single-shot entangling gate in the open quantum systems
      1. III.4.1 Entangling gate for two transmon systems
        1. III.4.1.1 Two-qubit entangling gate generation
        2. III.4.1.2 Two-qubit entangling gate characterization
      2. III.4.2 Entangling gate for three transmon systems
        1. III.4.2.1 Three-qubit entangling gate generation
        2. III.4.2.2 Three-qubit entangling gate characterization
    5. III.5 Chapter summary
  13. IV Implementing $X$-rotation gates in Grover’s algorithm and quantum process tomography of the entangling gates
    1. IV.1 Grover’s algorithm with X-rotation gates
      1. IV.1.1 Three-Qubit Grover’s Search Algorithm
      2. IV.1.2 Grover’s Search Algorithm for $n$-qubit
    2. IV.2 Implementing the $X$-rotation gate in Grover’s search algorithm
      1. IV.2.1 Grover’s algorithm for a two-qubit $X$-rotation gate
      2. IV.2.2 Grover’s algorithm for three-qubit $X$-rotation gate
    3. IV.3 Characterization of the single-shot entangling gate by quantum process tomography
      1. IV.3.1 Quantum process tomography (QPT)
      2. IV.3.2 Two-qubit entangling gate characterization
      3. IV.3.3 Three-qubit entangling gate characterization
    4. IV.4 Chapter summary
  14. V General Conclusion and Future Directions

Contents

##  Chapter General Introduction

> ”Computers are physical objects, and computations are physical processes.
> What computers can or cannot compute is determined by the laws of physics
> alone, and not by pure mathematics.”
>
> David Deutsch

Quantum information is an emerging field that includes several specialties
such as physics, engineering, chemistry, computer science, and mathematics.
The main aim of this integration is to know how the realization of a quantum
computer and the use of such a computer to perform specific calculations are
much faster than with a computer operating in a standard way (classical
computer)[1, 2]. It captures all of the operations regarding quantum
information processing exploiting the properties of quantum mechanics such as
the superposition of states, entanglement, and interference.

The important general idea of quantum information processing (QIP) is to
realize a quantum device- so-called a quantum computer which has started just
about the last century, exactly in the early 1980s when Feynman showed that a
classical machine would be unable to efficiently simulate a quantum system[3],
where the idea is that a classical bit can be replaced by a two-level quantum
system. Similar to a classical machine which can store its information in
strings of bits that can either be $0$ or $1$, a quantum computer uses a unit
of quantum information called a quantum bit or qubit, which is a two-level
quantum system. Still, unlike in classical physics, the qubit can exist in a
superposition of its computational basis states $\left|0\right\rangle$ and
$\left|1\right\rangle$ which can be formed by the ground and excited states of
an electron in an atom for example, which represents the phenomenon of quantum
superposition. Some examples mentioned in the physical realization are spin,
photons, ion traps, quantum dots, nuclear magnetic resonance (NMR),
superconducting qubits, etc.

Quantum information processing and computation is an application of quantum
information that includes quantum gates, quantum algorithms, quantum error-
correcting codes, quantum cryptography, etc, and represents a revolutionary
field of information processing. The first simple quantum algorithm using
quantum mechanics to solve essentially the determining problem of whether a
coin is fair or biased more efficiently than any classical algorithm was
proposed by David Deutsch in 1985[4]. In addition, Peter Shor 1994 invented an
efficient quantum algorithm that factorizes a large integer number[5], which
is exponentially more efficient than its classical counterpart in terms of
time. Another famous algorithm worth mentioning in this area was proposed by
Lov Grover in 1996[6], addresses searching in an unsorted large database which
is a hard mathematical problem in the classical approach. The computation time
of this search algorithm equals the square root of the fastest classical
algorithm time.

In the following, in the context of $QIP$ the quantum algorithms have been
demonstrated with many physical systems. For instance, the Deutsch-Jozsa
algorithm has been realized in NMR[7, 8, 9, 10, 11, 12], trapped ion[13],
photonic[14] systems, and superconducting devices[15, 16, 17], as well as for
further challenging Shor’s algorithm has only been demonstrated for the
factorization of the number $15$ with NMR techniques[18] and photons [19, 20,
21]. In the last decades, following the context of the building blocks of a
quantum device, the experimental realization of quantum error correction
protocols has been made in NMR[22], and then in linear optics [23], trapped
ions[24, 25], and also superconducting circuits[26, 27, 28].

Another interesting quantum algorithm in terms of physical realizations and
considered a good candidate for demonstrating the power of quantum information
processing and computation in view of its being easy to implement is Grover’s
algorithm. Grover’s algorithm is known as a search algorithm, which is a
quantum method that is used to search through unstructured databases using
iteration steps that are fewer than classical algorithms to determine the
unique input to a black box function (quantum Oracle) by increasing its
probability that it will be measured. This algorithm has been theoretically
and experimentally realized with its two- and multi-qubit case using the NMR
system[29, 30], trapped ions[31, 32, 33, 34], cavity quantum electrodynamics
(cavity QED)[35, 36, 37, 38]. Recently, Grover’s search algorithm was
experimentally realized using a scalable trapped atomic ion system for three
qubits and it was executed for one iteration[39].

In these physical systems such as NMR, trapped ions, and photonic, the
properties of the qubits are naturally occurring. In contrast, the
superconducting qubits are made up of artificial electronic structures that
consist of several circuit components, which are capacitors, inductors, and
the Josephson junction characterized by its inductance and capacitance.
Different of these elements allow the designing and engineering of their
properties to a large extent, such as the energy level structure and the
coupling of the qubits to their environment. Whereas some of these properties
can be tuned using magnetic fields. Several types of superconducting qubits
were proposed in different configurations using Josephson junctions such as
flux qubit[40], phase qubit[41], charge qubit[42], as well as the transmon
qubit as an interesting case in this thesis[43], which is a modified version
of the charge qubit.

It is worth noticing that the experimental realization for two-qubit Grover’s
algorithm has been demonstrated using a quantum device[16], including
transmon-type superconducting qubits coupled to a quantum electrodynamics
circuit (circuit QED). Circuit QED setup as proposed by Blais et al. [44] and
then realized by Wallraff et al. in the experiment[45], studying the
interaction between single microwave photons and superconducting qubits
behaving as artificial atoms, and a similar version of cavity quantum
electrodynamics (cavity QED), which is the fundamental coupling between a
photon and a two-level system. Additionally, many experimental realizations
have been successfully made with superconducting qubits in QED circuit in the
context of QIP: demonstration of a single qubit gate[46], generation and
measurement of two- and multi-qubit entanglement[48, 49, 50, 51, 52],
realizing multi-qubit quantum error correction[26, 27, 28].

Entanglement is the most interesting aspect of quantum theory[67], and is an
essential resource for QIP. Therefore, the creation and demonstration of
maximally entangled states are central for quantum communication and QIP, in
particular, a few examples that require entangling quantum systems in Bell and
Greenberger-Horne-Zeilinger (GHZ) states, quantum computation[1, 68], quantum
error correction[69, 70], quantum sense[71], and quantum metrology[72]. In the
past decade, significant progress has been witnessed in entanglement
engineering with superconducting devices: the preparation and measurement of
two-qubit entangled state(Bell states) based on the combination of single
rotation and controlled-phase gates, was realized by L. DiCarlo et al. in the
experiment[16]. In addition, the preparation of W- and GHZ-maximally-entangled
states for three and more qubits is based on the decomposition into sequences
of a single rotation and two-qubit controlled phase or iSWAP gates which were
realized in the experiments[48, 49, 50].

To fully characterize the performance of these quantum algorithms and gates,
it is necessary to use quantum process tomography (QPT)[1, 73, 74]. QPT is an
essential tool for reliable QIP, it allows us to predict the evolution of a
quantum state as it propagates through an imperfect quantum gate. In real
experiments, where the interaction between the quantum system and the
environment cannot be neglected, it is not possible to describe the evolution
of a quantum state by a unitary operator. Thus, QPT is a useful method for
experimentally describing a complete implementation of quantum gates. This
method is used to characterize the systems of one- and two-qubit quantum gates
in[75, 76, 77, 78, 79, 80, 81]. QPT has been used with superconducting systems
for the two-qubit entangling gate to characterize a square root i-SWAP
gate[62], and for three-qubit operations to characterize Toffoli and C-Phase
gates[64, 26].

Although the building blocks of quantum algorithms are constructed using basic
quantum logic gates as single- and two-qubit gates[82, 83, 84, 85, 86, 87],
however, building a multiqubit gate using only quantum basic gates will be
difficult owing to the number of basic gates drastically increasing with the
growing number of qubits. Therefore, the use of the direct implementation of a
multiqubit gate offers more efficiency than the combination of basic gates. In
addition, the realization of a multi-qubit gate using only a one-step
operation replaces the complex combination of one- and two-qubit gates. Over
the past years, using the direct implementation approach of the multiqubit
gate has drawn much interest. In this context, single-shot implementation of a
Toffoli gate of three qubits has been experimentally demonstrated in different
physical systems[88, 89, 91]. In addition, based on superconducting qubits
coupled to a resonator in circuit QED, many schemes have been previously
proposed for the one-step realization of multi-qubit gates[92, 93, 94, 95, 96,
97, 98, 99, 100, 101, 102, 103].

#### Outline of the thesis

This thesis focuses on the realization, and characterization of quantum gates
and algorithms using a one-step approach based on the superconducting quantum
device. Such a quantum device is constructed by transmon-type superconducting
qubits coupled to a superconducting resonator, driven by a classical field.
Using it, we propose a fast scheme to prepare a quantum register involving
many of the basic states in quantum superposition by realizing the
$X$-rotation gate for two and more qubits requiring only one-step operation.
In addition, using the same approach we realize the single-shot entangling
gate for two and more qubits, these gates are used to create a maximally
entangled state such as a two-qubit Bell-state and a three-qubit GHZ-state.
Using strictly numerical simulation under realistic assumptions about system
parameters, we investigate the quantum dynamics of these gates and show that
these schemes can be implemented with high fidelity.

Using the implemented x-rotation gates, we present an alternative way to
implement the two- and three-qubit Grover’s algorithm, which does not require
a series of single gates. Our implementation showed that the proposed scheme
allows for efficiently finding the correct state with a high $\succ 90\%$
fidelity. As well, we demonstrate by numerically simulating the use of QPT to
fully characterize the performance of a single-shot entangling gate for two
and more qubits and obtaining process fidelity greater than 93$\%$.

The structure of this thesis is organized as follows:

In the first chapter I, we will briefly discuss some of the basics of quantum
information processing and computing. This includes the mathematical and
geometric properties of quantum bits as well as the description of how they
can be manipulated and how a quantum computer differs from its classical
counterpart. We will also identify a universal set of quantum logic gates,
involving single- and multi-qubit gates, and how they contribute to building
quantum algorithms, for instance, the Grover and Deutsch Jozsa algorithms,
quantum entanglement protocols, and the preparation of superposition states.

In the second chapter II, we will discuss the physical implementation of
quantum bits using electrical circuits to realize superconducting qubits, and
we will review some types of these qubits such as the Cooper-pair box (charge
qubit), and the transmon qubit which is used in this thesis, as well as the
quantum harmonic $LC$ oscillator as an essential component in circuit QED.
There will also be a description about the interaction between superconducting
qubits and a superconducting transmission line or LC oscillator ( the
electromagnetic field) inside a circuit QED that is a similar version to a
cavity QED which is also discussed. As well, the dispersive regime needed for
QIP is discussed. Finally, we will conclude this chapter by associating
circuit QED (Quantum Electrodynamics) with QIP. In this section, we will
identify the single-qubit gates as well as C-Phase and iSWAP gates, which have
been experimentally realized using superconducting devices[16, 62].

Chapter III of this thesis is aimed to realizing an $X$-rotation gate for two
and three qubits which can be used for preparing superposition states and
others called entangling gates used for generating maximally entangled states
such as Bell and GHZ states, and thier implementation is based on a quantum
device comprising transmon qubits capacitively coupled with a resonator driven
by a microwave field using a one-step approach. The robustness of these gates
is demonstrated by including the effect of the decoherence of the transmon
systems and the resonator decay in a master equation formalism. For realistic
assumptions about the transmon-resonator system parameters, high fidelity will
be achieved on a quantum simulation. In the end, we will discuss the
experimental feasibility of the proposed schemes and compare them with the
latest experimental results within superconducting devices.

In the following, based on the implemented X-rotation gates in chapter III and
C-Phase gate we will propose practical and faster schemes for implementing a
two- and three-qubit Grover’s search with an implementation time signifcantly
shorter than the coherences time of superconducting qubits in chapter IV.
There will also be a brief discussion about some thoughts regarding the the
possibility of experimental realization of the proposed schemes. In addition
to the characterization of the single-shot entatngling gates for two and three
qubits in the chapter III we here fully characterize these gates using QPT
demonstrated by nemurical simulation and obtaining process fidelities greater
than 93$\%$.

Finally, this thesis will end with a conclusion and perspectives.

##  Chapter I Fundamental concepts of quantum information processing and
computation

This chapter is intended to briefly give the fundamental concepts of quantum
computation based on the book of Nielsen and Chuang[1], including the
mathematical and geometric properties of a qubit or a quantum bit. In
addition, it describes the quantum gates for single- and multi-qubit, as well
as the discussion of universal quantum gates needed to accomplish the
arbitrary quantum computation. At the end of this chapter, we also discuss
quantum algorithms exploiting quantum superposition and entanglement to
perform specific calculations. We first discuss the quantum parallelism, which
characterizes a method allowing quantum computers to evaluate a function f(x)
for a large number of x-values simultaneously. We will then use quantum
parallelism and other laws of quantum mechanics to move on to discuss some
quantum algorithms such as Grover and the Deatsch-Jozsa algorithm.

###  I.1 Quantum bit

In a classical computer, the fundamental unit of information is the so-called
binary digit or bit and can take one of two possible values $0$ or $1$. These
values physically correspond for instance to whether a capacitor is being
charged or discharged. Similarly, in a quantum computer two-level quantum
systems can form the quantum bit or qubit.

####  I.1.1 Definition

The fundamental unit of quantum information is the quantum bit or, more
simply, qubit, which can be in states $\left|0\right\rangle$ and
$\left|1\right\rangle$. These quantum states can be formed by any quantum
system with two different energy levels, for example of an appropriate system
is the spin of an electron in a static magnetic field. Unlike classical
computing in which the bit must be either 0 or 1, the qubit can be prepared in
what is called a ” quantum superposition” of both states
$\left|0\right\rangle$ and $\left|1\right\rangle$ at the same time, in quantum
computing. Mathematically, such a superposition state is represented by

$\left|\psi\right\rangle=\alpha\left({\begin{array}[]{*{20}{c}}1\\\
0\end{array}}\right)+\beta\left({\begin{array}[]{*{20}{c}}0\\\
1\end{array}}\right),$ (I.1)

where the numbers $\alpha$ and $\beta$ are complex numbers, which satisfy the
normalization condition
${\left|\alpha\right|^{2}}+{\left|\beta\right|^{2}}=1$. While the orthonormal
basis states $\left|0\right\rangle$ and $\left|1\right\rangle$ are known as
computational basis states. The measurement outcome of the qubit is either
$0$, with the probability ${\left|\alpha\right|^{2}}$ or $1$ , with the
probability ${\left|\beta\right|^{2}}$.

Figure I.1:  Bloch sphere representation of a qubit.

It is useful to geometrically represent the qubit state for a better
understanding of the action of quantum operations on the state. To find such a
visual interpretation, without losing any generality we may rewrite equation
(I.1) as

$\left|\psi\right\rangle=\cos\left({\frac{\theta}{2}}\right)\left|0\right\rangle+{e^{i\phi}}\sin\left({\frac{\theta}{2}}\right)\left|1\right\rangle,$
(I.2)

where $\theta\in\left[{0,\pi}\right]$ and $\phi\in\left[{0,2\pi}\right]$. The
ground and excited states $\left|0\right\rangle$ and $\left|1\right\rangle$
can represent the South and North poles of a sphere, respectively, and thus
each point on this so-called Bloch sphere represents a superposition state
(see figure I.1).

####  I.1.2 Many qubits or quantum register

A classical register is a set of $n$ bits that, in a sense, stores all numbers
between $0$ and ${2^{n}}-1$. A quantum register, like a classical register,
just is a system comprising many qubits. In the case of three quantum bits,
this yields the computational basis states involving the eight possible states
$\left\\{{\left|{000}\right\rangle,\left|{001}\right\rangle,\left|{010}\right\rangle,\left|{011}\right\rangle,\left|{100}\right\rangle,\left|{101}\right\rangle,\left|{110}\right\rangle,\left|{111}\right\rangle}\right\\}$
(where
$\left|{ijk}\right\rangle=\left|i\right\rangle\otimes\left|j\right\rangle\otimes\left|k\right\rangle$).
Like a single qubit, the three-qubit state is given by the linear combination
of the eight basis states

$\displaystyle\left|\psi\right\rangle$ $\displaystyle=$
$\displaystyle{\alpha_{000}}\left|{000}\right\rangle+{\alpha_{001}}\left|{001}\right\rangle+{\alpha_{010}}\left|{010}\right\rangle+{\alpha_{011}}\left|{011}\right\rangle+{\alpha_{100}}\left|{100}\right\rangle$
(I.3) $\displaystyle+$
$\displaystyle{\alpha_{101}}\left|{101}\right\rangle+{\alpha_{110}}\left|{110}\right\rangle+{\alpha_{111}}\left|{111}\right\rangle,$

where ${{\alpha_{ijk}}}$ are the complex numbers which satisfy the
normalization condition
${\sum\nolimits_{ijk}{\left|{{\alpha_{ijk}}}\right|}^{2}}=1$
($ijk=000,001,010,011,100,101,110,111$). An important property of qubits is
that they can be entangled with other qubits. Three quantum systems can be
entangled if the state of the total system cannot be written as a Kronecker
product of its parts. In the context of quantum computing, three qubits can be
entangled in two different ways by the states

$\left|{GHZ}\right\rangle=\frac{1}{{\sqrt{2}}}\left({\left|{000}\right\rangle+\left|{111}\right\rangle}\right),$
(I.4)
$\left|W\right\rangle=\frac{1}{{\sqrt{3}}}\left({\left|{001}\right\rangle+\left|{010}\right\rangle+\left|{100}\right\rangle}\right),$
(I.5)

###  I.2 Quantum gates and circuits

Besides storing information, we also mention that the processing of quantum
information is useful to build quantum computing, in order to move information
from one qubit to another. Such information processing is carried out by the
action of quantum gates on qubits or quantum registers. In analogy to
classical computing which through in it the circuit model is very useful for
computing processes and is also used to design and construct computing
hardware using different types of classical logic gates acting on some binary
input, for example, the NAND, NOR, AND, and OR gates, whereas quantum
computers can be interpreted in terms of a quantum circuit model comprising
quantum gates, which are applied to quantum register. In quantum computers, we
have two different types of quantum gates: single qubit and multi-qubit gates.
Single qubit gates operate only on one qubit, while multi-qubit gates operate
on two or more qubits. We will give examples of these types of operations in
detail in the rest of this section.

####  I.2.1 Single-qubit gates

An n-qubit gate can be represented as ${2^{n}}\times{2^{n}}$ unitary matrices.
Due to the single-qubit gate operating only on one qubit, thus single-qubit
gates can be represented by matrices of size ${2}\times{2}$.

### Rotation gates

Generally, the single quantum operation can be visualized by using the Bloch
sphere (shown in fgure I.1) as a rotation of the state around the x,y, and z
axes. Some of the single qubit gates which are used to generate the rotation
about the Bloch sphere axes can be described using Pauli matrices as follows:

$\displaystyle X={\sigma_{x}}=\left({\begin{array}[]{*{20}{c}}0&1\\\
1&0\end{array}}\right),$ (I.8) $\displaystyle
Y={\sigma_{y}}=\left({\begin{array}[]{*{20}{c}}0&{-i}\\\
i&0\end{array}}\right),$ (I.11) $\displaystyle
Z={\sigma_{z}}=\left({\begin{array}[]{*{20}{c}}1&0\\\
0&{-1}\end{array}}\right).$ (I.14)

When we apply these single-qubit gates to the computational basis states
$\left|0\right\rangle$ and $\left|1\right\rangle$, we easily get

$\displaystyle
X\left|0\right\rangle=\left|1\right\rangle,X\left|1\right\rangle=\left|0\right\rangle,$
(I.15) $\displaystyle
Y\left|0\right\rangle=i\left|1\right\rangle,Y\left|1\right\rangle=-i\left|0\right\rangle,$
(I.16) $\displaystyle
Z\left|0\right\rangle=\left|0\right\rangle,Z\left|1\right\rangle=-\left|1\right\rangle.$
(I.17)

Besides these rotation operations, we would like to mention the other
important single-qubit gates, which are also rotation operations, which are
given by

$\displaystyle{R_{x}}\left(\theta\right)={e^{-i\theta{{{\sigma_{x}}}\mathord{\left/{\vphantom{{{\sigma_{x}}}2}}\right.\kern-1.2pt}2}}}=\left({\begin{array}[]{*{20}{c}}{\cos\left({\frac{\theta}{2}}\right)}&{-i\sin\left({\frac{\theta}{2}}\right)}\\\
{-i\sin\left({\frac{\theta}{2}}\right)}&{\cos\left({\frac{\theta}{2}}\right)}\end{array}}\right),$
(I.20)
$\displaystyle{R_{y}}\left(\theta\right)={e^{-i\theta{{{\sigma_{y}}}\mathord{\left/{\vphantom{{{\sigma_{y}}}2}}\right.\kern-1.2pt}2}}}=\left({\begin{array}[]{*{20}{c}}{\cos\left({\frac{\theta}{2}}\right)}&{-\sin\left({\frac{\theta}{2}}\right)}\\\
{\sin\left({\frac{\theta}{2}}\right)}&{\cos\left({\frac{\theta}{2}}\right)}\end{array}}\right),$
(I.23)

$\displaystyle{R_{z}}\left(\theta\right)={e^{-i\theta{{\sigma
z}\mathord{\left/{\vphantom{{\sigma
z}2}}\right.\kern-1.2pt}2}}}=\left({\begin{array}[]{*{20}{c}}{{e^{-i{\theta\mathord{\left/{\vphantom{\theta
2}}\right.\kern-1.2pt}2}}}}&0\\\
0&{{e^{i{\theta\mathord{\left/{\vphantom{\theta
2}}\right.\kern-1.2pt}2}}}}\end{array}}\right),$ (I.26)

where $\theta$ is the angle of rotation. Based on these gates, we can identify
some other important single-qubit operations just by performing rotations of
$\theta=\mp\pi$ and $\theta=\mp{\pi\mathord{\left/{\vphantom{\pi
2}}\right.\kern-1.2pt}2}$, often mentioned as $\pi$-pulses and
$\mp{\pi\mathord{\left/{\vphantom{\pi 2}}\right.\kern-1.2pt}2}$-pulses,
respectively. We can identify the Pauli matrices by choosing $\theta=\pi$,
${R_{x}}\left(\pi\right)=-i{\sigma_{x}}$,
${R_{y}}\left(\pi\right)=-i{\sigma_{y}}$,
${R_{z}}\left(\pi\right)=-i{\sigma_{z}}$. Similarly, when choosing
$\theta={\pi\mathord{\left/{\vphantom{\pi 2}}\right.\kern-1.2pt}2}$ in the
equations (I.20) and (I.23), these operations map a state initially prepared
in the ground ($\left|0\right\rangle$) or an excited ($\left|1\right\rangle$)
state into an equal superposition of the states $\left|0\right\rangle$ and
$\left|1\right\rangle$ as follows

$\displaystyle{R_{x}}\left({\frac{\pi}{2}}\right)\left|0\right\rangle=\frac{1}{{\sqrt{2}}}\left({\left|0\right\rangle-i\left|1\right\rangle}\right),{R_{x}}\left({\frac{\pi}{2}}\right)\left|1\right\rangle=\frac{{-i}}{{\sqrt{2}}}\left({\left|0\right\rangle+i\left|1\right\rangle}\right),$
(I.27)
$\displaystyle{R_{y}}\left({\frac{\pi}{2}}\right)\left|0\right\rangle=\frac{1}{{\sqrt{2}}}\left({\left|0\right\rangle+\left|1\right\rangle}\right),{R_{y}}\left({\frac{\pi}{2}}\right)\left|1\right\rangle=\frac{{1}}{{\sqrt{2}}}\left({\left|0\right\rangle-\left|1\right\rangle}\right).$
(I.28)

In addition, it is worth mentioning that the similar single-qubit gate of
these operations, the Hadamard gate enables qubit interference which is
necessary for quantum algorithms and has no classical analog. It can be
described by a unitary matrix

$\displaystyle H=\frac{1}{{\sqrt{2}}}\left({\begin{array}[]{*{20}{c}}1&1\\\
1&{-1}\end{array}}\right).$ (I.31)

On the other hand, one can usually rewrite the Hadamard gate by decomposing it
into sequences of rotation operations around the x and y axes in the Bloch
sphere in the following form:

$\displaystyle
H={e^{i\frac{\pi}{2}}}{R_{x}}\left(\pi\right){R_{y}}\left({\frac{\pi}{2}}\right).$
(I.32)

Finally, we have to mention other important single qubit gates, S and T gates,
which are so-called the phase gates. These gates rotate the state vector
around the z-axis on the Bloch sphere by changing the dynamic phase of the
state

$\displaystyle S=\left({\begin{array}[]{*{20}{c}}1&0\\\
0&i\end{array}}\right),$ (I.35) $\displaystyle
T=\left({\begin{array}[]{*{20}{c}}1&0\\\
0&{{e^{i\frac{\pi}{4}}}}\end{array}}\right).$ (I.38)

For more details on how to implement the rotation gates for single- and multi-
qubit see chapters II and III.

####  I.2.2 Multi-qubit gates and universilaty

### Multi-qubit gates

In addition to single-qubit gates that can be applied to individual qubits,
there are also other important multi-qubit gates known as controlled-unitary
gates. In most of these gates, some qubits play a target role while some
qubits play the control role. Such controlled-unitary operations are performed
on the target qubit which will change depending on the state of the control
qubit. One of the most famous two-qubit gates is the controlled-not
operation($CNOT$). The action of the controlled-not operation is to invert the
state of the target qubit if the control qubit is $\left|1\right\rangle$ , and
the state of the target qubit does not change if the control qubit is
$\left|0\right\rangle$. Therefore, the quantum circuit (Figure (I.2)) and
matrix representation of this gate in the computational basis states of two-
qubits, $\left|{0,0}\right\rangle$, $\left|{0,1}\right\rangle$,
$\left|{1,0}\right\rangle$, and $\left|{1,1}\right\rangle$

$\displaystyle CNOT=\left({\begin{array}[]{*{20}{c}}1&0&0&0\\\ 0&1&0&0\\\
0&0&0&1\\\ 0&0&1&0\end{array}}\right).$ (I.43)

Figure I.2:  Quantum circuit representation for CNOT gate. Here $x,y=0$ or $1$
and $\oplus$ denotes the addition modulo2, while the vertical line between the
control and target qubit represents the circuit diagram of CNOT gate.

This gate is often used to generate the maximally entangled state, the Bell
states defined by
$\left|{{\Phi^{\pm}}}\right\rangle={{\left({\left|{00}\right\rangle\pm\left|{11}\right\rangle}\right)}\mathord{\left/{\vphantom{{\left({\left|{00}\right\rangle\pm\left|{11}\right\rangle}\right)}{\sqrt{2}}}}\right.\kern-1.2pt}{\sqrt{2}}}$
and
$\left|{{\psi^{\pm}}}\right\rangle={{\left({\left|{01}\right\rangle\pm\left|{10}\right\rangle}\right)}\mathord{\left/{\vphantom{{\left({\left|{01}\right\rangle\pm\left|{10}\right\rangle}\right)}{\sqrt{2}}}}\right.\kern-1.2pt}{\sqrt{2}}}$,
and thus CNOT is an entangling gate.

Another controlled-unitary operation of a two-qubit gate is called the CPhase
($CP$) gate which performs a controlled-Z operation. This is done if two
qubits are in $\left|1\right\rangle$. Thus, the matrix representation of this
gate (CP) is given by

$\displaystyle CP=\left({\begin{array}[]{*{20}{c}}1&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\ 0&0&0&{-1}\end{array}}\right).$ (I.48)

We note that the CNOT gate can be realized by combining $CP$ gate and two
Hadamard gates on both qubits.

Other important two-qubit gates are $iSWAP$ and $\sqrt{iSWAP}$, which are
perfect to generate maximally entangled states, and are defined by

$\displaystyle iSWAP=\left({\begin{array}[]{*{20}{c}}1&0&0&0\\\ 0&0&i&0\\\
0&i&0&0\\\ 0&0&0&1\end{array}}\right)and\leavevmode\nobreak\
\leavevmode\nobreak\ \sqrt{iSWAP}=\left({\begin{array}[]{*{20}{c}}1&0&0&0\\\
0&{\frac{1}{{\sqrt{2}}}}&{\frac{i}{{\sqrt{2}}}}&0\\\
0&{\frac{i}{{\sqrt{2}}}}&{\frac{1}{{\sqrt{2}}}}&0\\\
0&0&0&1\end{array}}\right).$ (I.57)

An example of a controlled-unitary operation for a three-qubit gate is a
Toffoli gate, which is essentially a controlled-controlled-X gate, flipping
the state of the target qubit only if both of the first two control qubits are
in $\left|1\right\rangle$

$\displaystyle Toffoli=\left({\begin{array}[]{*{20}{c}}1&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0\\\ 0&0&0&0&1&0&0&0\\\
0&0&0&0&0&1&0&0\\\ 0&0&0&0&0&0&0&1\\\ 0&0&0&0&0&0&1&0\end{array}}\right).$
(I.66)

In addition, we also have another three-qubit gate that is the Fredkin gate,
which performs a controlled SWAP operation, which exchanges the states of two
target qubits if only the first control qubit is in state
$\left|1\right\rangle$

$\displaystyle Fredkin=\left({\begin{array}[]{*{20}{c}}1&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0\\\ 0&0&0&0&1&0&0&0\\\
0&0&0&0&0&0&1&0\\\ 0&0&0&0&0&1&0&0\\\ 0&0&0&0&0&0&0&1\end{array}}\right).$
(I.75)

###  Universilaty

The concept of universality is associated with the ability to comprise any
computational algorithm with a set of simple gates. For instance, in classical
computing, the MAND gate is a gate that can be implemented using a combination
of AND and NOT gates and thus the NAND gate is referred to as a universal
gate. Similar to universal classical logic operations, there is also a set of
quantum gates that are universal, such combinations of some gates can realize
complex algorithms in quantum computing. However, unlike classical computing
in which a single gate is necessary, the university in quantum counterpart can
only be achieved by combining the arbitrary single qubit gates and cNOT gate,
as well as the set of single qubit gates and Toffoli gate. In general, at
least one multi-qubit gate is required in a universal set for quantum
computing.

####  I.2.3 Quantum circuit for two- and three-qubit entanglement

A quantum circuit is a set of gates applied to quantum registers or qubits, it
is used to design and construct computing hardware. As an example, we here
discuss the two quantum circuits that can be used to generate maximally
entangled states, the Bell state
($\left|{{\Phi^{+}}}\right\rangle={{\left({\left|{00}\right\rangle+\left|{11}\right\rangle}\right)}\mathord{\left/{\vphantom{{\left({\left|{00}\right\rangle+\left|{11}\right\rangle}\right)}{\sqrt{2}}}}\right.\kern-1.2pt}{\sqrt{2}}}$),
and the GHZ-state given in the equation (I.4). In order to generate the Bell
state, the single and two-qubit gates are required in the corresponding
quantum circuit as shown in Fig. (I.3). This circuit is read from left to
right. First, the single-qubit gate on the control qubit is applied as
depicted in Fig (I.3), followed by a CNOT gate to generate the entanglement.

Figure I.3:  Quantum circuit for creating the Bell state.

In addition, the GHZ protocol can be generated by using the single qubit gate
and two CNOT gates, as shown in the circuit diagram in Figure (I.4). Starting
in ground state $\left|{000}\right\rangle$ for all three qubits, we first
perform a rotation on the first qubit (with
$\theta={\pi\mathord{\left/{\vphantom{\pi 2}}\right.\kern-1.2pt}2}$) to put
the whole system in state
${{\left({\left|{000}\right\rangle+\left|{100}\right\rangle}\right)}\mathord{\left/{\vphantom{{\left({\left|{000}\right\rangle+\left|{100}\right\rangle}\right)}{\sqrt{2}}}}\right.\kern-1.2pt}{\sqrt{2}}}$.
To create GHZ-state, we need to map the second component of this state
${\left|{100}\right\rangle}$ to ${\left|{111}\right\rangle}$ without changing
the first component. This is exactly done by applying two CNOT gates to flip
the second and third qubit conditioned on the state of the first qubit. These
CNOTs can be applied in the following way; using the first qubit as the
control qubit for the first gate and the second qubit as the control qubit for
another gate.

Figure I.4:  Quantum circuit for creating the GHZ state.

###  I.3 Quantum algorithms

With access to a large number of quantum gates, we can now build more complex
quantum algorithms circuits by combining these gates. The power of these
algorithms lies in the fundamental laws of quantum mechanics such as
superposition and entanglement, which leads to performing calculations more
efficiently than their classical counterparts. In this section, we will
describe, in general, some quantum algorithms and their circuits diagram, they
are implementable in most quantum devices.

####  I.3.1 Quantum parallelism needed in an algorithm

One of the important difference between quantum and classical computers is
that quantum computers can store their information in superposition states.
The ability of quantum computing to accomplish a quantum operation on all
states at the same time, which is referred to as quantum parallelism, leads to
a significant speed-up advantage over classical ones. Bellow, we explain in
details how quantum parallelism works.

In general, we consider a quantum register involving $n+m$ qubits which takes
$n$ qubits as control qubits and $m$ qubits as a target qubits. Furthermore,
let us assume
$f\left(x\right):{\left\\{{0,1}\right\\}^{n}}\to{\left\\{{0,1}\right\\}^{m}}$
is a Boolean function and we suppose that we have an unitary transformation
${U_{f}}$ of size $\left({n+m}\right)\times\left({n+m}\right)$ in which we
compute the function $f(x)$ such that its action on the state
${\left|x\right\rangle_{n}}{\left|y\right\rangle_{m}}$ gives
${\left|x\right\rangle_{n}}{\left|{y\oplus
f\left(x\right)}\right\rangle_{m}}$, using a set of quantum gates. Instead of
applying ${U_{f}}$ on each individual input state, we first create a
superposition of all input states ${\left|x\right\rangle_{n}}$ using the
Hadamard gate as follows

$\displaystyle{H^{\left(1\right)}}\otimes{H^{\left(2\right)}}\otimes...{H^{\left(n\right)}}{\left|0\right\rangle_{n}}=\frac{1}{{\sqrt{{2^{n}}}}}\sum\limits_{x}{{{\left|x\right\rangle}_{n}}}.$
(I.76)

As shown in Figure (I.5), we then apply the unitary transformation ${U_{f}}$
on this superposition state and all qubits of the target register in the state
$\left|0\right\rangle$

$\displaystyle{U_{f}}\left({\frac{1}{{\sqrt{{2^{n}}}}}\sum\limits_{x}{{{\left|x\right\rangle}_{n}}{{\left|0\right\rangle}_{m}}}}\right)=\frac{1}{{\sqrt{{2^{n}}}}}\sum\limits_{x}{{{\left|x\right\rangle}_{n}}{{\left|{f\left(x\right)}\right\rangle}_{m}}}.$
(I.77)

Figure I.5:  Quantum circuit for evaluating $f(x)$. ${U_{f}}$ is the quantum
circuit which transforms input states
${\left|x\right\rangle_{n}}{\left|y\right\rangle_{m}}$ to
${\left|x\right\rangle_{n}}{\left|{y\oplus
f\left(x\right)}\right\rangle_{m}}$.

####  I.3.2 Grover quantum algorithm

Another class of quantum algorithms is for searching with quantum computing.
Consider the problem of searching for a specific element through an unordered
database of N elements. To classically do this, one can take up to N
evaluations because all elements of the database list must be checked. Thus,
the classical algorithm uses $O(N)$ operations. However, this problem can be
solved significantly more efficiently in quantum computing. Quantum search
algorithm, known as Grover’s algorithm, which enables one to search through an
unordered list with a high probability of success, requiring only
$O\left({\sqrt{N}}\right)$ operations. In this section, we describe Grover’s
algorithm steps step-by-step. We then represent the geometric visualization of
this algorithm to better understand its action.

### Steps and procedure

Suppose we have an unordered database of $N={2^{n}}$ elements (where $n$
denotes the number of qubits), represented with a ket,
$\left\\{{\left|0\right\rangle,\left|1\right\rangle,...\left|N\right\rangle}\right\\}$.
The general Grover algorithm implementation is divided into four steps as
follows(see figure I.6):

### Step$1$: Initialization

We initialize all qubits, by creating an equal superposition of all states,
that were in the state $\left|0\right\rangle$. This done by applying the
$n$-qubit Hadamard

$\displaystyle\left|S\right\rangle=\frac{1}{{\sqrt{{2^{n}}}}}\sum\limits_{x=1}^{{2^{n}}}{\left|x\right\rangle}.$
(I.78)

### Step$2$: Applying the oracle transformation

An oracle can be viewed as a black box that performs a quantum gate on a
quantum register that is not easily specified by universal gates. In Grover’s
algorithm, an oracle can work by applying a unitary transformation ${U_{O}}$
to the initialization qubit ($\left|s\right\rangle$) such that it flips the
sign of $x$ if and only if $x$ is a state we are looking for or the correct
state, with the following properties

$\displaystyle\left|x\right\rangle\mathop{\to}\limits^{{U_{O}}}{\left({-1}\right)^{f\left(x\right)}}\left|x\right\rangle,$
(I.79)

where $f(x)=1$ if $x$ is the correct quantum state and $f(x)=0$ otherwise. The
resulting state after applying an oracle is

$\displaystyle\left|\psi\right\rangle=-{\alpha_{x}}\left|x\right\rangle+{\alpha_{{x^{*}}}}\sum\limits_{{x^{*}}=1,{x^{*}}\neq
x}^{{2^{n}}}{\left|{{x^{*}}}\right\rangle}.$ (I.80)

### Step $3$: Amplification

The amplification, sometimes known as the inversion about mean operation, is
applied to the result quantum state after the application of an oracle, and
given by

$\displaystyle{U_{\phi}}=2\left|S\right\rangle\left\langle S\right|-I.$ (I.81)

This stage performs a reflection of the mean, thus increasing the amplitude of
the correct state. Then, the resulting state is

$\displaystyle\left|{{\psi^{*}}}\right\rangle=(2A+{\alpha_{x}})\left|x\right\rangle+(2A-{\alpha_{{x^{*}}}})\sum\limits_{{x^{*}}=1,{x^{*}}\neq
x}^{{2^{n}}}{\left|{{x^{*}}}\right\rangle},$ (I.82)

where $A$ is the amplitude of the maen vector $\left|\psi\right\rangle$
($A={1\mathord{\left/{\vphantom{1{{2^{n}}\left\\{{-{\alpha_{x}}+\left({{2^{n}}-1}\right){\alpha_{{x^{*}}}}}\right\\}}}}\right.\kern-1.2pt}{{2^{n}}\left\\{{-{\alpha_{x}}+\left({{2^{n}}-1}\right){\alpha_{{x^{*}}}}}\right\\}}}$).

### Step $4$: measurement

Finally, the algorithm output state is measured.

Figure I.6:  Evolution of amplitudes for each state during the Grover search
algorithm.

### Geometric visualization

We note that $G=\left({2\left|S\right\rangle\left\langle S\right|-I}\right)O$,
called the Grover iteration and it must be repeated many times owing to the
nature of the algorithm. We will here show that this operator can be
considered a rotation in the two-dimensional space spanned by the starting
vector $\left|S\right\rangle$(See Fig. I.7). We redefine the state vector in
equation (II.8) as follows

$\displaystyle\left|S\right\rangle=\sqrt{\frac{M}{N}}\left|t\right\rangle+\sqrt{\frac{{N-M}}{N}}\left|{\bar{t}}\right\rangle,$
(I.83)

with $\left|t\right\rangle$ and $\left|{\bar{t}}\right\rangle$ are the
normalized states
($\left|t\right\rangle=\frac{1}{{\sqrt{M}}}\left|x\right\rangle$ and
$\left|{\bar{t}}\right\rangle=\frac{1}{{\sqrt{N-M}}}\sum\limits_{{x^{*}}}{\left|{{x^{*}}}\right\rangle}$).

The effect of $G$ can be realized by applying successively the oracle
operation $O$ that performs a reflection about the state
$\left|t\right\rangle$ in the two-dimensional space defined by
$\left|t\right\rangle$ and $\left|{\bar{t}}\right\rangle$ and then the the
amlification ${U_{\phi}}=2\left|S\right\rangle\left\langle S\right|-I$ which
performs a reflection in the plane defined by $\left|t\right\rangle$ and
$\left|{\bar{t}}\right\rangle$, about the vector $\left|S\right\rangle$. Let’s
set $\cos({\theta\mathord{\left/{\vphantom{\theta
2}}\right.\kern-1.2pt}2})=\sqrt{{{\left({N-M}\right)}\mathord{\left/{\vphantom{{\left({N-M}\right)}N}}\right.\kern-1.2pt}N}}$,
and thus the equation (III.3) can be rewritten as
$\left|S\right\rangle=\sin\left({{\theta\mathord{\left/{\vphantom{\theta
2}}\right.\kern-1.2pt}2}}\right)\left|t\right\rangle+\cos({\theta\mathord{\left/{\vphantom{\theta
2}}\right.\kern-1.2pt}2})\left|{\bar{t}}\right\rangle$.

Figure I.7:  Geometric representation of Grover’s algorithm for a single
Grover iteration, $G$.

####  I.3.3 Deutsch-Jozsa quantum algorithm

One of the first and simple quantum algorithms that show that quantum
computing works better than classical computing is the Deutsch-Jozsa
algorithm[104], it demonstrates exponential acceleration compared to a
classical algorithm. A generalization of this algorithm to functions
$f\left(x\right):{\left\\{{0,1}\right\\}^{n}}\to{\left\\{{0,1}\right\\}}$ was
given by Deutsch and Jozsa and improved by Cleve, Ekert, Macchiavello, and
Mosca[105]. In the Deutsch-Jozsa algorithm, the function $f$ is either
constant in which $2^{n}$ values are all either equal to $0$ or to $1$ or
balanced in which exactly half of the values are equal to $0$ and the other
half to $1$.

The Deutsch -Jozsa algorithm is divided in general into four steps. These
steps are depicted in figure (I.8), let us follow the numbered states
according to this circuit:

### Step $1$

Deutsch-Jozsa algorithm starts with a quantum register of n qubits, we
initialize all the qubits of the first register in $\left|1\right\rangle$ and
of the second register in $\left|1\right\rangle$

$\displaystyle\left|{{\psi_{0}}}\right\rangle={\left|0\right\rangle^{\otimes
n}}\left|1\right\rangle.$ (I.84)

### Step $2$

We apply the Hadamard operator to each qubit of the registers, we get

$\displaystyle\left|{{\psi_{1}}}\right\rangle=\frac{1}{{\sqrt{{2^{n}}}}}\sum\limits_{x}{{{\left|x\right\rangle}_{n}}}\left[{\frac{{\left|0\right\rangle-\left|1\right\rangle}}{{\sqrt{2}}}}\right].$
(I.85)

### Step $3$

Next step, the function $f$ is evaluated using
${U_{f}}:\left|x\right\rangle\left|y\right\rangle\to\left|x\right\rangle\left|{y\oplus
f(x)}\right\rangle$, giving

$\displaystyle\left|{{\psi_{2}}}\right\rangle=\frac{1}{{\sqrt{{2^{n}}}}}{\sum\limits_{x}{\left({-1}\right)}^{f(x)}}{\left|x\right\rangle_{n}}\left[{\frac{{\left|0\right\rangle-\left|1\right\rangle}}{{\sqrt{2}}}}\right].$
(I.86)

### Step $4$

The last step is to apply the Hadamard operator on a state
$\left|x\right\rangle$ and the measurement.
We separately check the cases $x=0$ and $x=1$, we then see the Hadamard
transform for a single qubit
$H\left|x\right\rangle={{\sum\limits_{x}{{{\left({-1}\right)}^{xz}}\left|z\right\rangle}}\mathord{\left/{\vphantom{{\sum\limits_{x}{{{\left({-1}\right)}^{xz}}\left|z\right\rangle}}{\sqrt{2}}}}\right.\kern-1.2pt}{\sqrt{2}}}$,
and thus for $n$-qubits

$\displaystyle{H^{\otimes
n}}{\left|x\right\rangle_{n}}=\frac{1}{{\sqrt{{2^{n}}}}}\sum\limits_{z}{{{\left({-1}\right)}^{xz}}{{\left|z\right\rangle}_{n}}}.$
(I.87)

So by using this equation and (I.86), we easily find

$\displaystyle\left|{{\psi_{3}}}\right\rangle=\frac{1}{{\sqrt{{2^{n}}}}}\sum\limits_{z}{{{\left({-1}\right)}^{xz+f\left(x\right)}}{{\left|z\right\rangle}_{n}}}\left[{\frac{{\left|0\right\rangle-\left|1\right\rangle}}{{\sqrt{2}}}}\right].$
(I.88)

Figure I.8:  Quantum circuit implementing the general Deutsch-Jozsa algorithm.

###  I.4 Chapter summary

In this chapter, we have discussed the fundamental notions needed for quantum
information processing such as the mathematical and geometric of a qubit,
quantum superposition, and entanglement as well as single qubit gates. Such
notions and entangling two-qubit gates are required to build any simple
quantum processor. We have also seen how some two- and multi-qubit gates, for
instance, the $CNOT$, $CZ$, $iSWAP$ , $\sqrt{SWAP}$ and $Toffoli$, when
combined with the single rotation gates, can be realized as a universal set
for quantum computing. They permit the operation of some simple algorithms,
such as the Grover search and Deutsch-Jozsa. Over the next chapters, we will
suggest the implementation of some of these concepts with superconducting
qubit devices.

##  Chapter II Quantum superconducting circuits

Physically, Quantum bit or qubit implementations have taken many forms, for
instance, nuclear spins, trapped-ions, photons, and superconducting qubits. In
this chapter, we will describe how we use can electrical circuits to realize
superconducting qubits such as charge qubits and transmon qubits. An essential
building block to realizing these superconducting qubits is the Josephson
junction, the fundamental properties of which are discussed. We will also
describe the interaction between light and matter inside the cavity QED and
its analog called the circuit QED, in which the cavity is replaced by a
superconducting transmission line resonator capacitively coupled to a Cooper
box which plays the role of the two-level atom or the superconducting qubit in
the context of quantum information. We then discuss how the coupling between
transmon-type superconducting qubits and the transmission line resonator, can
be used to realize the single and two-qubit gates.

###  II.1 Superconducting qubits

One of the most popular and promising candidates for realizing scalable
universal quantum computing is superconducting qubits, based on Josephson
effect. The general principle of operation of these qubits includes a
Josephson junction, which is characterized by its nonlinear inductance and its
capacitance. Here, we introduce the Josephson junction as the key device for
realizing superconducting qubits. We then derive the Hamiltonian of the Cooper
pair box(Charge qubit) and the transmon-type superconducting, using the
quantization of electrical circuits to perform canonical quantization of our
circuits.

####  II.1.1 The Josephson Junction

Josephson junctions are based on the Josephson effect[106, 107]. A Josephson
junction consists of two superconductors separated by an insulating barrier
which limits the flow of supercurrent between the two superconductors, as
shown in the figure(II.1). Cooper pairs can coherently tunnel from one
superconductor to the other, with the supercurrent $I$ that is given by

$\displaystyle I={I_{c}}\sin\left(\phi\right),$ (II.1)

where ${I_{c}}$ is the critical current of the junction, and
$\phi={\theta_{2}}-{\theta_{1}}$ is a phase diﬀerence across the junction,
which is related to the potential $V$ between the two superconductors
according to

$\displaystyle\frac{{d\phi}}{{dt}}=\frac{{2\pi}}{{{\Phi_{0}}}}V,$ (II.2)

where
${\Phi_{0}}={h\mathord{\left/{\vphantom{h{2e}}}\right.\kern-1.2pt}{2e}}={2,07.10^{-15}}.T.{m^{-15}}$
is the magnetic flux quantum.

Figure II.1:  Schematic of a Josephson tunnel junction. It consists of two
superconductors separated by an insulating barrier. The two sides of the
junction are characterized by the number of Cooper pairs ${{n_{1}}}$,
${{n_{2}}}$ and the phases ${{\theta_{1}}}$, ${{\theta_{2}}}$. The charge
${n_{2}}-{n_{1}}$ and phase difference ${\theta_{2}}-{\theta_{1}}$ between the
two metals are the essential parameters describing the behavior of the
junction.

### Equivalent electric circuit of the Josephson junction

Physically, the Josephson junction is symbolized by a cross in a box and can
be modeled as an ideal Josephson element with the Josephson energy ${E_{J}}$
shunted by a capacitance ${C_{J}}$[108], as shown in Figure (II.2). We
introduce the number of Cooper pairs $N$ by $Q=2eN$, with $Q$ is the electrode
charge, and the voltage across the junction is given by $Q=CV$. We can then
determine the dynamics of a phase difference $\phi$ by using the Kirchhoff’s
rule for the circuit shown in Fig. (II.2.b) to give the differential equation

$\displaystyle{I_{ext}}$ $\displaystyle=$
$\displaystyle{I_{c}}\sin\phi+C\frac{{dV}}{{dt}}$ (II.3) $\displaystyle=$
$\displaystyle{I_{c}}\sin\phi+\frac{h}{{2e}}C\ddot{\phi}.$

From this equation, we readily find the Lagrangian of a Josephson junction

$\displaystyle
L=\frac{1}{2}\frac{{C{\hbar^{2}}}}{{4{e^{2}}}}{\ddot{\phi}^{2}}+{I_{c}}\frac{\hbar}{{2e}}\cos\phi+{I_{ext}}\frac{\hbar}{{2e}}\phi=K-U,$
(II.4)

where $K={1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}C{V^{2}}$ is the
kinetic term, and the potential energy of the system $U$ is given by

$\displaystyle
U=-\frac{{{I_{c}}\hbar}}{{2e}}\cos\phi-{I_{ext}}\frac{\hbar}{{2e}}\phi.$
(II.5)

Figure II.2:  Schematic of a Josephson tunnel junction. (a) The symbol of the
Josephson junction (Cross in a box) .(b) The equivalent electrical circuit of
the Josephson junction.

The dynamics of the circuit (Hamiltonian $H$) is associated with the
Lagrangian $L$ through

$\displaystyle H$ $\displaystyle=$ $\displaystyle p\dot{\phi}-L$ (II.6)
$\displaystyle=$
$\displaystyle\frac{1}{2}\frac{{{E_{C}}}}{{{\hbar^{2}}}}{p^{2}}-p\cos\phi-{E_{J}}\frac{{{I_{ext}}}}{{{I_{c}}}}\phi,$

where:
\- $p=-\frac{{\partial
H}}{{\partial\dot{\phi}}}=\frac{{C{\hbar^{2}}}}{{4{e^{2}}}}\dot{\phi}=\frac{\hbar}{{2e}}Q$
is the canonical momentum operator conjugate to $\phi$.
\- ${E_{C}}=\frac{{{{\left({2e}\right)}^{2}}}}{C}$ is the charging energy of a
Cooper pair, i.e. the energy required to increase the number of Cooper pairs
in the box.
\- ${E_{J}}=\frac{{{I_{0}}\hbar}}{{2e}}$ is the Josephson tunneling energy.
And the Hamiltonian equations of motion

$\displaystyle\dot{\phi}=\frac{{\partial H}}{{\partial
p}},\dot{p}=-\frac{{\partial H}}{{\partial\phi}}.$ (II.7)

####  II.1.2 SQUID

There is a variant of the Josephson junction called ”SQUID”
[109]((Superconducting Quantum Interference Device), which consists of two
ideally identical Josephson junctions in parallel coupled to a current source
as shown in figure (II.3).

Figure II.3:  Schematic of a SQUID. The two Josephson junctions form a loop,
which is crossed by a flux magnetic field.

According to the relation of Stockes, the magnetic flux through the loop in
the presence of a magnetic field which describes the vector potential
$\mathop{A}\limits^{\to}\left(r\right)$ is expressed by

$\displaystyle{\Phi_{ext}}=\oint_{C}{\mathop{A}\limits^{\to}}\left(r\right)dr.$
(II.8)

As known, the phase variation between two points in the case of a closed
superconducting loop is given by

$\displaystyle{\phi_{2}}-{\phi_{1}}=\oint_{C}{\mathop{A}\limits^{\to}}\left(r\right)dr=2k\pi,$
(II.9)

and thus by combining the equations (II.8) and (II.9), the magnetic flux
${\Phi_{ext}}$ can be expressed as follows

$\displaystyle{\Phi_{ext}}=\frac{{{\Phi_{0}}}}{{2\pi}}\left({{\phi_{2}}-{\phi_{1}}}\right).$
(II.10)

The Hamiltonian (II.6) becomes

$\displaystyle H=\frac{1}{2}\frac{{{E_{C}}}}{{{\hbar^{2}}}}{p^{2}}-{U_{T}},$
(II.11)

with ${U_{T}}={U_{1}}+{U_{2}}$: is the potential energy of the SQUID (the sum
of the Josephson energies of the two junctions), which is given by

$\displaystyle{U_{T}}=-{E_{J}}\left({\cos\left({{\phi_{1}}}\right)+\cos\left({{\phi_{2}}}\right)}\right)=-2{E_{J}}\cos\left({\frac{{{\phi_{1}}+{\phi_{1}}}}{2}}\right)\cos\left({\frac{{{\phi_{2}}-{\phi_{1}}}}{2}}\right).$
(II.12)

We assume that the phase
$\phi={{\left({{\phi_{1}}+{\phi_{2}}}\right)}\mathord{\left/{\vphantom{{\left({{\phi_{1}}+{\phi_{2}}}\right)}2}}\right.\kern-1.2pt}2}$.
Then the potential term is written in the form

$\displaystyle{U_{T}}$ $\displaystyle=$
$\displaystyle-{E_{J}}\cos\left({\pi\frac{{{\Phi_{ext}}}}{{{\Phi_{0}}}}}\right)\cos\left(\phi\right)$
(II.13) $\displaystyle=$
$\displaystyle-{E_{J}}\left({{\Phi_{ext}}}\right)\cos\left(\phi\right),$

where the effective Josephson energy
${E_{J}}\left({{\Phi_{ext}}}\right)=-{E_{J}}\cos\left({\pi{{{\Phi_{ext}}}\mathord{\left/{\vphantom{{{\Phi_{ext}}}{{\Phi_{0}}}}}\right.\kern-1.2pt}{{\Phi_{0}}}}}\right)$
can be tuned via a magnetic flux ${{\Phi_{ext}}}$ applied through the loop of
the SQUID.

####  II.1.3 Charge qubit( The Cooper pair box)

In the classical case, superconductivity is attributed to electron pairs,
which can exist within the island: the pairs can either be inside or outside
the island. However, if the tunnel barrier is large enough, as dictated by the
uncertainty principle, it becomes possible for a pair to simultaneously exist
both inside and outside the island, creating a superposition of $0$ and $1$.

The possible quantum bit, known as the ”Cooper pair box,” was first
theoretically proposed in ref.[110] and later experimentally achieved in ref.
[111]. Such the Cooper pair box consists of a superconducting island (green),
which is connected via a Josephson junction (blue) with capacitance $C_{J}$
and Josephson energy ${E_{J}}$ to a large superconducting electrode (orange)
as shown in figure II.4. The quantum variable is the number of superconducting
pairs that are attracted to the island by a gate potential ${V_{g}}$ applied
to the gate capacitance ${C_{g}}$.

Figure II.4:  a) Schematic of the Cooper pair box consisting of a Josephson
junction (blue) which connects an island (green) with a reservoir (orange). b)
The Josephson junction is replaced by a SQUID in which the qubit becomes more
controllable by a magnetic flux.

### Electric circuit corresponding to the Cooper pair box

The superconducting circuit consists either of a Josephson junction having a
Josephson energy ${E_{J}}$ and an intrinsic capacitance ${C_{J}}$ connected to
a source of potential via a capacitance ${C_{g}}$ as shown in the figure
(II.5a) or the loop SQUID in figure(II.5b).

We apply on the figure (II.5) the same procedure that we have seen in the
previous paragraphs (the Kirchhoff’s rule), we can easily identify the
following Lagrangian

$\displaystyle L=K-U,$ (II.14)

where $U=-{E_{J}}\cos\phi$ is the potential energy, which is already
determined in the previous sections and the kinetic energy $K$ is the sum of
the capacitive energies of the circuit, which is given by

Figure II.5:  a) Circuit diagram corresponding to the Cooper pair box
schematic in Fig. (II.4a). b) Circuit diagram corresponding to the Cooper pair
box schematic in Fig. (II.4b).

$\displaystyle K$ $\displaystyle=$
$\displaystyle\frac{1}{2}{C_{J}}V_{J}^{2}+\frac{1}{2}{C_{g}}{\left({{C_{J}}-{V_{g}}}\right)^{2}}$
(II.15) $\displaystyle=$
$\displaystyle\frac{{{C_{J}}+{C_{g}}}}{2}{\left({{V_{J}}-\frac{{{C_{g}}}}{{{C_{J}}+{C_{g}}}}{V_{g}}}\right)^{2}}+\frac{1}{2}{\left({{C_{g}}-\frac{{C_{g}^{2}}}{{{C_{J}}+{C_{g}}}}}\right)^{2}}.$

By using the relation in equation (II.2) and omitting a constant term
(independent of $\phi$ and ${\dot{\phi}}$) in equation (II.3) because it does
not influence the system dynamics, the Lagrangian can be written as

$\displaystyle
L=\frac{{{C_{\Sigma}}}}{2}{\left({\frac{\hbar}{{2e}}\dot{\phi}-\frac{{{C_{g}}}}{{{C_{\Sigma}}}}{V_{g}}}\right)^{2}}-{E_{J}}\cos\phi,$
(II.16)

where ${{C_{\Sigma}}={C_{J}}+{C_{g}}}$. Here the kinetic energy can be
controlled by the external gate voltage, whereas the potential energy, is
however fixed by the energy $E_{J}$.

As shown in figure (II.5b), the single Josephson junction is replaced by a
loop SQUID which consists of two identical junctions
(${E_{J,1}}={E_{J,2}}={{{E_{J}}}\mathord{\left/{\vphantom{{{E_{J}}}2}}\right.\kern-1.2pt}2}$),
in order to add additional control for potential energy via a magnetic flux.
The potential term in the Lagrangian (II.16) can be replaced by the potential
term (II.16), the corresponding Lagrangian is then given by

$\displaystyle
L=\frac{{{C_{\Sigma}}}}{2}{\left({\frac{\hbar}{{2e}}\dot{\phi}-\frac{{{C_{g}}}}{{{C_{\Sigma}}}}{V_{g}}}\right)^{2}}-{E_{J}}\left({{\Phi_{ext}}}\right)\cos\phi.$
(II.17)

The canonical momentum operator corresponding to (II.15) with respect to
$\dot{\phi}$ is given by $p={{\partial K}\mathord{\left/{\vphantom{{\partial
K}{\partial\dot{\phi}=\left({{\hbar\mathord{\left/{\vphantom{\hbar{2e}}}\right.\kern-1.2pt}{2e}}}\right)Q}}}\right.\kern-1.2pt}{\partial\dot{\phi}=\left({{\hbar\mathord{\left/{\vphantom{\hbar{2e}}}\right.\kern-1.2pt}{2e}}}\right)Q}}$,
(with $Q=\left({{C_{J}}+{C_{g}}}\right){V_{J}}-{C_{g}}{V_{g}}=-2eN$), which
implies
$\left({{C_{J}}+{C_{g}}}\right)V_{J}^{2}={{{{{\left({2e}\right)}^{2}}}\mathord{\left/{\vphantom{{{{\left({2e}\right)}^{2}}}{\left({{C_{J}}+{C_{g}}}\right)\left({N-{{{C_{g}}{N_{g}}}\mathord{\left/{\vphantom{{{C_{g}}{N_{g}}}{2e}}}\right.\kern-1.2pt}{2e}}}\right)}}}\right.\kern-1.2pt}{\left({{C_{J}}+{C_{g}}}\right)\left({N-{{{C_{g}}{N_{g}}}\mathord{\left/{\vphantom{{{C_{g}}{N_{g}}}{2e}}}\right.\kern-1.2pt}{2e}}}\right)}}^{2}}$.

Finally, we can identify the Hamiltonian of the following system:

$\displaystyle
H=\frac{{{E_{c}}}}{2}{\left({N-{N_{g}}}\right)^{2}}-{E_{J}}\left({{\Phi_{ext}}}\right)\cos\phi,$
(II.18)

where
${E_{c}}={{{{(2e)}^{2}}}\mathord{\left/{\vphantom{{{{(2e)}^{2}}}{\left({{C_{J}}+{C_{g}}}\right)}}}\right.\kern-1.2pt}{\left({{C_{J}}+{C_{g}}}\right)}}$
and
${N_{g}}={{{C_{g}}{V_{g}}}\mathord{\left/{\vphantom{{{C_{g}}{V_{g}}}{2e}}}\right.\kern-1.2pt}{2e}}$.

We consider that the electrostatic energy is dominant compared to the
Josephson energy(${E_{C}}\gg{E_{J}}$), it is called the charge qubit regime.
To quantitatively describe the Cooper pair box, we define the operator
$\hat{N}$ which is associated with the number of Cooper pairs in the base
$\left|N\right\rangle$ as follows

$\displaystyle\hat{N}=\sum\limits_{N\in Z}{N\left|N\right\rangle\left\langle
N\right|}.$ (II.19)

The phase difference $\phi$ across the Josephson junction presents
fluctuations related to $N$ by the following uncertainty relation: $\Delta
N.\Delta\phi={1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}$, assuming
that the fluctuations of the number of excess Cooper pairs $N$ is very small
to that of $\phi$, the canonical commutation relation
$\left[{N,\phi}\right]=i$ and
$N=i{\partial\mathord{\left/{\vphantom{\partial{\partial\phi}}}\right.\kern-1.2pt}{\partial\phi}}$,
and thus the basis $\left|\phi\right\rangle$ can be introduced as

$\displaystyle\left|\phi\right\rangle=\sum\limits_{N\in
Z}{{e^{-iN\phi}}\left|N\right\rangle},$ (II.20)

and by using the Fourier transform[112], the number states
$\left|N\right\rangle$ can be laid to the phase difference
${\left|\phi\right\rangle}$ states

$\displaystyle\left|N\right\rangle=\frac{1}{{2\pi}}\int_{0}^{2\pi}{d\phi{e^{iN\phi}}\left|\phi\right\rangle}.$
(II.21)

From equation (II.21), we notice the following identity
${e^{i\phi}}\left|N\right\rangle\approx\left|{N+1}\right\rangle$, and we have
$\cos\phi={{\left({{e^{i\phi}}+{e^{-i\phi}}}\right)}\mathord{\left/{\vphantom{{\left({{e^{i\phi}}+{e^{-i\phi}}}\right)}2}}\right.\kern-1.2pt}2}={{\left({\left|N\right\rangle\left\langle{N+1}\right|+\left|{N+1}\right\rangle\left\langle
N\right|}\right)}\mathord{\left/{\vphantom{{\left({\left|N\right\rangle\left\langle{N+1}\right|+\left|{N+1}\right\rangle\left\langle
N\right|}\right)}2}}\right.\kern-1.2pt}2}$. We can then write the Hamiltonian
of the Cooper pair box in the matrix form

$\displaystyle
H=\frac{{{E_{C}}}}{2}\sum\limits_{N}{{{\left({N-{N_{g}}}\right)}^{2}}\left|N\right\rangle\left\langle{N+1}\right|-\frac{{{E_{J}}}}{2}\sum\limits_{N}{\left({\left|N\right\rangle\left\langle{N+1}\right|+\left|{N+1}\right\rangle\left\langle
N\right|}\right)}}.$ (II.22)

We can diagonalize the Hamiltonian (II.22) and obtain the energy spectrum as a
function of the gate charge $N_{g}$, which will be composed of a set of
parabolas centered around the integer values of $N_{g}$. We see that for half-
integer values of $N_{g}$
(${N_{g}}=N+{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}$ , $N\in{\rm
Z}$) the state of charge is twice degenerate, more precisely the states
$\left|N\right\rangle$ and $\left|{N+1}\right\rangle$ have the same energy.
The Cooper box in the ${E_{C}}\gg{E_{J}}$ regime is a good two-level system,
especially if we skew the grid voltage so that we have $N_{g}$ $=$ 0.5. At
this point, the quantum states are the superposition of the symmetric and
anti-symmetric of a state corresponding to $N$ Cooper pairs and $N+1$ pairs of
Cooper on the superconducting island, we have
$\left|0\right\rangle={{\left({\left|{N+1}\right\rangle-\left|N\right\rangle}\right)}\mathord{\left/{\vphantom{{\left({\left|{N+1}\right\rangle-\left|N\right\rangle}\right)}{\sqrt{2}}}}\right.\kern-1.2pt}{\sqrt{2}}}$
and
$\left|1\right\rangle={{\left({\left|{N+1}\right\rangle+\left|N\right\rangle}\right)}\mathord{\left/{\vphantom{{\left({\left|{N+1}\right\rangle+\left|N\right\rangle}\right)}{\sqrt{2}}}}\right.\kern-1.2pt}{\sqrt{2}}}$
as indicated in figure(II.6).

The energy levels of the Hamiltonian (II.22) being very anharmonic, we can
restrict to only two levels and thus we can consider the circuit as a quantum
bit.

For $N=0$ the Hamiltonian (II.22) becomes

$\displaystyle H=\frac{{{E_{C}}}}{2}N_{g}^{2}\left|0\right\rangle\left\langle
0\right|-\frac{{{E_{J}}}}{2}\left({\left|0\right\rangle\left\langle
1\right|+\left|1\right\rangle\left\langle 0\right|}\right).$ (II.23)

For $N=1$

$\displaystyle
H=\frac{{{E_{C}}}}{2}{\left({1-{N_{g}}}\right)^{2}}\left|1\right\rangle\left\langle
1\right|-\frac{{{E_{J}}}}{2}\left({\left|1\right\rangle\left\langle
2\right|+\left|2\right\rangle\left\langle 1\right|}\right).$ (II.24)

From equations (II.23) and (II.24), one can determine the elements of the
matrix of (II.22) in the states of charge
$\left\\{{\left|0\right\rangle,\left|1\right\rangle}\right\\}$ as follows:

$\displaystyle
H=-\frac{{{E_{C}}}}{2}\left({1-2{N_{g}}}\right){\sigma_{z}}-\frac{{{E_{J}}}}{2}{\sigma_{x}},$
(II.25)

where ${\sigma_{z}}$ and ${\sigma_{x}}$ are the Pauli matrices. The
Hamiltonian of the charge qubit is therefore written in the following form

$\displaystyle
H=-\frac{1}{2}{B_{z}}{\sigma_{z}}-\frac{1}{2}{B_{x}}{\sigma_{x}},with\leavevmode\nobreak\
{B_{z}}=\frac{{{E_{C}}}}{2}\left({1-2{N_{g}}}\right);{B_{x}}={E_{J}}.$ (II.26)

The corresponding eigenvalues and eigenstates of this Hamiltonian are given by

$\displaystyle{E_{0}}=-\frac{1}{2}\sqrt{B_{z}^{2}+B_{x}^{2}};\leavevmode\nobreak\
{E_{1}}=\frac{1}{2}\sqrt{B_{z}^{2}+B_{x}^{2}},$ (II.27)
$\displaystyle\left|+\right\rangle=\cos\theta\left|\uparrow\right\rangle+\sin\theta\left|\downarrow\right\rangle;\leavevmode\nobreak\
\left|-\right\rangle=-\sin\theta\left|\uparrow\right\rangle+\cos\theta\left|\downarrow\right\rangle,$
(II.28)

with the expression of $\theta$ is given by the following relation
$\theta=\left({{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}\right){\tan^{-1}}\left[{{{2{E_{J}}}\mathord{\left/{\vphantom{{2{E_{J}}}{{E_{C}}\left({1-2{N_{g}}}\right)}}}\right.\kern-1.2pt}{{E_{C}}\left({1-2{N_{g}}}\right)}}}\right]=\left({{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}\right){\tan^{-1}}\left[{{{{B_{x}}}\mathord{\left/{\vphantom{{{B_{x}}}{{B_{z}}}}}\right.\kern-1.2pt}{{B_{z}}}}}\right]$.

Figure II.6:  Energy spectrum of the charge qubit for the ratio as a function
of gate charge $N_{g}$. The
${E\mathord{\left/{\vphantom{E{{E_{C}}}}}\right.\kern-1.2pt}{{E_{C}}}}$ is
lower, the first two levels are isolated from the third for values of $N_{g}$
near the half-integer. a) the bold curve designates the energy spectrum of the
Hamiltonian with $E_{C}$ = 0.2, the fine curve with
${E\mathord{\left/{\vphantom{E{{E_{C}}}}}\right.\kern-1.2pt}{{E_{C}}}}=0$. b)
the energy spectrum for $0\leq{N_{g}}\leq 1$.

####  II.1.4 Transmon qubit

The difference between the transmon qubit and the charge qubit is that the
transmon qubit is independent of the gate charge by going to another regime
where ${E_{J}}\gg{E_{C}}$, which is so-called the transmon regime. This is
done at the level of the equivalent circuit of the superconducting qubits by
inroducing the addition of a large shunting capacitance placed in parallel
with the Josephson junction to increase the overall capacitance as illustrated
in figure (II.7).

Figure II.7:  The equivalent circuit of the Transmon qubit, an additional
capacitance (Cs) has been added in parallel to the junction.

The increased
${{{E_{J}}}\mathord{\left/{\vphantom{{{E_{J}}}{{E_{C}}}}}\right.\kern-1.2pt}{{E_{C}}}}$
ratio makes the Cooper box less sensitive to gate charge and its
anharmonicity, leading to the flattening energy bands, whereas the transmon
frequency can be tuned by applying the magnetic flux through the loop of
SQUID. Such energy bands are illustrated in figure (II.8), where three
eigenenergies ${E_{k}}$ (k=1,2,3) of the Cooper pair box Hamiltonian (II.22)
are shown as a function of gate charge $N_{g}$ for different values of the
ratio
${{{E_{J}}}\mathord{\left/{\vphantom{{{E_{J}}}{{E_{C}}}}}\right.\kern-1.2pt}{{E_{C}}}}$.
The sensitivity to charge noise is defined as the charge dispersion, which is
the transition energy between two neighboring energy levels

$\displaystyle{\in_{k}}={E_{k,k+1}}\left({{N_{g}}=0.5}\right)-{E_{k,k+1}}\left({{N_{g}}=0}\right).$
(II.29)

Owing to the charge dispersion decreases exponentially fast as the ratio
${{{E_{J}}}\mathord{\left/{\vphantom{{{E_{J}}}{{E_{C}}}}}\right.\kern-1.2pt}{{E_{C}}}}$
is increased as shown in Ref. [43]

$\displaystyle{\in_{k}}\infty{e^{-i\sqrt{{{8{E_{J}}}\mathord{\left/{\vphantom{{8{E_{J}}}{{E_{C}}}}}\right.\kern-1.2pt}{{E_{C}}}}}}}.$
(II.30)

The transmon Hamiltonian is the same as that of the hamiltonian (II.18), with
only change being the charging energy ${E_{C}}$:

$\displaystyle
H=\frac{{{E_{C}}}}{2}{\left({N-{N_{g}}}\right)^{2}}-{E_{J}}\left({{\Phi_{ext}}}\right)\cos\phi,$
(II.31)

where
${E_{C}}={{{{\left({2e}\right)}^{2}}}\mathord{\left/{\vphantom{{{{\left({2e}\right)}^{2}}}{\left({{C_{J}}+{C_{g}}+{C_{s}}}\right)}}}\right.\kern-1.2pt}{\left({{C_{J}}+{C_{g}}+{C_{s}}}\right)}}$
is the charging energy. Since
${{{E_{J}}}\mathord{\left/{\vphantom{{{E_{J}}}{{E_{C}}\gg
1}}}\right.\kern-1.2pt}{{E_{C}}\gg 1}}$, we can consider that the fluctuations
in $\phi$ in the Hamiltonian (II.31), are very weak and make a development of
the cosine of the tunnel energy term
$-{E_{J}}\left({{\Phi_{ext}}}\right)\cos\phi\simeq-{E_{J}}\left({{\Phi_{ext}}}\right)+\left({{{{E_{J}}\left({{\Phi_{ext}}}\right)}\mathord{\left/{\vphantom{{{E_{J}}\left({{\Phi_{ext}}}\right)}2}}\right.\kern-1.2pt}2}}\right){\phi^{2}}-\left({{{{E_{J}}\left({{\Phi_{ext}}}\right)}\mathord{\left/{\vphantom{{{E_{J}}\left({{\Phi_{ext}}}\right)}{24}}}\right.\kern-1.2pt}{24}}}\right){\phi^{4}}$.
The Hamiltonian (II.31) becomes

$\displaystyle
H=\frac{{{E_{C}}}}{2}{\left({N-{N_{g}}}\right)^{2}}-{E_{J}}\left({{\Phi_{ext}}}\right)+\frac{{{E_{J}}\left({{\Phi_{ext}}}\right)}}{2}{\phi^{2}}-\frac{{{E_{J}}\left({{\Phi_{ext}}}\right)}}{{24}}{\phi^{4}}.$
(II.32)

Figure II.8:  Eigenenergies for three energy levels ($k=0,1,2$) of the Cooper
pair box Hamiltonian (II.31) for different ratios
${{{E_{J}}}\mathord{\left/{\vphantom{{{E_{J}}}{{E_{C}}}}}\right.\kern-1.2pt}{{E_{C}}}}$.
For very large
${{{E_{J}}}\mathord{\left/{\vphantom{{{E_{J}}}{{E_{C}}}}}\right.\kern-1.2pt}{{E_{C}}}}$,
the spectrum practically becomes that of the harmonic oscillator. The system
gets less sensitive to gate charge when the
${{{E_{J}}}\mathord{\left/{\vphantom{{{E_{J}}}{{E_{C}}}}}\right.\kern-1.2pt}{{E_{C}}}}$
is increased from the charging regime with
${{{E_{J}}}\mathord{\left/{\vphantom{{{E_{J}}}{{E_{C}}}}}\right.\kern-1.2pt}{{E_{C}}}}=0.5$
(a) to the transmon regime with
${{{E_{J}}}\mathord{\left/{\vphantom{{{E_{J}}}{{E_{C}}}}}\right.\kern-1.2pt}{{E_{C}}}}=50$
(c). The figure used from Ref.[66].

In figure (II.8), we observe that in the Transmon system, the energy levels
are independent of the charging shift $N_{g}$. Therefore, we can omit this
term from the Hamiltonian. Using the gauge transformation
${U_{t}}={e^{i{N_{g}}\phi}}$, and thus

$\displaystyle
H\approx{U_{t}}HU_{t}^{+}=\frac{{{E_{C}}}}{2}{N^{2}}+\frac{{{E_{J}}}}{2}{\phi^{2}}-{E_{J}}-\frac{{{E_{J}}}}{{24}}{\phi^{4}},$
(II.33)

the first part of this Hamiltonian is clearly that of a harmonic oscillator:

$\displaystyle{H_{1}}=\frac{{\sqrt{8{E_{J}}{E_{C}}}}}{2}\left({{{\left\\{{{{\left({\frac{{{E_{C}}}}{{8{E_{J}}}}}\right)}^{\frac{1}{4}}}N}\right\\}}^{2}}+{{\left\\{{{{\left({\frac{{{E_{J}}}}{{8{E_{C}}}}}\right)}^{\frac{1}{4}}}\phi}\right\\}}^{2}}}\right).$
(II.34)

We can then introduce the bosonic variables
${b^{+}}=\left({{1\mathord{\left/{\vphantom{1{\sqrt{2}}}}\right.\kern-1.2pt}{\sqrt{2}}}}\right)\left({i{{\left({{{8{E_{C}}}\mathord{\left/{\vphantom{{8{E_{C}}}{{E_{J}}}}}\right.\kern-1.2pt}{{E_{J}}}}}\right)}^{{1\mathord{\left/{\vphantom{14}}\right.\kern-1.2pt}4}}}N+{{\left({{{{E_{J}}}\mathord{\left/{\vphantom{{{E_{J}}}{8{E_{C}}}}}\right.\kern-1.2pt}{8{E_{C}}}}}\right)}^{{1\mathord{\left/{\vphantom{14}}\right.\kern-1.2pt}4}}}\phi}\right)$
and
$b=\left({{1\mathord{\left/{\vphantom{1{\sqrt{2}}}}\right.\kern-1.2pt}{\sqrt{2}}}}\right)\left({-i{{\left({{{8{E_{C}}}\mathord{\left/{\vphantom{{8{E_{C}}}{{E_{J}}}}}\right.\kern-1.2pt}{{E_{J}}}}}\right)}^{{1\mathord{\left/{\vphantom{14}}\right.\kern-1.2pt}4}}}N+{{\left({{{{E_{J}}}\mathord{\left/{\vphantom{{{E_{J}}}{8{E_{C}}}}}\right.\kern-1.2pt}{8{E_{C}}}}}\right)}^{{1\mathord{\left/{\vphantom{14}}\right.\kern-1.2pt}4}}}\phi}\right)$,
and therefore we have
$N=-\left({{i\mathord{\left/{\vphantom{i2}}\right.\kern-1.2pt}2}}\right){\left({{{{E_{J}}}\mathord{\left/{\vphantom{{{E_{J}}}{2{E_{C}}}}}\right.\kern-1.2pt}{2{E_{C}}}}}\right)^{{1\mathord{\left/{\vphantom{14}}\right.\\\
\kern-1.2pt}4}}}\\\ \left({{b^{+}}-b}\right)$ and
$\phi={\left({{{2{E_{C}}}\mathord{\left/{\vphantom{{2{E_{C}}}{{E_{J}}}}}\right.\kern-1.2pt}{{E_{J}}}}}\right)^{{1\mathord{\left/{\vphantom{14}}\right.\kern-1.2pt}4}}}\left({{b^{+}}+b}\right)$
where $\left[{b,{b^{+}}}\right]=1$,
${b^{+}}=\sum\limits_{J}{\sqrt{J+1}\left|{J+1}\right\rangle\left\langle
J\right|}$. The Hamiltonian (II.33) becomes

$\displaystyle
H=\sqrt{8{E_{J}}{E_{C}}}\left({{b^{+}}b+\frac{1}{2}}\right)-{E_{J}}-\frac{{{E_{C}}}}{{12}}{\left({{b^{+}}+b}\right)^{4}}.$
(II.35)

The corresponding eigen-energies are separated by the energy quantum
$\sqrt{8{E_{J}}{E_{C}}}$.

In the transmon regime, the charging energy determines the non-linearity of
the oscillator through its anharmonicity $\alpha$

$\displaystyle\alpha={E_{12}}-{E_{01}}=-{E_{C}}.$ (II.36)

Finally, the flat energy bands enable the assignment of a unique frequency to
each energy level of the states $\left|J\right\rangle$. So the Hamiltonian is
written in the form

$\displaystyle
H=\hbar\sum\limits_{J}{{\omega_{J}}\left|J\right\rangle\left\langle
J\right|}.$ (II.37)

For the first two levels, the Hamiltonian in equation (II.37) becomes

$\displaystyle H=\hbar\frac{{{\omega_{q}}}}{2}{\sigma_{z}},$ (II.38)

where ${\omega_{q}}={\omega_{2}}-{\omega_{1}}$ is the qubit transition
frequency, ${\sigma_{z}}=\left|0\right\rangle\left\langle
0\right|-\left|1\right\rangle\left\langle 1\right|$.

###  II.2 Cavity quantum electrodynamics

Cavity quantum electrodynamics is the physical theory describing the
interaction between light and matter schematically represented in Fig. (II.9),
which studies the interaction between atoms represented by a two-level system
and the quantized electromagnetic mode inside a cavity. Cavity QED is
interesting in fundamental physics, as well as for applications in quantum
information processing.

Figure II.9:  QED cavity architecture includes an electromagnetic field mode
of decay rate $\kappa$ coupled to a two-level system of decay rate $\gamma$
and the transition time ${t_{transit}}$ with a coupling force $g$.

####  II.2.1 Quantification of the field in a cavity

We consider a cavity of length L of two perfect mirrors as shown in Fig.
(II.10). Maxwell’s equations:

$\displaystyle\nabla.E=0,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \nabla.B=0,$ (II.39)
$\displaystyle\nabla\times E=-\frac{{\partial B}}{{\partial
t}},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \nabla\times B={\mu_{0}}{\varepsilon_{0}}\frac{{\partial
B}}{{\partial t}},$ (II.40)

where ${\mu_{0}}$ and ${\varepsilon_{0}}$ are the vacuum permeability and
permittivity. The electric field is polarized along the x axis,
$\vec{E}\left({r,t}\right)={E_{x}}\left({z,t}\right){\vec{e}_{x}}$. The
boundary conditions impose the following decomposition
${E_{x}}\left({z,t}\right)={\left[{{{2{\omega^{2}}}\mathord{\left/{\vphantom{{2{\omega^{2}}}{\left({V{\varepsilon_{0}}}\right)}}}\right.\kern-1.2pt}{\left({V{\varepsilon_{0}}}\right)}}}\right]^{{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}}q\left(t\right)\sin\left({kz}\right)$,
with
${k_{n}}={{n\pi}\mathord{\left/{\vphantom{{n\pi}L}}\right.\kern-1.2pt}L}$($n\succ
0$), ${\omega_{n}}=c{k_{n}}$, and $V$ is the effective volume of the cavity,
and $q(t)$ is the amplitude of normal mode $n$. From Maxwell’s equations
(II.39) and (II.40), we readily get the expression of the magnetic field
$\vec{B}\left({r,t}\right)={B_{y}}\left({z,t}\right){\vec{e}_{y}}$,
${B_{y}}\left({z,t}\right)=\left({{{{\mu_{0}}{\varepsilon_{0}}}\mathord{\left/{\vphantom{{{\mu_{0}}{\varepsilon_{0}}}k}}\right.\kern-1.2pt}k}}\right)\left({{{2{\omega^{2}}}\mathord{\left/{\vphantom{{2{\omega^{2}}}{V{\varepsilon_{0}}}}}\right.\kern-1.2pt}{V{\varepsilon_{0}}}}}\right)p\left(t\right)\cos\left({kz}\right)$,
where $p\left(t\right)=\dot{q}\left(t\right)$ is the canonical momentum
operator conjugate to $q$. By using Poynting’s theorem, we obtain the energy
of the electromagnetic field in the cavity:

Figure II.10:  An electromagnetic cavity of length $L$, the electric field
must therefore be zero at the extremities of the cavity.

$\displaystyle
H=\frac{1}{2}\int{dV\left({{\varepsilon_{0}}E_{x}^{2}\left({z,t}\right)+\frac{1}{{{\mu_{0}}}}B_{y}^{2}\left({z,t}\right)}\right)}.$
(II.41)

We replace the expressions of the electric and magnetic field as mentioned
above in this expression, then using the trigonometric relations
${\sin^{2}}\left({kz}\right)={{\left({1-\cos\left({2kz}\right)}\right)}\mathord{\left/{\vphantom{{\left({1-\cos\left({2kz}\right)}\right)}2}}\right.\kern-1.2pt}2}$,
${\cos^{2}}\left({kz}\right)={{\left({1+\cos\left({2kz}\right)}\right)}\mathord{\left/{\vphantom{{\left({1+\cos\left({2kz}\right)}\right)}2}}\right.\kern-1.2pt}2}$,
we finally find the following expression

$\displaystyle H=\frac{1}{2}\left({{p^{2}}+{\omega^{2}}{q^{2}}}\right).$
(II.42)

By moving from the classical variables $p$ and $q$ to their quantum operators,
which are given by

$\displaystyle\tilde{p}=i\sqrt{\frac{\omega}{2}}\left({{a^{+}}-a}\right);\leavevmode\nobreak\
\tilde{q}=i\sqrt{\frac{1}{{2\omega}}}\left({{a^{+}}+a}\right).$ (II.43)

where $\tilde{p}$ and $\tilde{q}$ obey the canonical commutation relation
$\left[{\tilde{q},\tilde{p}}\right]=i\hbar$, the quantum Hamiltonian of the
field is written in the following form

$\displaystyle H=\hbar\omega\left({{a^{+}}a+\frac{1}{2}}\right).$ (II.44)

####  II.2.2 Jaynes-Cummings model

We now consider the atom, described as a two-level system whose Hamiltonian,
${H_{atom}}=\left({{{\hbar\Omega}\mathord{\left/{\vphantom{{\hbar\Omega}2}}\right.\kern-1.2pt}2}}\right){\sigma_{z}}$,
as indicated in the diagram(II.11).

Figure II.11:  Two level atom.

We can now study the interaction between the two-level atom and the quantized
field. The Hamiltonian system is divided into three parts

$\displaystyle H={H_{atom}}+{H_{field}}+{H_{{\mathop{\rm int}}er}},$ (II.45)

where ${H_{atom}}$ is the Hamiltonian of a two-level atom,
${H_{field}}=\hbar{\omega_{r}}{a^{+}}a$ is the Hamiltonian of the quantized
field and ${H_{{\mathop{\rm int}}er}}$ is the interaction Hamiltonian between
the atom and the quantized field.

####  II.2.3 Approximation of the dipole moment and the rotating wave

The coupling between an atom and an electromagnetic wave is dominated by the
coupling of the electric field with the electric dipole. The interaction
Hamiltonian can therefore be written as

$\displaystyle{H_{{\mathop{\rm int}}er}}=-\tilde{d}.\tilde{E},$ (II.46)

where the dipole operator is written as $\tilde{d}=d{\sigma_{x}}$, and from
the expression of the electric field and equation (II.43). We therefore can
write the field operator

$\displaystyle\tilde{E}={\left({\frac{\omega}{{V{\varepsilon_{0}}}}}\right)^{\frac{1}{2}}}\sin\left({kz}\right)\left({{a^{+}}+a}\right).$
(II.47)

Finally, the interaction Hamiltonian (II.46) becomes

$\displaystyle{H_{{\mathop{\rm int}}er}}=\hbar
g\left({{\sigma^{+}}+\sigma}\right)\left({a+{a^{+}}}\right),$ (II.48)

where
$g={{g{\varepsilon_{rms}}}\mathord{\left/{\vphantom{{g{\varepsilon_{rms}}}\hbar}}\right.\kern-1.2pt}\hbar}$(${\varepsilon_{rms}}=\sqrt{{\omega\mathord{\left/{\vphantom{\omega{V{\varepsilon_{0}}}}}\right.\kern-1.2pt}{V{\varepsilon_{0}}}}}\sin\left({kz}\right)$),
while ${\sigma^{-}}=\left|\uparrow\right\rangle\left\langle\downarrow\right|$
and ${\sigma^{+}}=\left|\downarrow\right\rangle\left\langle\uparrow\right|$
are Pauli matrices expressed in terms of the ground $\left|0\right\rangle$ and
excited $\left|1\right\rangle$ atom states. The Hamiltonian of the system
(II.45) then becomes

$\displaystyle
H=\hbar{\omega_{r}}{a^{+}}a+\frac{{\hbar\Omega}}{2}{\sigma_{z}}+\hbar
g\left({{\sigma^{+}}+{\sigma^{-}}}\right)\left({{a^{+}}+a}\right).$ (II.49)

The Jaynes-Cummings Hamiltonian can be derived from the Hamiltonian (II.49) by
using the rotating wave approximation (RWA). The RWA is valid in the regime
where $g\ll{\omega_{r}},\Omega$. The terms
${\sigma^{+}}a\left({{\sigma^{-}}{a^{+}}}\right)$ correspond to the processes
of excitation (de-excitation) of the atom and a photon is created(annihilated)
in the cavity mode. The two other terms corresponding to the excitation (de-
excitation) of both the atom and the cavity field mode do not conserve energy.
In the RWA regime, these two terms can be neglected. Within the framework of
this approximation, we obtain the Hamiltonian corresponding to the Jaynes-
Cummings model[113]

$\displaystyle
H=\hbar{\omega_{r}}{a^{+}}a+\frac{{\hbar\Omega}}{2}{\sigma_{z}}+\hbar
g\left({a{\sigma^{+}}+{a^{+}}{\sigma^{-}}}\right).$ (II.50)

We can transform this expression into a more practical form, by introducing
the following excitation number operator
$N\simeq{a^{+}}a+{{{\sigma_{z}}}\mathord{\left/{\vphantom{{{\sigma_{z}}}2}}\right.\kern-1.2pt}2}$,
the Hamiltonian (II.50) can be rewritten as

$\displaystyle H=\hbar{\omega_{r}}N+\frac{{\hbar\Delta}}{2}{\sigma_{z}}+\hbar
g\left({a{\sigma^{+}}+{a^{+}}{\sigma^{-}}}\right),$ (II.51)

where $\Delta=\Omega-{\omega_{r}}$ is the atom-cavity detuning. We can easily
demonstrate the following commutation relation $\left[{H,N}\right]=0$. The
stationary states of the system without coupling ($g=0$) are

$\displaystyle\left|{\downarrow,n}\right\rangle=\left|\downarrow\right\rangle\otimes\left|n\right\rangle,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\left|{\uparrow,n}\right\rangle=\left|\uparrow\right\rangle\otimes\left|n\right\rangle.$
(II.52)

The interaction Hamiltonian $H_{inter}$ couples these two-by-two states
$\left|{\uparrow,n}\right\rangle$ with $\left|{\downarrow,n+1}\right\rangle$.
Only the fundamental level $\left|{\uparrow,0}\right\rangle$ is not coupled to
any other state

$\displaystyle{H_{{\mathop{\rm int}}er}}\left|{\uparrow,0}\right\rangle=0,$
(II.53) $\displaystyle{H_{{\mathop{\rm
int}}er}}\left|{\downarrow,0}\right\rangle=\hbar
g\left|{\uparrow,1}\right\rangle,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ {H_{{\mathop{\rm
int}}er}}\left|{\uparrow,1}\right\rangle=\hbar
g\left|{\downarrow,0}\right\rangle,$ (II.54) $\displaystyle{H_{{\mathop{\rm
int}}er}}\left|{\downarrow,n}\right\rangle=\hbar
g\sqrt{n+1}\left|{\uparrow,n+1}\right\rangle,{H_{{\mathop{\rm
int}}er}}\left|{\uparrow,n+1}\right\rangle=\hbar
g\sqrt{n+1}\left|{\downarrow,n}\right\rangle.$ (II.55)

In the double states
$\left\\{{\left|{\downarrow,n}\right\rangle,\left|{\uparrow,n+1}\right\rangle}\right\\}$,
the Hamiltonian (II.51) is then written

$\displaystyle
H=\hbar{\omega_{r}}\left({n+1}\right)\left({\begin{array}[]{*{20}{c}}1&0\\\
0&1\end{array}}\right)+\frac{\hbar}{2}\left({\begin{array}[]{*{20}{c}}{-\Delta}&{2g\sqrt{n+1}}\\\
{2g\sqrt{n+1}}&\Delta\end{array}}\right).$ (II.60)

The eigen-energies of the Hamiltonian (II.51) are

$\displaystyle{E_{+,n}}=\hbar{\omega_{r}}\left({n+1}\right)+\frac{\hbar}{2}\sqrt{4g^{2}\left({n+1}\right)+{\Delta^{2}}},{E_{-,n}}=\hbar{\omega_{r}}\left({n+1}\right)-\frac{\hbar}{2}\sqrt{4g^{2}\left({n+1}\right)+{\Delta^{2}}},$
(II.61)

and the corresponding states are given by

$\displaystyle\left|{+,n}\right\rangle=\cos{\theta_{n}}\left|{\downarrow,n}\right\rangle+\sin{\theta_{n}}\left|{\uparrow,n+1}\right\rangle,\left|{-,n}\right\rangle=-\sin{\theta_{n}}\left|{\downarrow,n}\right\rangle+\cos{\theta_{n}}\left|{\uparrow,n+1}\right\rangle,$
(II.62)

while the egen-energy of the ground state $\left|{\uparrow,0}\right\rangle$ is

$\displaystyle\frac{{-\hbar\Delta}}{2},\leavevmode\nobreak\
\leavevmode\nobreak\ and\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
{\theta_{n}}=\left({\frac{1}{2}}\right){\tan^{-1}}\left({\frac{{2g\sqrt{n+1}}}{2}}\right).$
(II.63)

####  II.2.4 Resonance regime and dispersive regime

Resonance regime ( $\Delta=0$)

In the resonant case, characterized by $\Delta=0$, the eigenstates are the
combinations of symmetric and anti-symmetric of the states
$\left|{\downarrow,n}\right\rangle$ and
$\left|{\uparrow,n+1}\right\rangle$($\left|{\pm,n}\right\rangle=\left({{1\mathord{\left/{\vphantom{1{\sqrt{2}}}}\right.\kern-1.2pt}{\sqrt{2}}}}\right)\left({\left|{\uparrow,1}\right\rangle\pm\left|{\downarrow,0}\right\rangle}\right)$),
and the interaction term lifts the degeneracy of these states. This situation
is illustrated in Figure (II.12).

Figure II.12:  Energy diagram in the resonant case.

Dispersive regime ( $\Delta\gg g$)

We now describe the dispersive case in which the atom is operated at a
frequency for which the detuning $\Delta$ between atom and cavity are large.
In this case, no exchanging excitations between atom and cavity, and the
eigenstates take the form

$\displaystyle\left|{+,n}\right\rangle\simeq\left|{\downarrow,n}\right\rangle+\frac{{g\sqrt{n+1}}}{\Delta}\left|{\uparrow,n+1}\right\rangle,\left|{-,n}\right\rangle\simeq-\frac{{g\sqrt{n+1}}}{\Delta}\left|{\downarrow,n}\right\rangle+\left|{\uparrow,n+1}\right\rangle.$
(II.64)

As shown in figure (II.13), the energies of the eigenstates are shifted by the
coupling. In this regime, the energy exchange between the qubit and the
resonator field is difficult. By applying the unitary transformation,
$U=\exp(\lambda\left({a{\sigma^{+}}-{a^{+}}{\sigma^{-}}}\right)$ with
$\lambda={g\mathord{\left/{\vphantom{g{\Delta\ll
1}}}\right.\kern-1.2pt}{\Delta\ll 1}}$, it is possible to obtain an
approximate effective Hamiltonian

$\displaystyle{H_{eff}}=UH{U^{+}}\approx\hbar\left({{\omega_{r}}+\frac{{{g^{2}}}}{\Delta}{\sigma_{z}}}\right){a^{+}}a+\frac{\hbar}{2}\left({\Omega+\frac{{{g^{2}}}}{\Delta}}\right){\sigma_{z}}.$
(II.65)

Figure II.13:  Energy diagram in the dispersive regime.

###  II.3 Circuit quantum electrodynamics

Circuit quantum electrodynamics (circuit QED) is an analog of cavity quantum
electrodynamics, which is proposed by Blais[44]. Such a circuit QED is
realized in an electrical circuit by coupling capacitively a superconducting
qubit to a coplanar resonator (playing the role of the cavity) or the quantum
harmonic LC oscillator.

####  II.3.1 Quantum harmonic LC oscillator

The quantum LC oscillator is the simplest quantum circuit as shown in figure
(II.14), it consists of a capacitor $C$ in series with an inductance $L$. By
using the Kirchhoff’s rule for quantum circuit shown in Fig. (II.14), we find
the following equation of motion

$\displaystyle L\frac{{d\dot{Q}}}{{dt}}+\frac{Q}{C}=0.$ (II.66)

Using the following Euler-Lagrangian equation
${{d\left({{{\partial{L_{r}}}\mathord{\left/{\vphantom{{\partial{L_{r}}}{\partial\dot{Q}}}}\right.\kern-1.2pt}{\partial\dot{Q}}}}\right)}\mathord{\left/{\vphantom{{d\left({{{\partial
L}\mathord{\left/{\vphantom{{\partial{L_{r}}}{\partial\dot{Q}}}}\right.\kern-1.2pt}{\partial\dot{Q}}}}\right)}{dt-{{\partial{L_{r}}}\mathord{\left/{\vphantom{{\partial{L_{r}}}{\partial
Q}}}\right.\kern-1.2pt}{\partial
Q}}=0}}}\right.\kern-1.2pt}{dt-{{\partial{L_{r}}}\mathord{\left/{\vphantom{{\partial
L}{\partial Q}}}\right.\kern-1.2pt}{\partial Q}}=0}}$, we can determine the
Lagrangian of the system

$\displaystyle{L_{r}}=\frac{L}{2}{\dot{Q}^{2}}-\frac{{{Q^{2}}}}{{2C}}.$
(II.67)

The corresponding conjugate moment
$p={{\partial{L_{r}}}\mathord{\left/{\vphantom{{\partial{L_{r}}}{\partial\dot{Q}}}}\right.\kern-1.2pt}{\partial\dot{Q}}}=L\dot{Q}$.
Therefore, the Hamiltonian is written in the following form

Figure II.14:  Energy diagram in the dispersive regime.

$\displaystyle H=\frac{\Phi}{{2L}}+\frac{Q}{{2C}}.$ (II.68)

The flux $\Phi$ in the inductor can be chosen as a position variable, whereas
the charge $Q$ of the capacitor can be seen as its conjugate variable,
$\left[{\Phi,Q}\right]=i\hbar$. We introduce ${a^{+}}$ and $a$ to diagonalize
the hamiltonian with the quantum harmonic oscillator. $Q$ and $\Phi$ can thus
be written as
$a=-i{\Phi\mathord{\left/{\vphantom{\Phi{\sqrt{2\hbar{\omega_{r}}L}+{Q\mathord{\left/{\vphantom{Q{\sqrt{2\hbar{\omega_{r}}C}}}}\right.\kern-1.2pt}{\sqrt{2\hbar{\omega_{r}}C}}}}}}\right.\kern-1.2pt}{\sqrt{2\hbar{\omega_{r}}L}+{Q\mathord{\left/{\vphantom{Q{\sqrt{2\hbar{\omega_{r}}C}}}}\right.\kern-1.2pt}{\sqrt{2\hbar{\omega_{r}}C}}}}}$
and
${a^{+}}=i{\Phi\mathord{\left/{\vphantom{\Phi{\sqrt{2\hbar{\omega_{r}}L}+{Q\mathord{\left/{\vphantom{Q{\sqrt{2\hbar{\omega_{r}}C}}}}\right.\kern-1.2pt}{\sqrt{2\hbar{\omega_{r}}C}}}}}}\right.\kern-1.2pt}{\sqrt{2\hbar{\omega_{r}}L}+{Q\mathord{\left/{\vphantom{Q{\sqrt{2\hbar{\omega_{r}}C}}}}\right.\kern-1.2pt}{\sqrt{2\hbar{\omega_{r}}C}}}}}$.
We then plug these expressions in equation (II.68) we get

$\displaystyle H=\hbar{\omega_{r}}\left({{a^{+}}a+\frac{1}{2}}\right),$
(II.69)

where
${\omega_{r}}={1\mathord{\left/{\vphantom{1{\sqrt{LC}}}}\right.\kern-1.2pt}{\sqrt{LC}}}$.

####  II.3.2 Transmission line (Coplanar waveguide)

The aim of this section is to obtain a quantum description of the
electromagnetic field inside a transmission line. We use the standard approach
of modeling the line as a set of inductors in parallel with capacitors as
shown (II.15).

We assume inductance $l$ and capacitance $c$ per unit the length of the line.
Each segment of the line of the length $dx$ has an inductance $ldx$ in
parallel with a capacitance $cdx$, as shown in figure(II.16). We apply
Kirchhoff’s rule to this figure, and we find the following equation
$V\left({x,t}\right)=ldx{{\partial
I\left({x,t}\right)}\mathord{\left/{\vphantom{{\partial
I\left({x,t}\right)}{\partial
t+V\left({x+dx,t}\right)}}}\right.\kern-1.2pt}{\partial
t+V\left({x+dx,t}\right)}}$ (i.e
${{V\left({x+dx,t}\right)}\mathord{\left/{\vphantom{{V\left({x+dx,t}\right)}{dx-{{V\left({x,t}\right)}\mathord{\left/{\vphantom{{V\left({x,t}\right)}{dx=l{{\partial
I\left({x,t}\right)}\mathord{\left/{\vphantom{{\partial
I\left({x,t}\right)}{\partial t}}}\right.\kern-1.2pt}{\partial
t}}}}}\right.\kern-1.2pt}{dx=l{{\partial
I\left({x,t}\right)}\mathord{\left/{\vphantom{{\partial
I\left({x,t}\right)}{\partial t}}}\right.\kern-1.2pt}{\partial
t}}}}}}}\right.\kern-1.2pt}{dx-{{V\left({x,t}\right)}\mathord{\left/{\vphantom{{V\left({x,t}\right)}{dx=l{{\partial
I\left({x,t}\right)}\mathord{\left/{\vphantom{{\partial
I\left({x,t}\right)}{\partial t}}}\right.\kern-1.2pt}{\partial
t}}}}}\right.\kern-1.2pt}{dx=l{{\partial
I\left({x,t}\right)}\mathord{\left/{\vphantom{{\partial
I\left({x,t}\right)}{\partial t}}}\right.\kern-1.2pt}{\partial t}}}}}}$).
Finally, we find the following constitutive relation

Figure II.15:  Representation of the transmission line by a set of inductors
in parallel with capacitors.

$\displaystyle
l{\partial_{t}}I\left({x,t}\right)+{\partial_{x}}V\left({x,t}\right)=0.$
(II.70)

Figure II.16:  Representation of an inductor in parallel with the capacitor.

We choose the generalized flux as the degree of freedom instead of choosing
current or voltage, the generalized flux is defined as
$\Phi\left({x,t}\right)=\int{V\left({x,t}\right)dx}$. By replacing the
expression of V (x, t) in equation (II.70), we find the expression of the
current
$I\left({x,t}\right)=\left({{1\mathord{\left/{\vphantom{1l}}\right.\kern-1.2pt}l}}\right){\partial_{x}}\Phi\left({x,t}\right)$.
The Lagrangian of the system is given by

$\displaystyle{L_{g}}=\int_{\frac{{-d}}{2}}^{\frac{d}{2}}{dxL\left({x,t}\right)=\int_{\frac{{-d}}{2}}^{\frac{d}{2}}{dx\left[{\frac{c}{2}{{\left({{\partial_{t}}\Phi}\right)}^{2}}-\frac{1}{{2l}}{{\left({{\partial_{x}}\Phi}\right)}^{2}}}\right]}}.$
(II.71)

The associated Euler-Lagrange equations are those of a wave propagating at the
speed
$\nu={1\mathord{\left/{\vphantom{1{\sqrt{lc}}}}\right.\kern-1.2pt}{\sqrt{lc}}}$.
By using boundary conditions
($\Phi\left({{d\mathord{\left/{\vphantom{d2}}\right.\kern-1.2pt}2},t}\right)=\Phi\left({-{d\mathord{\left/{\vphantom{d2}}\right.\kern-1.2pt}2},t}\right)=0$),
$\Phi\left({x,t}\right)$ can be written under the form

$\displaystyle\Phi\left({x,t}\right)=\sqrt{\frac{2}{d}}\sum\limits_{{k_{0}}=1}{{\phi_{{k_{0}}}}\left(t\right)}\cos\left({\frac{{{k_{0}}\pi
x}}{d}}\right)+\sqrt{\frac{2}{d}}\sum\limits_{{k_{e}}=2}{{\phi_{{k_{e}}}}\left(t\right)}\sin\left({\frac{{{k_{e}}\pi
x}}{d}}\right).$ (II.72)

We replace (II.27) in (II.70), the Lagrangian $L_{g}$ is then written in terms
of the new variables
${\phi_{k}}\left(t\right)\left({k={k_{0}},{k_{e}}}\right)$

$\displaystyle{L_{g}}=\sum\limits_{k}{\frac{c}{2}}{\left({{{\dot{\phi}}_{k}}}\right)^{2}}-\frac{1}{{2l}}{\left({\frac{{k\pi}}{d}}\right)^{2}}{\left({{\phi_{k}}}\right)^{2}}.$
(II.73)

It is the Lagrangian of the sum of decoupled harmonic oscillators. The
calculation of the moments ${p_{k}}$ combined with the variables
${{\phi_{k}}}$: ${p_{k}}=c{\dot{\phi}_{k}}$ by the Legendre transformation,
then the commutation relation is
$\left[{{a_{k}},{a^{+}}_{k^{\prime}}}\right]={\delta_{k,k^{\prime}}}$. The
Hamiltonian of the resonator is then written

$\displaystyle
H=\sum\limits_{k}{{\omega_{k}}\left({a_{k}^{+}{a_{k}}+\frac{1}{2}}\right)},$
(II.74)

where
${\omega_{k}}=\left({{{k\pi}\mathord{\left/{\vphantom{{k\pi}d}}\right.\kern-1.2pt}d}}\right)\nu$,
${a_{k}}=-i\sqrt{\left({{{k\pi}\mathord{\left/{\vphantom{{k\pi}{2\hbar}}}\right.\kern-1.2pt}{2\hbar}}}\right)\sqrt{{c\mathord{\left/{\vphantom{cl}}\right.\kern-1.2pt}l}}}{\tilde{\phi}_{k}}+\sqrt{\left({{d\mathord{\left/{\vphantom{d{2\hbar
k\pi}}}\right.\kern-1.2pt}{2\hbar
k\pi}}}\right)\sqrt{{l\mathord{\left/{\vphantom{lc}}\right.\kern-1.2pt}c}}}{\tilde{p}_{k}}$
and
$a_{k}^{+}=i\sqrt{\left({{{k\pi}\mathord{\left/{\vphantom{{k\pi}{2\hbar}}}\right.\kern-1.2pt}{2\hbar}}}\right)\sqrt{{c\mathord{\left/{\vphantom{cl}}\right.\kern-1.2pt}l}}}{\tilde{\phi}_{k}}+\sqrt{\left({{d\mathord{\left/{\vphantom{L{2\hbar
k\pi}}}\right.\kern-1.2pt}{2\hbar
k\pi}}}\right)\sqrt{{l\mathord{\left/{\vphantom{lc}}\right.\kern-1.2pt}c}}}{\tilde{p}_{k}}$.

####  II.3.3 Coupling a transmon-type superconducting qubit to a coplanar
waveguide resonator

The coupling between the transmon-type superconducting qubit and the coplanar
waveguide resonator is an electrostatic capacitive interaction. Such a
transmon is located at the center of a transmission line resonator, as shown
in Fig.(II.17b). As seen in Fig. (II.17a), the transmon qubit is capacitively
coupled to the resonator in the equivalent circuit of the transmon-resonator
system (figure II.17b), it is composed of two Josephson junctions, which are
hijacked by two large capacitances $C_{g}$ and $C_{B}$. These large capacities
lead to low charge energy,
${E_{c}}={{{{\left({2e}\right)}^{2}}}\mathord{\left/{\vphantom{{{{\left({2e}\right)}^{2}}}{{C_{t}}}}}\right.\kern-1.2pt}{{C_{t}}}}$
with $C_{t}$ = $C_{B}$ \+ $C_{g}$ \+ $C_{J}$ which distinguishes the transmon
from the charge qubit.

The coupling between the transmon and the resonator is demonstrated by
following an approach similar to that used for dissipation in the cavity. The
standing wave in the resonator (Fig. II.17(b)) induces tension between the two
islands. We consider a capacitive coupling between these two devices, the
interaction Hamiltonian takes the following form

$\displaystyle{H_{{\mathop{\rm
int}}er}}=2\frac{{{C_{g}}}}{{{C_{t}}}}eV_{rms}^{0}\tilde{N}\left({a+{a^{+}}}\right),$
(II.75)

where
${\omega_{r}}={1\mathord{\left/{\vphantom{1{\sqrt{{L_{r}}{C_{r}}}}}}\right.\kern-1.2pt}{\sqrt{{L_{r}}{C_{r}}}}}$
is the resonator frequency , and
$V_{rms}^{0}=\sqrt{{{\hbar{\omega_{r}}}\mathord{\left/{\vphantom{{\hbar{\omega_{r}}}{{C_{r}}}}}\right.\kern-1.2pt}{{C_{r}}}}}$.
The total Hamiltonian of the system is then written

Figure II.17:  Circuit QED for the transmon qubit: (a) The transmon qubit is
coupled to the $LC$ resonator through a gate capacitance $C_{d}$. The two
Josephson junctions, with capacitance $C_{J}$ and Josephson energy $E_{J}$,
are in parallel with an additional large capacitance $C_{B}$. (b) The transmon
device consists of a transmon qubit coupled to a transmission line. The figure
used from Ref.[43].

$\displaystyle
H=\frac{{{E_{c}}}}{2}{\left({N-{N_{g}}}\right)^{2}}-{E_{J}}\cos\phi+\hbar{\omega_{r}}{a^{+}}a+2\beta
eV_{rms}^{0}\tilde{N}\left({a+{a^{+}}}\right),$ (II.76)

where
$\beta={{{C_{g}}}\mathord{\left/{\vphantom{{{C_{g}}}{{C_{t}}}}}\right.\kern-1.2pt}{{C_{t}}}}$.
We can rewrite the Hamiltonian in terms of the eigenstates of the transmon
$\left|i\right\rangle$, we then obtain the generalized Jaynes-Cummings
Hamiltonian

$\displaystyle H=\sum\limits_{j}{{\omega_{j}}\left|j\right\rangle\left\langle
j\right|+\hbar{\omega_{r}}{a^{+}}a+\hbar\sum\limits_{ij}{{g_{ij}}\left|i\right\rangle}}\left\langle
j\right|\left({a+{a^{+}}}\right),$ (II.77)

where $\hbar{g_{ij}}=2\beta eV_{rms}^{0}\left\langle
i\right|\tilde{N}\left|j\right\rangle$ are nearest-neighbor energy levels of
the coupling, with
$\left|{\left\langle{j+1}\right|\tilde{N}\left|j\right\rangle}\right|\approx\sqrt{{{\left({j+1}\right)}\mathord{\left/{\vphantom{{\left({j+1}\right)}2}}\right.\kern-1.2pt}2}}{\left({{{{E_{J}}}\mathord{\left/{\vphantom{{{E_{J}}}{8{E_{c}}}}}\right.\kern-1.2pt}{8{E_{c}}}}}\right)^{{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}}$.

We omit the terms corresponding to a double excitation and a double
desexcitation, we obtain the generalized Hamiltonian of Jaynes-Cummings

$\displaystyle
H=\hbar\sum\limits_{j}{{\omega_{j}}\left|j\right\rangle\left\langle
j\right|+\hbar{\omega_{r}}{a^{+}}a+\hbar\sum\limits_{i}{{g_{i,i+1}}\left|i\right\rangle\left\langle{i+1}\right|+h.c}}.$
(II.78)

With sufficient anharmonicity for a transmon system, we can keep only the
first two levels, the Hamiltonian (II.78) becomes Jaynes-Cummings in equation
(II.49).

Similar to cavity quantum electrodynamics, we will be able to describe a
dispersive regime in circuit quantum electrodynamics. This is done when the
detuning $\Delta={\omega_{q}}-{\omega_{r}}$ between the transmon and resonator
is large. We diagonalize the Hamiltonian (II.78), by applying the unitary
transformation $D={e^{S-{S^{+}}}}$, with
$S=\sum\limits_{i}{{\beta_{i}}a\left|{i+1}\right\rangle\left\langle i\right|}$
(${\beta_{i}}={{{g_{i,i+1}}}\mathord{\left/{\vphantom{{{g_{i,i+1}}}{\left({{\omega_{i,i+1}}-{\omega_{r}}}\right)}}}\right.\kern-1.2pt}{\left({{\omega_{i,i+1}}-{\omega_{r}}}\right)}}$).
We use the ”Baker-Campbell-Hausdorff[63]” relation and keeping the terms up to
${{{g^{2}}_{i,i+1}}\mathord{\left/{\vphantom{{{g^{2}}_{i,i+1}}{\Delta_{i}^{2}}}}\right.\kern-1.2pt}{\Delta_{i}^{2}}}$
(${\Delta_{i}}={\omega_{i,i+1}}-{\omega_{r}}$), we get

$\displaystyle{H_{eff}}=DH{D^{+}}$ $\displaystyle=$
$\displaystyle\hbar\sum\limits_{j}{{\omega_{j}}\left|i\right\rangle\left\langle
i\right|+\hbar{\omega_{r}}{a^{+}}a+\hbar\sum\limits_{i}{{\chi_{i,i+1}}\left|{i+1}\right\rangle\left\langle{i+1}\right|}}-\hbar{\chi_{01}}\left|0\right\rangle\left\langle
0\right|$ (II.79) $\displaystyle+$
$\displaystyle\hbar\sum\limits_{i=1}{\left({{\chi_{i-1,i}}-{\chi_{i,i+1}}}\right){a^{+}}a\left|i\right\rangle\left\langle
i\right|+\hbar\sum\limits_{i}{{\eta_{i}}aa\left|{i+2}\right\rangle}}\left\langle
i\right|+h.c,$

where
${\chi_{i,i+1}}={{g_{i,i+1}^{2}}\mathord{\left/{\vphantom{{g_{i,i+1}^{2}}{\left({{\omega_{i,i+1}}-{\omega_{r}}}\right)}}}\right.\kern-1.2pt}{\left({{\omega_{i,i+1}}-{\omega_{r}}}\right)}}$
and
${\eta_{i}}={{{g_{i,i+1}}{g_{i+1,i+2}}}\mathord{\left/{\vphantom{{{g_{i,i+1}}{g_{i+1,i+2}}}{\left[{2\left({{\omega_{i+1}}-{\omega_{i}}-{\omega_{r}}}\right)\left({{\omega_{i+2}}-{\omega_{i+1}}-{\omega_{r}}}\right)}\right]}}}\right.\kern-1.2pt}{\left[{2\left({{\omega_{i+1}}-{\omega_{i}}-{\omega_{r}}}\right)\left({{\omega_{i+2}}-{\omega_{i+1}}-{\omega_{r}}}\right)}\right]}}$.

The last term ${\eta_{i}}$ corresponding to two-photon transitions can be
neglected from ${\chi_{i,i+1}}$. Then, equation (II.79) becomes

$\displaystyle{H_{eff}}=\frac{\hbar}{2}\underbrace{\left({{\omega_{01}}+{\chi_{01}}}\right)}_{{{\omega^{\prime}}_{01}}}{\sigma_{z}}+\hbar\underbrace{\left({{\omega_{r}}-\frac{1}{2}{\chi_{12}}}\right)}_{{{\omega^{\prime}}_{r}}}{a^{+}}a+\hbar\underbrace{\left({{\chi_{01}}-\frac{{{\chi_{12}}}}{2}}\right)}_{\chi}{\sigma_{z}}{a^{+}}a.$
(II.80)

Finally, the dispersive Hamiltonian can be written as

$\displaystyle{H_{eff}}=\hbar\frac{{{{\omega^{\prime}}_{01}}}}{2}{\sigma_{z}}+\hbar\left({{{\omega^{\prime}}_{r}}+\chi{\sigma_{z}}}\right){a^{+}}a.$
(II.81)

###  II.4 Quantum gates with superconducting device

####  II.4.1 Single-qubit gates

#####  $X$-rotation gate for one qubit

Applying the classical field polarized according to ${e_{x}}$,