H_{\pazocal{O}}(\omega,t)=\hbar\left(\omega_{0}-\omega\right)H_{0}+\hbar\omega_{\text{T}}H_{\pazocal{O},\text{T}}(\omega,t)\text{
. }$ (313)

where
$H_{\pazocal{O},\text{T}}(\omega,t)=\pazocal{O}(t)H_{\text{T}}(\omega,t)\pazocal{O}^{\dagger}(t)$.
Now, we separately consider two specific cases:

$\bullet$ _Far-from resonance situation
$|\omega_{0}-\omega|\gg|\omega_{\text{T}}|$_: In this case, the term
$\hbar\omega_{\text{T}}H_{\pazocal{O},\text{T}}(\omega,t)$ in Eq. (313) works
as a perturbation. Therefore the set of eigenvectors of
$H_{\pazocal{O}}(\omega,t)$ reads

$\displaystyle|E^{\pazocal{O}}_{n}(t)\rangle$
$\displaystyle=|E_{n}^{0}\rangle+\pazocal{O}\left(||\hbar\omega_{\text{T}}H_{\text{T}}(\omega,t)||\right)\text{
, }$ (314)

where we have used that the energy gaps $\tilde{E}_{n}^{0}-\tilde{E}_{k}^{0}$
of the Hamiltonian $\hbar\left(\omega_{0}-\omega\right)H_{0}$ are identical to
energy gaps $E_{n}^{0}-E_{k}^{0}$ of $\hbar\omega_{0}H_{0}$ and
$||\hbar\omega_{\text{T}}H_{\pazocal{O},\text{T}}(\omega,t)||=||\hbar\omega_{\text{T}}H_{\text{T}}(\omega,t)||$.
Thus, from Eqs. (311) and (314) we conclude, for any eigenstate
$|E_{k}{(t)}\rangle$, that

$\displaystyle\langle
E^{\pazocal{O}}_{m}(t)|\pazocal{O}(t)|E_{k}(t)\rangle\approx
e^{i\frac{\omega}{\omega_{0}}\frac{E_{k}^{0}}{\hbar}t}\delta_{mk}\text{ , }$
(315)

so that we get $|\langle
E^{\pazocal{O}}_{m}(t)|\pazocal{O}(t)|E_{k}(t)\rangle|=\text{constant}$,
$\forall m$, $\forall t\in[t_{0},\tau]$.

$\bullet$ _Resonance situation $|\omega_{0}-\omega|\ll|\omega_{\text{T}}|$_:
Now, we have a more subtle situation. Firstly, we can use Eqs. (311) and (312)
to write

$\displaystyle\pazocal{O}(t)|E_{n}(t)\rangle$
$\displaystyle=e^{i\frac{\omega}{\omega_{0}}\frac{E_{n}^{0}}{\hbar}t}|E_{n}^{0}\rangle+\pazocal{O}\left(||\hbar\omega_{\text{T}}H_{\text{T}}(\omega,t)||\right)\text{
, }$ (316) $\displaystyle\int_{t_{0}}^{t}\theta_{n}(\xi)d\xi$
$\displaystyle=-\frac{E_{n}^{0}}{\hbar}(t-t_{0})+\pazocal{O}\left(||\hbar\omega_{\text{T}}H_{\text{T}}(\omega,t)||\right)\text{
, }$ (317)

so that

$\displaystyle\langle
E^{\pazocal{O}}_{m}(t)|\pazocal{O}(t)|E_{k}(t)\rangle\approx
e^{i\frac{\omega}{\omega_{0}}\frac{E_{k}^{0}}{\hbar}t}\langle
E^{\pazocal{O}}_{m}(t)|E^{0}_{k}\rangle\text{ . }$ (318)

Now, it is possible to see that if $|\langle
E^{\pazocal{O}}_{m}(t)|E^{0}_{k}\rangle|=|\langle
E^{\pazocal{O}}_{m}(t_{0})|E^{0}_{k}\rangle|$, $\forall t\in[t_{0},\tau]$,
then the theorem is satisfied. By applying this results to the Hamiltonian in
Eq. (296) we can see that for $|\omega_{0}-\omega|\ll|\omega_{\text{T}}|$ we
get $H_{\pazocal{O}}(\omega,t)\propto\cos(\omega t)\sigma{x}-\sin(\omega
t)\sigma{y}$ and

$\displaystyle|E^{\pazocal{O}}_{\pm}(t)\rangle=\frac{|1\rangle\pm e^{i\omega
t}|0\rangle}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}}}\text{ . }$ (319)

Since $|E^{0}_{k}\rangle=|k\rangle$, with $k=\\{0,1\\}$, we conclude that
$|\langle E^{\pazocal{O}}_{m}(t)|E^{0}_{k}\rangle|=|\langle
E^{\pazocal{O}}_{m}(t_{0})|E^{0}_{k}\rangle|$.

##  Chapter 7 Experimental setup

##  Appendix 7.A The Ytterbium trapped ion experimental setup

In this section we present more details on the first system used to
experimentally study some theoretical results presented in this thesis. In
particular, we are interested here in providing a detailed discussion about
the main elements used in this thesis, so that the readers could better
understand our results. For a complete discussion, we indicate the PhD thesis
by Chang-Kang Hu [74] (thesis in Chinese, read it moderately!).

####  7.A.1 Foundations of ion trapping and qubit initialization

In general, the trapping mechanism of neutral or charged particles is
dependent on the particle to be trapped, among many other factors, so here we
provide a discussion on the particular system used in this thesis [74].
However, independent on the system, to trap a particle (charged or not) we use
electric and magnetic fields. A general result, called Earnshaw’s theorem
[168], allows us to identify the minimum requirement to trap a particle in a
stable way, more specifically, a charged particle cannot be held in a stable
equilibrium by using static fields. Even though very general, the
demonstration of this result is very simple and it immediately comes from the
Gauss equations for electric and magnetic static fields
$\vec{\nabla}\cdot\vec{E}=0$ and $\vec{\nabla}\cdot\vec{B}=0$. These equations
tell us that the electric/magnetic field flux going through a region (volume
$V$) is null, so a charged particle under action of this field will undergoes
a non-confining state.

It is reasonable to think about the trapping task for slow motion particles
(less energetic), otherwise we need strong fields to trap them around a
specific position. For atomic beam that comes from a heat gas, we need to
reduce the beam temperature by using a procedure called Doppler cooling [169,
170]. The Doppler cooling is a technique used to slow down the motion of each
atom by using a laser beam tuned slightly below at the natural resonance
frequency of the atom energy level. In a Ytterbium ion 171Yb+, the relevant
energy levels populated during the cooling procedure are shown in Fig. 17 [74,
83]. For the ion 171Yb+ the Doppler cooling is achieved through a $369.5$ nm
laser beam, where the cooling is obtained from the optical transition cycle
${}^{2}S_{1/2}\\!\rightleftarrows\\!^{2}P_{1/2}$ [83]. This cycle transition
happens because the system at state ${}^{2}S_{1/2}$ absorbs a photon and it
becomes excited at the state ${}^{2}P_{1/2}$, then the atom spontaneously
emits a photon in a aleatory direction and uniformly probability distribution.
In particular, if the system decays to state the ${}^{2}S_{1/2}$, the system
remain in cooling cycle. However, as shown in Fig. 17, during the transition
cycle ${}^{2}S_{1/2}\\!\rightleftarrows\\!^{2}P_{1/2}$ there is a branching
ratio $R$ for population decay from ${}^{2}P_{1/2}$ state to the
${}^{2}D_{3/2}$, where the decay probability has been theoretically studied in
Ref. [171], with experimentally verification in Refs. [83, 172]. In order to
sent the system back to the cooling cycle, a light at $935.2$ nm is used to
promote transitions ${}^{2}D_{3/2}\\!\rightleftarrows\\!^{3}D[3/2]_{1/2}$,
where the system can quickly decay from ${}^{3}D[3/2]_{1/2}$ to
${}^{2}S_{1/2}$. Therefore, by driving the system through a many cooling
cycles process we can achieve the cooling limit (minimal temperature). This
limit is known as Doppler cooling limit and given it predicts the minimal
temperature $T_{\text{D}}=\hbar\Gamma/2k_{\text{B}}$, where $\Gamma$ is a
characteristic parameter associated to system (transition linewidth) and
$k_{\text{B}}$ is the Boltzmann’s constant [173, 174, 175, 176, 177]. For the
Ytterbium ion 171Yb+ we have $\Gamma=2\pi\times 19.6$ MHz, so that we can find
$T_{\text{D}}\approx 468.5\leavevmode\nobreak\ \mu$K. Thus, since we are
interested in applications in quantum information processing, after Doppler
cooling step a single ion is trapped by the radio frequency electric field.

Figure 17: Relevant energy levels of the Ytterbium ion 171Yb+ used in the
experimental implementations in this thesis.

After the cooling and trapping process, the qubit is initialized and then can
be manipulated in order to process quantum information. Our qubit is encoded
within energy subspace ${}^{2}S_{1/2}$ spanned by the states
${}^{2}S_{1/2}|F=0,m=0\rangle$, ${}^{2}S_{1/2}|F=1,m=0\rangle$ and
${}^{2}S_{1/2}|F=1,m=\pm 1\rangle$, where $F$ denotes the total angular
momentum of the atom and $m_{F}$ is its projection along the quantization
axis. In absence of an external fields, the subspace $F=1$ is degenerate.
Therefore, when we apply an external static magnetic field $\vec{B}$ the
Zeeman effect in order to highlight the hyperfine structure splitting, so that
an amount of energy $\delta E\propto||\vec{B}||$ separates the energy
eigenstates with $F=1$, as shown in Fig. 2. Thus, we identify subspace of two
states that can be used as a two-level system, i.e., as our desired qubit. The
Zeeman shift in transition frequency is given by $\delta_{\text{Z}}=1.4$ MHz/G
and the transition frequency of the qubit reads as
$\omega_{\text{s}}^{0}\approx 12.64$ GHz. It is important to mention that the
frequency $\omega_{\text{s}}^{0}$ is smoothly affected by the external
magnetic field as
$\omega_{\text{s}}=\omega^{0}_{\text{s}}+\delta_{\text{2z}}$, where
$\delta_{\text{2z}}\approx 310.8B^{2}$ Hz is the second order Zeeman shift
($B$ is the intensity of the static magnetic field in Gauss) [74]. Thus, in
the experiments considered in this thesis, an additional shift about
$2\pi\times 10$ KHz is added. Therefore, the qubit is encoded in states
$|0\rangle\\!\equiv\\!^{2}S_{1/2}|F=0,m=0\rangle$ and
$|1\rangle\\!\equiv\\!^{2}S_{1/2}|F=1,m=0\rangle$, where the energy gap reads
as $E_{01}=\hbar\omega_{\text{s}}$.

####  7.A.2 Trapped ion qubit dynamics

Given the natural configuration of the system and the set of states spanned by
${}^{2}S_{1/2}|F=0,m=0\rangle$ and ${}^{2}S_{1/2}|F=1,m=0\rangle$, with
energies $E_{0}$ and $E_{1}$, respectively, we can define the bare Hamiltonian
for our qubit as

$\displaystyle H_{0}=E_{0}|0\rangle\langle 0|+E_{1}|1\rangle\langle
1|=\frac{E_{01}}{2}\sigma_{z}=\frac{\hbar\omega_{\text{s}}}{2}\sigma_{z}\text{
, }$ (320)

where $\sigma_{z}=|1\rangle\langle 1|-|0\rangle\langle 0|$ is the Pauli
matrix. As schematically shown in Fig. 2, by using an arbitrary waveform
generator (AWG) we can manipulate the qubit through a unitary dynamics by
driving the system with an magnetic field
$\vec{B}_{\text{un}}(t)=\vec{B}_{0}\cos(\omega t+\phi)$, one uses the dipole
approximation to write the contribution of this interaction to energy of the
system as $H_{1}=-\vec{\mu}\cdot\vec{B}_{\text{un}}(t)$, where
$\vec{\mu}=\vec{\mu}_{01}|0\rangle\langle 1|+\vec{\mu}_{10}|1\rangle\langle
0|$ [124, 178]. Here we use the subscript “un” to denote the field associated
with a unitary dynamics, later we shall see that the same AWG can be used to
design a non-unitary dynamics through a field $\vec{B}_{\text{n-un}}(t)$.
Therefore, we find

$\displaystyle H_{1}(t)=-\vec{\mu}_{01}\cdot\vec{B}_{0}\cos(\omega
t+\phi)|0\rangle\langle 1|-\vec{\mu}_{10}\cdot\vec{B}_{0}\cos(\omega
t+\phi)|1\rangle\langle 0|\text{ . }$ (321)

Then, for simplicity we use that $\vec{\mu}_{01}=\vec{\mu}_{10}=\vec{\mu}$, so
that the total Hamiltonian $H(t)=H_{0}+H_{1}$ reads as

$\displaystyle
H(t)=\frac{\hbar\omega_{\text{s}}}{2}\sigma_{z}+\hbar\omega_{\text{R}}\cos(\omega
t+\phi)\sigma_{x}\text{ , }$ (322)

where we defined the Rabi frequency
$\omega_{\text{R}}=-\vec{\mu}_{01}\cdot\vec{B}_{0}/\hbar$ and the Pauli matrix
$\sigma_{x}=|0\rangle\langle 1|+|1\rangle\langle 0|$. Now, by using
perturbation theory we can write the dynamical equation to the coefficients
$c_{0}(t)$ and $c_{1}(t)$ associated with the evolved state

$\displaystyle|\psi(t)\rangle=c_{0}(t)|0\rangle+c_{1}(t)|1\rangle\text{ . }$
(323)

In fact, by writing
$|\psi_{\text{I}}(t)\rangle=e^{iH_{0}t/\hbar}|\psi(t)\rangle$ we get the
dynamics for $|\psi_{\text{I}}(t)\rangle$ as

$\displaystyle|\dot{\psi}_{\text{I}}(t)\rangle=i\hbar
V_{\text{I}}(t)|\psi_{\text{I}}(t)\rangle\text{ , }$ (324)

with $V_{\text{I}}(t)=e^{iH_{0}t/\hbar}H_{1}(t)e^{-iH_{0}t/\hbar}$. As a first
remark, note that $[H_{1}(t_{1}),H_{1}(t_{2})]=0$ for any $(t_{1},t_{2})$,
therefore we can also write $[V_{\text{I}}(t_{1}),V_{\text{I}}(t_{2})]=0$ for
any $(t_{1},t_{2})$. This tells us that the solution for the above equation
reads as

$\displaystyle|\psi_{\text{I}}(t)\rangle=U_{\text{I}}(t)|\psi_{\text{I}}(0)\rangle\text{
, }$ (325)

with

$\displaystyle
U_{\text{I}}(t)=\exp\left[\frac{1}{i\hbar}\int_{0}^{t}V_{\text{I}}(\xi)d\xi\right]\text{
. }$ (326)

We use that

$\displaystyle V_{\text{I}}(t)$
$\displaystyle=e^{iH_{0}t/\hbar}H_{1}(t)e^{-iH_{0}t/\hbar}=\frac{\hbar\omega_{\text{R}}}{2}\left(e^{i(\omega
t+\phi)}+e^{-i(\omega
t+\phi)}\right)e^{iH_{0}t/\hbar}\sigma_{x}e^{-iH_{0}t/\hbar}$
$\displaystyle=\frac{\hbar\omega_{\text{R}}}{2}\left(e^{i(\omega
t+\phi)}+e^{-i(\omega
t+\phi)}\right)e^{i\omega_{\text{s}}t/2}\left(|0\rangle\langle
1|+|1\rangle\langle 0|\right)e^{-i\omega_{\text{s}}t/2}$
$\displaystyle=\frac{\hbar\omega_{\text{R}}}{2}\left(e^{i(\omega
t+\phi)}+e^{-i(\omega
t+\phi)}\right)\left(e^{-i\omega_{\text{s}}t}|0\rangle\langle
1|+e^{i\omega_{\text{s}}t}|1\rangle\langle 0|\right)$
$\displaystyle=\frac{\hbar\omega_{\text{R}}}{2}\left(e^{i(\omega_{-}t+\phi)}+e^{-i(\omega_{+}t+\phi)}\right)|0\rangle\langle
1|+\frac{\hbar\omega_{\text{R}}}{2}\left(e^{i(\omega_{+}t+\phi)}+e^{-i(\omega_{-}t+\phi)}\right)|1\rangle\langle
0|\text{ , }$ (327)

where $\omega_{\pm}=\omega-\omega_{\text{s}}$. Therefore, by integrating the
above expression

$\displaystyle\int_{0}^{t}V_{\text{I}}(\xi)d\xi$
$\displaystyle=\frac{\hbar\omega_{\text{R}}}{2}\left[\frac{-ie^{i\phi}\left(e^{i\omega_{-}t}-1\right)}{\omega_{-}}+\frac{ie^{-i\phi}\left(e^{-i\omega_{+}t}-1\right)}{\omega_{+}}\right]|0\rangle\langle
1|$
$\displaystyle+\frac{\hbar\omega_{\text{R}}}{2}\left[\frac{-ie^{i\phi}\left(e^{i\omega_{+}t}-1\right)}{\omega_{+}}+\frac{ie^{-i\phi}\left(e^{-i\omega_{-}t}-1\right)}{\omega_{-}}\right]|1\rangle\langle
0|\text{ . }$ (328)

Thus, for a highly oscillating external field, we can approximate the above
expression to

$\displaystyle\int_{0}^{t}V_{\text{I}}(\xi)d\xi$
$\displaystyle\approx\frac{\hbar\omega_{\text{R}}}{2}\left[\frac{-ie^{i\phi}\left(e^{i\omega_{-}t}-1\right)}{\omega_{-}}|0\rangle\langle
1|+\frac{ie^{-i\phi}\left(e^{-i\omega_{-}t}-1\right)}{\omega_{-}}|1\rangle\langle
0|\right]\text{ . }$ (329)

For this reason, in our analysis we can neglect highly oscillating terms, as
we shall use soon. Now, by writing the Hamiltonian $H(t)$ in rotating frame by
using $R(t)=e^{i\omega t\sigma_{z}/2}$ we find

$\displaystyle H_{R}$
$\displaystyle=R(t)H(t)R^{\dagger}(t)+i\hbar\dot{R}(t)R^{\dagger}(t)=\frac{\hbar\Delta}{2}\sigma_{z}+R(t)H_{1}(t)R^{\dagger}(t)$
$\displaystyle=\frac{\hbar\Delta}{2}\sigma_{z}+\frac{\hbar\omega_{\text{R}}}{2}\left(e^{i(\omega
t+\phi)}+e^{-i(\omega t+\phi)}\right)\left(e^{-i\omega t}\sigma_{-}+e^{i\omega
t}\sigma_{+}\right)\text{ , }$ (330)

where we define $\Delta=\omega_{\text{s}}-\omega$ and we already used that
$e^{i\omega t\sigma_{z}/2}\sigma_{x}e^{-i\omega t\sigma_{z}/2}=e^{-i\omega
t}\sigma_{-}+e^{i\omega t}\sigma_{+}$, where $\sigma_{-}=|0\rangle\langle 1|$
and $\sigma_{+}=\sigma_{-}^{\dagger}$. Now, by neglecting the highly
oscillating terms, it is possible to proof that we get

$\displaystyle H_{R}$
$\displaystyle=\frac{\hbar\Delta}{2}\sigma_{z}+\frac{\hbar\omega_{\text{R}}}{2}\left(e^{i\phi}\sigma_{-}+e^{-i\phi}\sigma_{+}\right)\text{
. }$ (331)

Now, by using that $\sigma_{\pm}=(1/2)(\sigma_{x}\pm i\sigma_{y})$, we find

$\displaystyle H_{R}$
$\displaystyle=\frac{\hbar\Delta}{2}\sigma_{z}+\frac{\hbar\omega_{\text{R}}}{2}\left(\cos\phi\sigma_{x}+\sin\phi\sigma_{y}\right)\text{
. }$ (332)

Therefore, the above Hamiltonian can be used to implement an arbitrary
dynamics of the qubit encoded as shown in Fig. 2.

####  7.A.3 State detection and tomography process

To read out the outcome of some operation in the trapped ion system used in
this thesis, it is used a ion fluorescence technique [179]. In Ytterbium
trapped ion systems, this technique is applied by brightening the ion with
light at $369.53$ nm, where we promote transitions from the state
${}^{2}S_{1/2}|F=1\rangle$ to ${}^{2}P_{1/2}|F=0\rangle$ [83]. This process
allows us to determinate the population in energy level
${}^{2}S_{1/2}|F=1\rangle$, since the state ${}^{2}S_{1/2}|F=0\rangle$ is a
dark state for the laser at $369.53$ nm. After repeating this procedure many
times, it is possible to accurately estimate the probability in bright state
${}^{2}S_{1/2}|F=1\rangle$ and the dark one. If some population is present in
state ${}^{2}S_{1/2}|F=0\rangle$, few photons are detected due to the
background scattering, and the dark counting of the photomultiplier tube (a
high efficiency detector used to measure the ion fluorescence during the state
detection). On the other hand, a large number of photons are scattered by the
system when the system is in state ${}^{2}S_{1/2}|F=1\rangle$. These scattered
photons are collected by a numerical aperture $NA=0.4$ objective lens. After a
processing of the captured photons, it is possible to estimate that the
measurement fidelity is around $99.4\%$ [43].

In terms of state tomography, the above procedure is used to estimate the
values of the quantities $|\alpha|$ and $|\beta|$ of an unknown state
$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$. Therefore, to measure the
system state we need to follow the tomography standard process where we need
to measure a number of different parameters in order to reconstruct the
density matrix of the system [150]. In our case, the state tomography is
obtained from three parameters, since an arbitrary density matrix for a two-
level system can be written as in Eq. (18).

####  7.A.4 Controllable dephasing in trapped ion systems

In order to simulate the coupling of our qubit with a dephasing reservoir, it
is used frequency modulation (FM) method where a Gaussian noise is introduced
by mixing the field $\vec{B}_{1}(t)$. The experimental is sketched in Fig. 10.
After this process, the new magnetic field reads as (the experiment considered
in this thesis sets $\phi=0$ in Eq. (332).)

$\displaystyle\vec{B}_{\text{n-un}}(t)=\vec{B}_{0}\cos\left[\omega
t+C\eta(t)t\right]\text{ , }$ (333)

where the noise source is encoded in the function $\eta(t)=A_{0}g(t)$, with
$A$ being the average amplitude of the noise, $g(t)$ is a random analog
voltage signal and $C$ is the modulation depth supported by the commercial
microwave generator E8257D. Here we want to demonstrate that this Gaussian
noise can be used to implement the dynamics given by

$\displaystyle\dot{\rho}(t)=\frac{1}{i\hbar}[H(t),\rho(t)]+\gamma(t)\left[\sigma_{z}\rho(t)\sigma_{z}-\rho(t)\right]\text{
, }$ (334)

with $\gamma(t)$ being the desired decohering rate. To this end, we consider
here the time-independent Hamiltonian
$H^{\prime}=\hbar\tilde{\omega}\sigma_{x}$ obtained from Eq. (332) with
$\Delta=0$, $\omega_{\text{R}}=\tilde{\omega}$ and $\phi=0$. That is, a
resonant microwave with Rabi frequency $\tilde{\omega}$. Therefore,
experimentally we can proof that the dynamics is given by the above equation
in two different ways: i) by accompanying the dynamics of the system and ii)
by computing the process matrix $\chi$ that describes the quantum process.
Here we will consider both protocols.

By taking into account a particular dynamics of a system driven by the
Hamiltonian $H^{\prime}$, from Eq. (334) we can write the Bloch equations for
$\rho(t)$ as (with $\gamma(t)=\gamma_{0}$)

$\displaystyle\dot{r}_{x}(t)$ $\displaystyle=-2\gamma_{0}r_{x}(t)\text{ , }$
(335) $\displaystyle\dot{r}_{y}(t)$
$\displaystyle=-2\gamma_{0}\left[r_{z}(t)+r_{y}(t)\right]\text{ , }$ (336)
$\displaystyle\dot{r}_{z}(t)$ $\displaystyle=2\gamma_{0}r_{y}(t)\text{ , }$
(337)

whose solution for the initial condition $|\psi(0)\rangle=|0\rangle$ reads as

$\displaystyle r_{x}(t)$ $\displaystyle=0\text{ , }\quad
r_{y}(t)=\frac{2\omega
e^{-\gamma_{0}t}}{\mathchoice{{\hbox{$\displaystyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=9.02667pt,depth=-7.22137pt}}}{{\hbox{$\textstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=9.02667pt,depth=-7.22137pt}}}{{\hbox{$\scriptstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=8.51111pt,depth=-6.80891pt}}}{{\hbox{$\scriptscriptstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=8.51111pt,depth=-6.80891pt}}}}\sin\left(t\mathchoice{{\hbox{$\displaystyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=9.02667pt,depth=-7.22137pt}}}{{\hbox{$\textstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=9.02667pt,depth=-7.22137pt}}}{{\hbox{$\scriptstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=8.51111pt,depth=-6.80891pt}}}{{\hbox{$\scriptscriptstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule height=8.51111pt,depth=-6.80891pt}}}\right)\text{ , }$ (338)
$\displaystyle r_{z}(t)$
$\displaystyle=-e^{-\gamma_{0}t}\left[\cos\left(t\mathchoice{{\hbox{$\displaystyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=9.02667pt,depth=-7.22137pt}}}{{\hbox{$\textstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=9.02667pt,depth=-7.22137pt}}}{{\hbox{$\scriptstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=8.51111pt,depth=-6.80891pt}}}{{\hbox{$\scriptscriptstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=8.51111pt,depth=-6.80891pt}}}\right)+\frac{\gamma_{0}\sin\left(t\mathchoice{{\hbox{$\displaystyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=9.02667pt,depth=-7.22137pt}}}{{\hbox{$\textstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=9.02667pt,depth=-7.22137pt}}}{{\hbox{$\scriptstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=8.51111pt,depth=-6.80891pt}}}{{\hbox{$\scriptscriptstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=8.51111pt,depth=-6.80891pt}}}\right)}{\mathchoice{{\hbox{$\displaystyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=9.02667pt,depth=-7.22137pt}}}{{\hbox{$\textstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=9.02667pt,depth=-7.22137pt}}}{{\hbox{$\scriptstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=8.51111pt,depth=-6.80891pt}}}{{\hbox{$\scriptscriptstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule height=8.51111pt,depth=-6.80891pt}}}}\right]\text{ . }$
(339)

Therefore, by defining the projector $\mathcal{P}_{1}=|1\rangle\langle 1|$
computing the population in state excited state $|1\rangle$ we get

$\displaystyle
p_{1}(t)=\text{Tr}[\mathcal{P}_{1}\rho(t)]=\frac{1}{2}-\frac{1}{2}e^{-\gamma_{0}t}\left[\cos\left(t\mathchoice{{\hbox{$\displaystyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=9.02667pt,depth=-7.22137pt}}}{{\hbox{$\textstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=9.02667pt,depth=-7.22137pt}}}{{\hbox{$\scriptstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=8.51111pt,depth=-6.80891pt}}}{{\hbox{$\scriptscriptstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=8.51111pt,depth=-6.80891pt}}}\right)+\frac{\gamma_{0}\sin\left(t\mathchoice{{\hbox{$\displaystyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=9.02667pt,depth=-7.22137pt}}}{{\hbox{$\textstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=9.02667pt,depth=-7.22137pt}}}{{\hbox{$\scriptstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=8.51111pt,depth=-6.80891pt}}}{{\hbox{$\scriptscriptstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=8.51111pt,depth=-6.80891pt}}}\right)}{\mathchoice{{\hbox{$\displaystyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=9.02667pt,depth=-7.22137pt}}}{{\hbox{$\textstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=9.02667pt,depth=-7.22137pt}}}{{\hbox{$\scriptstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=8.51111pt,depth=-6.80891pt}}}{{\hbox{$\scriptscriptstyle\sqrt{4\tilde{\omega}^{2}-\gamma_{0}\,}$}\lower
0.4pt\hbox{\vrule height=8.51111pt,depth=-6.80891pt}}}}\right]\text{ , }$
(340)

where in the limit $\gamma_{0}\rightarrow 0$ we recover the well-known
equation $p_{n}(t)|_{\gamma_{0}=0}=\sin^{2}(\tilde{\omega}t)$. As we can see,
the behavior of $p_{n}(t)$ is a damped oscillating function that converges to
$1/2$ in the regime $\gamma_{0}t\gg 1$, where the damping rate is dictated by
the parameter $\gamma_{0}$.

Figure 18: Dephasing rate controlled by the amplitude of noise. (a) Rabi
oscillations between states $|0\rangle$ and $|1\rangle$ under different noise
intensities. From top to bottom, the noise amplitude $A_{0}$ is set to $0.4$
V, $0.8$ V, $1.2$ V, $1.6$ V and $2.0$ V, with the corresponding damping rates
$\gamma$ are approximately $180$ Hz, $649$ Hz, $1391$ Hz, $2441$ Hz and $3785$
Hz, respectively. (b) Dephasing rate as a function of the noise amplitude.
Points are measured data. A linear fit is obtained. Without driving noise
(noise amplitude is zero), the dephasing rate of the qubit is fitted as 3.03
Hz, which is caused by the magnetic fluctuation in the laboratory.

To show how to simulate the non-unitary dynamics in Eq. (334), the
experimental realization it was considered with the parameter
$\tilde{\omega}=2\pi\times 8.44$ KHz and the modulation depth $C$ as $96.00$
KHz/V, here both parameters $\tilde{\omega}$ and $C$ are kept fixed for all
the experiments. Thus, by adjusting the amplitude noise $A_{0}$, different
dynamics were implemented. We experimentally evaluated the quantity $p_{1}(t)$
to get the Rabi oscillations of the system for each dynamics during a total
time interval of $\Delta t_{\text{exp}}=0.5$ ms. The result are shown in Fig.
18a for the Rabi oscillations between states $|0\rangle$ and $|1\rangle$. For
different values of the amplitude $A_{0}$, it is possible to see different
behaviors, each one of them associated with distinct values of $\gamma_{0}$.
Moreover, it is possible to experimentally evaluate the relation between the
dephasing rate $\gamma$ and the noise amplitude $A_{0}$, where we used the
graph in Fig. 18b to this end. By evaluating the square root of $\gamma$ we
find a linear behavior as $A_{0}$ increases, so through the linear regression
method we find (approximately)

$\displaystyle\mathchoice{{\hbox{$\displaystyle\sqrt{\gamma\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{\gamma\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{\gamma\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\gamma\,}$}\lower
0.4pt\hbox{\vrule
height=2.15277pt,depth=-1.72223pt}}}=29.806861A_{0}+1.739986\text{ , }$ (341)

and this show that a perfect control of decoherence rate in Eq. (334) is
obtained from a precise control of amplitude noise.

As a second, and complementary, analysis of the decohering dynamics it was
used the tomography process to reconstruct the process matrix $\chi$. Let us
now give brief introduction to the matrix $\chi$ used to completely describe a
physical process. A general physical process is adequately characterized by
the equation [150]

$\displaystyle\rho(0)\rightarrow\rho(x)=\pazocal{E}_{x}[\rho(0)]=\sum\nolimits_{n=1}^{N}E_{n}\rho(0)E_{n}^{\dagger}\text{
, }$ (342)

where $E_{n}$ is a complete set of $N$ operation elements that satisfy
$\sum_{n=1}^{N}E_{n}^{\dagger}E_{n}=\mathbbm{1}$, because
$\pazocal{E}_{x}[\bullet]$ should be a trace-preserving operation. In general
we have a specific set $\\{E_{n}\\}$ for each process considered, so for this
reason it is convenient to consider a fixed basis of $K$ operators $F_{n}^{k}$
so that we can write $E_{n}=\sum_{k=1}^{K}e_{nk}\bar{E}_{k}$. Therefore, we
get

$\displaystyle\pazocal{E}_{x}[\rho(0)]=\sum\nolimits_{n=1}^{N}\left(\sum\nolimits_{k=1}^{K}e_{nk}\bar{E}_{k}\right)\rho(0)\left(\sum\nolimits_{l=1}^{K}e^{\ast}_{nl}\bar{E}^{\dagger}_{l}\right)=\sum\nolimits_{k,l=1}^{K}\chi_{kl}\bar{E}_{k}\rho(0)\bar{E}^{\dagger}_{l}\text{
. }$ (343)

Figure 19: Process matrix obtained by process tomography, where we plot the
real and imaginary parts of $\chi$ obtained from the experimental measured
data ($\chi_{\text{exp}}$) and numerical simulation ($\chi_{\text{the}}$).

Therefore, since the set is early defined, the above equation tells that the
process is characterized by a matrix with elements $\chi_{kl}$. Thus, by
following this process we adopted the reference basis for a single-qubit
tomography process as $\bar{E}_{1}=\mathbbm{1}$, $\bar{E}_{2}=\sigma_{x}$,
$\bar{E}_{3}=\sigma_{y}$, $\bar{E}_{4}=\sigma_{z}$ in the case where the
dynamics is simulated with $\gamma=2.5$ KHz, where we set the total evolution
time as $0.24$ ms (here, the noise amplitude is 1.62 V and the modulation
depth is 96.00 KHz/V). The resulting estimated process matrix is shown in Fig.
19. Then, the fidelity between the simulated and desired process can be
obtained by evaluating the fidelity between theoretical and experimental
process matrices defined as

$\displaystyle\pazocal{F}({\chi_{\text{exp}}},{\chi_{\text{the}}})=\left[\text{Tr}\mathchoice{{\hbox{$\displaystyle\sqrt{\mathchoice{{\hbox{$\displaystyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule height=2.15277pt,depth=-1.72223pt}}}\text{
}\chi_{\text{the}}\mathchoice{{\hbox{$\displaystyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule height=2.15277pt,depth=-1.72223pt}}}\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{\mathchoice{{\hbox{$\displaystyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule height=2.15277pt,depth=-1.72223pt}}}\text{
}\chi_{\text{the}}\mathchoice{{\hbox{$\displaystyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule height=2.15277pt,depth=-1.72223pt}}}\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{\mathchoice{{\hbox{$\displaystyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\textstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=2.10971pt,depth=-1.68779pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule height=1.50694pt,depth=-1.20557pt}}}\text{
}\chi_{\text{the}}\mathchoice{{\hbox{$\displaystyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\textstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=2.10971pt,depth=-1.68779pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule height=1.50694pt,depth=-1.20557pt}}}\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\mathchoice{{\hbox{$\displaystyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=2.15277pt,depth=-1.72223pt}}}{{\hbox{$\textstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=2.15277pt,depth=-1.72223pt}}}{{\hbox{$\scriptstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=1.50694pt,depth=-1.20557pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule height=1.07639pt,depth=-0.86111pt}}}\text{
}\chi_{\text{the}}\mathchoice{{\hbox{$\displaystyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=2.15277pt,depth=-1.72223pt}}}{{\hbox{$\textstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=2.15277pt,depth=-1.72223pt}}}{{\hbox{$\scriptstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule
height=1.50694pt,depth=-1.20557pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\chi_{\text{exp}}\,}$}\lower
0.4pt\hbox{\vrule height=1.07639pt,depth=-0.86111pt}}}\,}$}\lower
0.4pt\hbox{\vrule height=2.15277pt,depth=-1.72223pt}}}\right]^{2}\text{ . }$
(344)

For example, when the amplitude of the noise is set to $1.54$ V, the process
fidelities are measured as $\pazocal{F}_{t_{1}}=99.27\%$,
$\pazocal{F}_{t_{2}}=99.50\%$, $\pazocal{F}_{t_{3}}=99.72\%$,
$\pazocal{F}_{t_{4}}=99.86\%$ and $\pazocal{F}_{t_{5}}=99.87\%$, at times
$t_{1}=0.08$ ms, $t_{2}=0.16$ ms, $t_{3}=0.24$ ms, $t_{4}=0.32$ ms and
$t_{5}=0.40$ ms, respectively. Thus, the dephasing channel can be precisely
controlled as desired and it can support the scheme to implement the time-
dependent dephasing in experiment. A point to be highlighted here is the case
where $A_{0}=0$, situation in which the system evolves under its natural
dephasing rate $\gamma_{\text{nd}}$. From Eq. (341) one finds that
$\gamma_{\text{nd}}\approx 1.74^{2}\approx 3.03$ Hz. Thus, we can see that, if
we change the parameter $A$, which we can done with high controllability, the
quantity $\gamma$ can be efficiently controlled. On the other hand, if we need
a time-dependent rate $\gamma(t)$, we just need to consider a way to vary
$A_{0}$ as a function $A(t)$. To this end, we use a second channel (CH2) of
the AWG to perform amplitude modulation (AM) of the Gaussian noise. The
temporal dependence of $A(t)$ is achieved by programming the channel (CH2) to
change the noise amplitude along the evolution.

##  Appendix 7.B Nuclear Magnetic Resonance experimental setup

In NMR experimental implementations for quantum information processing, we
encode the quantum bit into degree of freedom associated with the nuclear spin
of atoms in molecules. The NMR qubits used in this thesis are the carbon
nuclei (13C) and the hydrogen nuclei (1H). However, independent on the nucleon
used in our discussion, it is possible to derive the Hamiltonian used to drive
the system and, consequently, implement quantum gates on the qubits. By
applying a strong static magnetic field $\vec{B}_{0}=B_{0}\hat{\text{z}}$
along some direction (let’s say the direction $Z$), we are choosing an
quantization axes and different the spin states (spin up
$|\\!\\!\uparrow\rangle$ and spin down $|\\!\\!\downarrow\rangle$) of the
nuclei set a energy splitting of energy given by $\Delta
E=\hbar\omega_{\text{L}}$, so that we can write the internal Hamiltonian as

$H_{\text{Zee}}=\frac{\hbar\omega_{\text{L}}}{2}\left(|\\!\\!\uparrow\rangle\langle\uparrow\\!\\!|-|\\!\\!\downarrow\rangle\langle\downarrow\\!\\!|\right)=\frac{\hbar\omega_{\text{L}}}{2}\sigma_{z}=\hbar\omega_{\text{L}}I_{z}\text{
, }$ (345)

called Zeeman Hamiltonian due to the Zeeman interaction occurring between the
magnetic dipole moment of the nuclei and the local magnetic fields. The
quantity $\omega_{\text{L}}$ is the Larmor frequency and it is a
characteristic parameter of the nuclei used in experiment; it is associated
with the gyromagnetic ratio of the nucleus $\gamma_{0}$ as
$\omega_{\text{L}}=-\gamma_{0}B_{0}$.

Therefore, the qubit is encoded in states $|\\!\\!\uparrow\rangle$ and
$|\\!\\!\downarrow\rangle$ as $|0\rangle=|\\!\\!\downarrow\rangle$ and
$|1\rangle=|\\!\\!\uparrow\rangle$. The qubit is driven by an transverse
rotating field called radio-frequency (RF) field $\vec{B}_{\text{rf}}(t)$
given by

$\vec{B}_{\text{rf}}(t)=B_{1}\left[\cos(\omega t)\hat{\text{x}}+\sin(\omega
t)\hat{\text{y}}\right]\text{ , }$ (346)

where $\omega$ is the rotating frequency of the field
$\vec{B}_{\text{rf}}(t)$. Therefore, the single-qubit Hamiltonian is a
combination of the Zeeman and RF Hamiltonian, this gives

$H(t)=\hbar\omega_{0}I_{z}+\hbar\omega_{1}\left[\cos(\omega
t)I_{y}+\sin(\omega t)I_{y}\right]\text{ , }$ (347)

with $I_{j}=(1/2)\sigma_{j}$ and $\omega_{1}=-\gamma_{0}B_{1}$. In general,
since the field $|\vec{B}_{1}|$ is much less intense than $|\vec{B}_{0}|$, it
is convenient to consider the qubit dynamics in rotating frame as done in
Appendix 6.A. By doing that, we find the driven Hamiltonian in rotating frame
as

$H=\hbar(\omega_{0}-\omega)I_{z}+\hbar\omega_{1}I_{x}\text{ . }$ (348)

where the term $H_{\text{Zee}}^{\text{eff}}=\hbar(\omega_{0}-\omega)I_{z}$ is
the effective Zeeman Hamiltonian. Therefore, when we set $\omega_{0}=\omega$
we find the resulting Hamiltonian in rotating frame at resonance as

$H_{x}=\hbar\omega_{1}I_{x}\text{ . }$ (349)

Then, by letting the system evolve under this Hamiltonian we get the evolution
operator given by

$U_{x}(\Delta t)=\exp\left(-\frac{i}{\hbar}H_{x}\Delta
t\right)=\exp\left(-i\omega_{1}I_{x}\Delta t\right)=R_{x}(\varpi)\text{ , }$
(350)

where we defined the angle $\varpi=\omega_{1}\Delta t$, with $\Delta t$ being
the pulse duration of the RF field. Therefore, we can adjust the time interval
of each RF pulse in order to select an arbitrary value for $\varpi$. After
such pulse, the system is again governed by the effective Zeeman Hamiltonian,
through a “free evolution”. In this case, the evolution operator becomes

$U_{\text{f}}(\Delta
t)=\exp\left(-\frac{i}{\hbar}H_{\text{Zee}}^{\text{eff}}\Delta
t\right)=\exp\left(-i(\omega_{0}-\omega)I_{z}\Delta
t\right)=R_{z}(\zeta)\text{ , }$ (351)

where we define the angle $\zeta=(\omega_{0}-\omega)\Delta t$. Therefore, by
using pulses $R_{x}(\varpi)$ and free evolution $R_{z}(\zeta)$ we can
implement any single-qubit rotation [152].

In addition, the Hamiltonian that describes a two-qubit system here is
obtained by considering the nuclear Zeeman effect, chemical shifts (due to the
orbital motion of nearby electrons) and $J$-coupling (interaction between the
nuclear magnetic dipole moments of neighbor nuclei), which are represented by
the following secular Hamiltonian (for spin-$\frac{1}{2}$ nuclei)

$\pazocal{H}=\hbar\omega_{0}^{(1)}I_{z}^{(1)}+\hbar\omega_{0}^{(2)}I_{z}^{(2)}+2\pi\hbar
J_{12}I_{z}^{(1)}I_{z}^{(2)}$ (352)

where $I_{z}^{(n)}=(1/2)\sigma_{z}^{(n)}$ is the nuclear spin operators along
the quantization axes for the $n$-th nuclei. Therefore, since the $J$-coupling
can be used to implement any controlled phase gate [31], it is possible to
implement universal quantum computation with these number of single and two-
qubit gates discussed here.

####  7.B.1 Pulses composition for adiabatic and TQD quantum gates

From a Trotter decomposition approach for a time-dependent quantum Hamiltonian
(see, e.g., Ref. [150]), we use the pulse sequence shown in Fig. 16a to
implement each Hamiltonian in Eqs. (280a), (280b) and (280c). The algorithm is
composed by rotations $(\vartheta)_{\varphi_{n}}$ and free evolutions. The
rotations are implemented of an angle $\vartheta$ around a direction
$\hat{\varphi}_{n}=\cos\varphi_{n}\hat{\text{x}}+\sin\varphi_{n}\hat{\text{y}}$,
whose the rotation operator reads [150]

$(\vartheta)_{\varphi_{n}}=\exp\left[-\frac{i\vartheta}{2}\hat{\varphi}_{n}\cdot\vec{\sigma}\right]\text{
, }$ (353)

where
$\vec{\sigma}=\sigma_{x}\hat{\text{x}}+\sigma_{y}\hat{\text{y}}+\sigma_{z}\hat{\text{z}}$,
with $\sigma_{n}$ being the Pauli matrices. Each direction $\varphi_{n}$ of
the rotations in Fig. 16a-I are obtained from

$\varphi_{1}=\pi+\chi_{n}\text{ , }\quad\varphi_{2}=\varphi_{4}=\chi_{n}\text{
, }\quad\varphi_{3}=\frac{\pi}{2}+\chi_{n}\text{ , }$ (354)

with

$\chi_{n}=\sum_{\ell=n+1}^{N}\left[-\frac{4\pi\nu\tau}{N-1}\cos\left(\pi\frac{t_{k}}{\tau}\right)+\frac{\pi}{2}\right]\text{
. }$ (355)

The free evolution used in circuits of the Figs. 16a-I and 16a-II appear as a
consequence of the natural spin evolution of the nuclei in our chloroform
molecule, so that the evolution operator reads as

$U_{\text{free}}(\Delta t)=e^{-\frac{i}{\hbar}H_{\text{free}}\Delta t}\text{ ,
}$ (356)

where

$\displaystyle
H_{\text{free}}=\hbar\frac{J}{2}\sigma_{z}^{(\text{C})}\sigma_{z}^{(\text{H})}\text{
. }$ (357)

In our experiment, the interaction between the spins is constant with strength
$J=215$ Hz, so that we need to manipulate some additional parameters in order
to implement different values of $\tau$ of the interaction term in Eqs. (280).
To this end, we use a free evolution for coupled spins so that we can map the
parameter $\tau$ from two parameter $J$ and the time intervals $\Delta t$ of
each free evolutions. This relation is obtained as

$\displaystyle\Delta t_{1,n}$
$\displaystyle=\frac{\tau}{2(N-1)}\left[\frac{4\nu}{J}\sin\left(\pi\frac{t_{k}}{\tau}\right)+1\right]\text{
, }\quad$ $\displaystyle\Delta t_{2,n}$ $\displaystyle=\frac{\tau}{N-1}-\Delta
t_{1,n}\text{ , }\quad$ $\displaystyle\Delta t_{3,n}$
$\displaystyle=\frac{\tau}{2(N-1)}\left(\frac{1}{J\tau}+1\right)\text{ ,
}\quad$ $\displaystyle\Delta t_{4,n}$ $\displaystyle=\frac{\tau}{N-1}-\Delta
t_{3,n}\text{ . }$ (358)

For Fig. 16a-III, the Hamiltonian is constant and the time interval is then
given by

$\Delta t_{1}=\frac{J\tau+1}{2J}\quad\text{ and }\quad\Delta t_{2}=\tau-\Delta
t_{1}\text{ . }$ (359)

Therefore, we can choose the value of transitionless total evolution time
$\tau$ according with the parameter $\Delta t$, for a constant value of $J$.

##  Chapter 8 Adiabatic dynamics in open system

##  Appendix 8.A Evolution operator for adiabatic dynamics in open system

Here we study some properties of the evolution operator $\pazocal{U}(t,t_{0})$
as derived in Eq. (135). First, given the evolution operator
$\pazocal{U}(t,t_{0})$, let us show a necessary and sufficient condition to
find $\pazocal{U}^{-1}(t,t_{0})$ so that
$\pazocal{U}(t,t_{0})\pazocal{U}^{-1}(t,t_{0})=\mathbbm{1}$. To this, let us
define

$\displaystyle\pazocal{U}^{-1}(t,t_{0})=\sum_{\alpha=1}^{N}\pazocal{U}_{\alpha}^{-1}(t,t_{0})\text{
, }$ (360)

with each operator $\pazocal{U}_{\alpha}^{-1}(t,t_{0})$ obtained from

$\displaystyle\pazocal{U}_{\alpha}^{-1}(t,t_{0})=e^{-\int_{t_{0}}^{t}\lambda_{\alpha}(\xi)d\xi}\sum_{n_{\alpha}=1}^{N_{\alpha}}\sum_{m_{\alpha}=1}^{N_{\alpha}}\tilde{u}_{n_{\alpha}m_{\alpha}}(t)|\pazocal{D}_{\alpha}^{m_{\alpha}}(t_{0})\rangle\rangle\langle\langle\pazocal{E}_{\alpha}^{n_{\alpha}}(t)|\text{
, }$ (361)

with parameters $\tilde{u}_{n_{\alpha}m_{\alpha}}(t)$ to be determined. This
definition is convenient because we can write

$\displaystyle\pazocal{U}_{\beta}(t,t_{0})\pazocal{U}_{\alpha}^{-1}(t,t_{0})=\delta_{\alpha\beta}\pazocal{U}_{\beta}(t,t_{0})\pazocal{U}_{\alpha}^{-1}(t,t_{0})\text{
, }$ (362)

where we use the orthogonality relation between right- and left-hand side
quasi-eigenvectors. Now, we write

$\displaystyle\pazocal{A}_{1}$
$\displaystyle=\pazocal{U}(t,t_{0})\pazocal{U}^{-1}(t,t_{0})=\sum_{\alpha=1}^{N}\sum_{\beta=1}^{N}\pazocal{U}_{\alpha}(t,t_{0})\pazocal{U}_{\beta}^{-1}(t,t_{0})=\sum_{\alpha=1}^{N}\pazocal{U}_{\alpha}(t,t_{0})\pazocal{U}_{\alpha}^{-1}(t,t_{0})\text{
, }$ (363)

where we already used the Eq. (362). Thus

$\displaystyle\pazocal{A}_{1}$
$\displaystyle=\sum_{\alpha=1}^{N}\left[\left(\sum_{n_{\alpha}=1}^{N_{\alpha}}\sum_{m_{\alpha}=1}^{N_{\alpha}}u_{n_{\alpha}m_{\alpha}}(t)|\pazocal{D}_{\alpha}^{n_{\alpha}}(t)\rangle\rangle\langle\langle\pazocal{E}_{\alpha}^{m_{\alpha}}(t_{0})|\right)\left(\sum_{j_{\alpha}=1}^{N_{\alpha}}\sum_{k_{\alpha}=1}^{N_{\alpha}}\tilde{u}_{j_{\alpha}k_{\alpha}}(t)|\pazocal{D}_{\alpha}^{j_{\alpha}}(t_{0})\rangle\rangle\langle\langle\pazocal{E}_{\alpha}^{k_{\alpha}}(t)|\right)\right]$
$\displaystyle=\sum_{\alpha=1}^{N}\left[\sum_{n_{\alpha}=1}^{N_{\alpha}}\sum_{m_{\alpha}=1}^{N_{\alpha}}\sum_{j_{\alpha}=1}^{N_{\alpha}}\sum_{k_{\alpha}=1}^{N_{\alpha}}u_{n_{\alpha}m_{\alpha}}(t)\tilde{u}_{j_{\alpha}k_{\alpha}}(t)|\pazocal{D}_{\alpha}^{n_{\alpha}}(t)\rangle\rangle\left(\langle\langle\pazocal{E}_{\alpha}^{m_{\alpha}}(t_{0})|\pazocal{D}_{\alpha}^{j_{\alpha}}(t_{0})\rangle\rangle\right)\langle\langle\pazocal{E}_{\alpha}^{k_{\alpha}}(t)|\right]$
$\displaystyle=\sum_{\alpha=1}^{N}\left[\sum_{n_{\alpha}=1}^{N_{\alpha}}\sum_{j_{\alpha}=1}^{N_{\alpha}}\sum_{k_{\alpha}=1}^{N_{\alpha}}u_{n_{\alpha}j_{\alpha}}(t)\tilde{u}_{j_{\alpha}k_{\alpha}}(t)|\pazocal{D}_{\alpha}^{n_{\alpha}}(t)\rangle\rangle\langle\langle\pazocal{E}_{\alpha}^{k_{\alpha}}(t)|\right]\text{
. }$ (364)

Now, computing the matrix elements
$\langle\langle\pazocal{E}_{\eta}^{p_{\eta}}(t)|\pazocal{A}_{1}|\pazocal{D}_{\nu}^{m_{\nu}}(t)\rangle\rangle$,
we get (to simplify the notation, from now on we will omit the time-dependence
of the coefficients $u$ and $\tilde{u}$)

$\displaystyle\langle\langle\pazocal{E}_{\eta}^{p_{\eta}}(t)|\pazocal{A}|\pazocal{D}_{\nu}^{m_{\nu}}(t)\rangle\rangle$
$\displaystyle=\sum_{\alpha=1}^{N}\left[\sum_{n_{\alpha}=1}^{N_{\alpha}}\sum_{j_{\alpha}=1}^{N_{\alpha}}\sum_{k_{\alpha}=1}^{N_{\alpha}}u_{n_{\alpha}j_{\alpha}}\tilde{u}_{j_{\alpha}k_{\alpha}}\langle\langle\pazocal{E}_{\eta}^{p_{\eta}}(t)|\pazocal{D}_{\alpha}^{n_{\alpha}}(t)\rangle\rangle\langle\langle\pazocal{E}_{\alpha}^{k_{\alpha}}(t)|\pazocal{D}_{\nu}^{m_{\nu}}(t)\rangle\rangle\right]$
$\displaystyle=\sum_{\alpha=1}^{N}\left[\sum_{n_{\alpha}=1}^{N_{\alpha}}\sum_{j_{\alpha}=1}^{N_{\alpha}}\sum_{k_{\alpha}=1}^{N_{\alpha}}u_{n_{\alpha}j_{\alpha}}\tilde{u}_{j_{\alpha}k_{\alpha}}\delta_{p_{\eta}n_{\alpha}}\delta_{k_{\alpha}m_{\nu}}\delta_{\eta\alpha}\delta_{\alpha\nu}\right]$
$\displaystyle=\sum_{\alpha=1}^{N}\left[\sum_{j_{\alpha}=1}^{N_{\alpha}}u_{p_{\eta}j_{\alpha}}\tilde{u}_{j_{\alpha}m_{\nu}}\delta_{\eta\alpha}\delta_{\alpha\nu}\right]=\delta_{\eta\nu}\sum_{j_{\nu}=1}^{N_{\nu}}u_{p_{\eta}j_{\nu}}\tilde{u}_{j_{\nu}m_{\nu}}\text{
. }$ (365)

Therefore, to obtain $\pazocal{A}_{1}=\mathbbm{1}$ the coefficients need to
satisfy

$\displaystyle\sum_{j_{\nu}=1}^{N_{\nu}}u_{p_{\nu}j_{\nu}}\tilde{u}_{j_{\nu}m_{\nu}}=\delta_{p_{\nu}m_{\nu}}\text{
. }$ (366)

A second important property of the operator $\pazocal{U}(t,t_{0})$ is related
with the Jordan block form of the Lindbladian. To proof it, let us consider a
operator $\pazocal{A}_{2}$ given by
$\pazocal{A}_{2}=\pazocal{U}^{-1}(t,t_{0})\mathbbm{L}(t)\pazocal{U}(t,t_{0})$,
so we have

$\displaystyle\pazocal{A}_{2}$
$\displaystyle=\pazocal{U}^{-1}(t,t_{0})\mathbbm{L}(t)\pazocal{U}(t,t_{0})=\sum_{\alpha=1}^{N}\sum_{\beta=1}^{N}\underbrace{\pazocal{U}_{\alpha}^{-1}(t,t_{0})\mathbbm{L}(t)\pazocal{U}_{\beta}(t,t_{0})}_{\pazocal{A}_{2}^{\alpha\beta}}=\sum_{\alpha=1}^{N}\sum_{\beta=1}^{N}\pazocal{A}_{2}^{\alpha\beta}\text{
. }$ (367)

Now let us analyze the each matrix $\pazocal{A}_{2}^{\alpha\beta}$

$\displaystyle\pazocal{A}_{2}^{\alpha\beta}$
$\displaystyle=\pazocal{U}_{\alpha}^{-1}(t,t_{0})\mathbbm{L}(t)\pazocal{U}_{\beta}(t,t_{0})$
$\displaystyle=e^{\int_{t_{0}}^{t}\lambda_{\beta}(\xi)-\lambda_{\alpha}(\xi)d\xi}\sum_{j_{\alpha}=1}^{N_{\alpha}}\sum_{k_{\alpha}=1}^{N_{\alpha}}\sum_{n_{\beta}=1}^{N_{\beta}}\sum_{p_{\beta}=1}^{N_{\beta}}\tilde{u}_{j_{\alpha}k_{\alpha}}u_{n_{\beta}p_{\beta}}\langle\langle\pazocal{E}_{\alpha}^{k_{\alpha}}(t)|\mathbbm{L}(t)|\pazocal{D}_{\beta}^{n_{\beta}}(t)\rangle\rangle|\pazocal{D}_{\alpha}^{j_{\alpha}}(t_{0})\rangle\rangle\langle\langle\pazocal{E}_{\beta}^{p_{\beta}}(t_{0})|$
$\displaystyle=e^{\int_{t_{0}}^{t}\lambda_{\beta}(\xi)-\lambda_{\alpha}(\xi)d\xi}\sum_{j_{\alpha}=1}^{N_{\alpha}}\sum_{k_{\alpha}=1}^{N_{\alpha}}\sum_{n_{\beta}=1}^{N_{\beta}}\sum_{p_{\beta}=1}^{N_{\beta}}\tilde{u}_{j_{\alpha}k_{\alpha}}u_{n_{\beta}p_{\beta}}\lambda_{\alpha}(t)\langle\langle\pazocal{E}_{\alpha}^{k_{\alpha}}(t)|\pazocal{D}_{\beta}^{n_{\beta}}(t)\rangle\rangle|\pazocal{D}_{\alpha}^{j_{\alpha}}(t_{0})\rangle\rangle\langle\langle\pazocal{E}_{\beta}^{p_{\beta}}(t_{0})|$
$\displaystyle+e^{\int_{t_{0}}^{t}\lambda_{\beta}(\xi)-\lambda_{\alpha}(\xi)d\xi}\sum_{j_{\alpha}=1}^{N_{\alpha}}\sum_{k_{\alpha}=1}^{N_{\alpha}}\sum_{n_{\beta}=1}^{N_{\beta}}\sum_{p_{\beta}=1}^{N_{\beta}}\tilde{u}_{j_{\alpha}k_{\alpha}}u_{n_{\beta}p_{\beta}}\langle\langle\pazocal{E}_{\alpha}^{k_{\alpha}}(t)|\pazocal{D}_{\beta}^{(n_{\beta}-1)}(t)\rangle\rangle|\pazocal{D}_{\alpha}^{j_{\alpha}}(t_{0})\rangle\rangle\langle\langle\pazocal{E}_{\beta}^{p_{\beta}}(t_{0})|\text{
, }$

where we already used the quasi-eigenvalue equation. Thus, due orthogonality
of the basis we write

$\displaystyle\pazocal{A}_{2}^{\alpha\beta}$
$\displaystyle=e^{\int_{t_{0}}^{t}\lambda_{\beta}(\xi)-\lambda_{\alpha}(\xi)d\xi}\sum_{j_{\alpha}=1}^{N_{\alpha}}\sum_{k_{\alpha}=1}^{N_{\alpha}}\sum_{n_{\beta}=1}^{N_{\beta}}\sum_{p_{\beta}=1}^{N_{\beta}}\tilde{u}_{j_{\alpha}k_{\alpha}}u_{n_{\beta}p_{\beta}}\lambda_{\alpha}(t)\delta_{\alpha\beta}\delta_{k_{\alpha}n_{\beta}}|\pazocal{D}_{\alpha}^{j_{\alpha}}(t_{0})\rangle\rangle\langle\langle\pazocal{E}_{\beta}^{p_{\beta}}(t_{0})|$
$\displaystyle+e^{\int_{t_{0}}^{t}\lambda_{\beta}(\xi)-\lambda_{\alpha}(\xi)d\xi}\sum_{j_{\alpha}=1}^{N_{\alpha}}\sum_{k_{\alpha}=1}^{N_{\alpha}}\sum_{n_{\beta}=1}^{N_{\beta}}\sum_{p_{\beta}=1}^{N_{\beta}}\tilde{u}_{j_{\alpha}k_{\alpha}}u_{n_{\beta}p_{\beta}}\delta_{\alpha\beta}\delta_{k_{\alpha}(n_{\beta}-1)}|\pazocal{D}_{\alpha}^{j_{\alpha}}(t_{0})\rangle\rangle\langle\langle\pazocal{E}_{\beta}^{p_{\beta}}(t_{0})|\text{
, }$

and due the $\delta_{\alpha\beta}$, we can write

$\displaystyle\pazocal{A}_{2}$
$\displaystyle=\sum_{\alpha=1}^{N}\sum_{j_{\alpha}=1}^{N_{\alpha}}\sum_{k_{\alpha}=1}^{N_{\alpha}}\sum_{n_{\alpha}=1}^{N_{\alpha}}\sum_{p_{\alpha}=1}^{N_{\alpha}}\tilde{u}_{j_{\alpha}k_{\alpha}}u_{n_{\alpha}p_{\alpha}}\lambda_{\alpha}(t)\delta_{k_{\alpha}n_{\alpha}}|\pazocal{D}_{\alpha}^{j_{\alpha}}(t_{0})\rangle\rangle\langle\langle\pazocal{E}_{\alpha}^{p_{\alpha}}(t_{0})|$
$\displaystyle+\sum_{\alpha=1}^{N}\sum_{j_{\alpha}=1}^{N_{\alpha}}\sum_{k_{\alpha}=1}^{N_{\alpha}}\sum_{n_{\alpha}=1}^{N_{\alpha}}\sum_{p_{\alpha}=1}^{N_{\alpha}}\tilde{u}_{j_{\alpha}k_{\alpha}}u_{n_{\alpha}p_{\alpha}}\delta_{k_{\alpha}(n_{\alpha}-1)}|\pazocal{D}_{\alpha}^{j_{\alpha}}(t_{0})\rangle\rangle\langle\langle\pazocal{E}_{\alpha}^{p_{\alpha}}(t_{0})|$
$\displaystyle=\sum_{\alpha=1}^{N}\sum_{j_{\alpha}=1}^{N_{\alpha}}\sum_{n_{\alpha}=1}^{N_{\alpha}}\sum_{p_{\alpha}=1}^{N_{\alpha}}\left(\tilde{u}_{j_{\alpha}n_{\alpha}}u_{n_{\alpha}p_{\alpha}}\lambda_{\alpha}(t)+\tilde{u}_{j_{\alpha}(n_{\alpha}-1)}u_{n_{\alpha}p_{\alpha}}\right)|\pazocal{D}_{\alpha}^{j_{\alpha}}(t_{0})\rangle\rangle\langle\langle\pazocal{E}_{\alpha}^{p_{\alpha}}(t_{0})|\text{
. }$ (368)

By computing the matrix elements of $\pazocal{A}_{2}$ in basis
$\\{|\pazocal{D}_{\alpha}^{j_{\alpha}}(t_{0})\rangle\rangle,\langle\langle\pazocal{E}_{\alpha}^{p_{\alpha}}(t_{0})|\\}$,
we get

$\displaystyle\langle\langle\pazocal{E}_{\eta}^{g_{\eta}}(t_{0})|\pazocal{A}_{2}|\pazocal{D}_{\nu}^{l_{\nu}}(t_{0})\rangle\rangle$
$\displaystyle=\sum_{\alpha=1}^{N}\sum_{j_{\alpha}=1}^{N_{\alpha}}\sum_{n_{\alpha}=1}^{N_{\alpha}}\sum_{p_{\alpha}=1}^{N_{\alpha}}\left(\tilde{u}_{j_{\alpha}n_{\alpha}}u_{n_{\alpha}p_{\alpha}}\lambda_{\alpha}(t)+\tilde{u}_{j_{\alpha}(n_{\alpha}-1)}u_{n_{\alpha}p_{\alpha}}\right)\delta_{\eta\alpha}\delta_{g_{\eta}j_{\alpha}}\delta_{\alpha\nu}\delta_{p_{\alpha}l_{\nu}}$
$\displaystyle=\sum_{n_{\eta}=1}^{N_{\eta}}\left(\tilde{u}_{g_{\eta}n_{\eta}}u_{n_{\eta}l_{\nu}}\lambda_{\eta}(t)+\tilde{u}_{g_{\eta}(n_{\eta}-1)}u_{n_{\eta}l_{\nu}}\right)\delta_{\eta\nu}\text{
. }$ (369)

As a first result, we can see that
$\langle\langle\pazocal{E}_{\eta}^{g_{\eta}}(t_{0})|\pazocal{A}_{2}|\pazocal{D}_{\nu}^{l_{\nu}}(t_{0})\rangle\rangle=0$
for any element outside some block, it means that $\pazocal{A}_{2}$ is block
diagonal in this basis. Thus, the elements of a single block are

$\displaystyle\langle\langle\pazocal{E}_{\nu}^{g_{\nu}}(t_{0})|\pazocal{A}_{2}|\pazocal{D}_{\nu}^{l_{\nu}}(t_{0})\rangle\rangle$
$\displaystyle=\lambda_{\nu}(t)\sum_{n_{\nu}=1}^{N_{\nu}}\tilde{u}_{g_{\nu}n_{\nu}}u_{n_{\nu}l_{\nu}}+\sum_{n_{\nu}=1}^{N_{\nu}}\tilde{u}_{g_{\nu}(n_{\nu}-1)}u_{n_{\nu}l_{\nu}}\text{
. }$ (370)

The above equation provides us some new conditions on the coefficients which
allows us to identify the elements
$\langle\langle\pazocal{E}_{\nu}^{g_{\nu}}(t_{0})|\pazocal{A}_{2}|\pazocal{D}_{\nu}^{l_{\nu}}(t_{0})\rangle\rangle$
as the elements of the Jordan-Block form for $\mathbbm{L}(t)$. In fact, if we
impose that

$\displaystyle\sum_{n_{\nu}=1}^{N_{\nu}}\tilde{u}_{l_{\nu}(n_{\nu}-1)}u_{n_{\nu}l_{\nu}}=0\quad\text{
and
}\quad\sum_{n_{\nu}=1}^{N_{\nu}}\tilde{u}_{l_{\nu}n_{\nu}}u_{n_{\nu}l_{\nu}}=1\text{
, }$ (371)

we find the diagonal elements of a single block as

$\displaystyle\langle\langle\pazocal{E}_{\nu}^{l_{\nu}}(t_{0})|\pazocal{A}_{2}|\pazocal{D}_{\nu}^{l_{\nu}}(t_{0})\rangle\rangle$
$\displaystyle=\lambda_{\nu}(t)\text{ . }$ (372)

It is important to remark that the second condition in Eq. (371) does not
imply in condition in Eq. (366). In case we are interested in get a Jordan
block decomposition, the first element after a diagonal element should be $1$,
that is, a third condition should provides

$\displaystyle\langle\langle\pazocal{E}_{\nu}^{l_{\nu}}(t_{0})|\pazocal{A}_{2}|\pazocal{D}_{\nu}^{l_{\nu}+1}(t_{0})\rangle\rangle=1\quad\text{
and
}\quad\langle\langle\pazocal{E}_{\nu}^{N_{\nu}}(t_{0})|\pazocal{A}_{2}|\pazocal{D}_{\nu}^{N_{\nu}+1}(t_{0})\rangle\rangle=0\text{
, }$ (373)

where the last condition is associated with last element in Jordan block.
However, the last equation is automatically satisfied because
$|\pazocal{D}_{\nu}^{N_{\nu}+1}(t_{0})\rangle\rangle=0$, for construction.
Therefore, we just need to have

$\displaystyle\lambda_{\nu}(t)\sum_{n_{\nu}=1}^{N_{\nu}}\tilde{u}_{l_{\nu}n_{\nu}}u_{n_{\nu}(l_{\nu}+1)}+\sum_{n_{\nu}=1}^{N_{\nu}}\tilde{u}_{l_{\nu}(n_{\nu}-1)}u_{n_{\nu}(l_{\nu}+1)}=1\text{
, }$ (374)

whose a possible solution is

$\displaystyle\sum_{n_{\nu}=1}^{N_{\nu}}\tilde{u}_{l_{\nu}n_{\nu}}u_{n_{\nu}(l_{\nu}+1)}=0\quad\text{
and
}\quad\sum_{n_{\nu}=1}^{N_{\nu}}\tilde{u}_{l_{\nu}(n_{\nu}-1)}u_{n_{\nu}(l_{\nu}+1)}=1\text{
. }$ (375)

##  Chapter 9 Complementary discussion on transitionless quantum driving

##  Appendix 9.A Transitionless quantum driving in closed systems

As a first task, let us proof the Theorem 4.1. To this end, consider a
$D$-dimensional quantum system and adopt as a measure of energy cost the
Hamiltonian Hilbert-Schmidt norm, which reads

$\displaystyle\Sigma_{\text{tqd}}\left(\tau\right)=\frac{1}{\tau}\int_{0}^{\tau}\mathchoice{{\hbox{$\displaystyle\sqrt{\text{Tr}\left[{H_{\text{tqd}}}^{2}\left(t\right)\right]\,}$}\lower
0.4pt\hbox{\vrule
height=8.63776pt,depth=-6.91023pt}}}{{\hbox{$\textstyle\sqrt{\text{Tr}\left[{H_{\text{tqd}}}^{2}\left(t\right)\right]\,}$}\lower
0.4pt\hbox{\vrule
height=8.63776pt,depth=-6.91023pt}}}{{\hbox{$\scriptstyle\sqrt{\text{Tr}\left[{H_{\text{tqd}}}^{2}\left(t\right)\right]\,}$}\lower
0.4pt\hbox{\vrule
height=6.0722pt,depth=-4.85779pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\text{Tr}\left[{H_{\text{tqd}}}^{2}\left(t\right)\right]\,}$}\lower
0.4pt\hbox{\vrule height=4.70554pt,depth=-3.76445pt}}}\text{ }dt\text{ ,}$
(376)

Then, we obtain

$\displaystyle
H_{\text{tqd}}^{2}\left(t\right)=\hbar^{2}\sum_{n=1}^{D}\left[\frac{}{}|\dot{E}_{n}(t)\rangle\langle\dot{E}_{n}(t)|+\theta_{n}^{2}\left(t\right)|E_{n}(t)\rangle\langle
E_{n}(t)|+i\theta_{n}\left(t\right)\left(|E_{n}(t)\rangle\langle\dot{E}_{n}(t)|-|\dot{E}_{n}(t)\rangle\langle
E_{n}(t)|\right)\right]\text{ .}$ (377)

By taking the trace of $H_{\text{tqd}}^{2}\left(t\right)$ in Eq. (377), we
have

$\displaystyle\text{Tr}\left[H_{\text{tqd}}^{2}\left(t\right)\right]=\sum_{m=1}^{D}\langle
E_{m}(t)|H_{\text{tqd}}^{2}\left(t\right)|E_{m}(t)\rangle=\hbar^{2}\sum_{n=1}^{D}\left[\langle\dot{E}_{n}(t)|\dot{E}_{n}(t)\rangle+\theta_{n}^{2}\left(t\right)+2i\theta_{n}\left(t\right)\langle\dot{E}_{n}(t)|E_{n}(t)\rangle\right]\text{
.}$ (378)

Then

$\displaystyle\Sigma_{\text{\text{tqd}}}\left(\tau\right)=\frac{1}{\tau}\int_{0}^{\tau}\mathchoice{{\hbox{$\displaystyle\sqrt{\sum_{n=1}^{D}\left[\langle\dot{E}_{n}(t)|\dot{E}_{n}(t)\rangle+\Gamma_{n}(\theta_{n})\right]\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\textstyle\sqrt{\sum_{n=1}^{D}\left[\langle\dot{E}_{n}(t)|\dot{E}_{n}(t)\rangle+\Gamma_{n}(\theta_{n})\right]\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\scriptstyle\sqrt{\sum_{n=1}^{D}\left[\langle\dot{E}_{n}(t)|\dot{E}_{n}(t)\rangle+\Gamma_{n}(\theta_{n})\right]\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\sum_{n=1}^{D}\left[\langle\dot{E}_{n}(t)|\dot{E}_{n}(t)\rangle+\Gamma_{n}(\theta_{n})\right]\,}$}\lower
0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}\text{ }dt\text{ ,}$ (379)

where we have
$\Gamma_{n}(\theta_{n})=\theta_{n}^{2}\left(t\right)+2i\theta_{n}\left(t\right)\langle\dot{E}_{n}(t)|E_{n}(t)\rangle$.
We can now find out the functions $\theta_{n}\left(t\right)$ that minimize the
energy cost in transitionless evolutions. For this end, we minimize the
quantity $\Sigma_{\text{tqd}}\left(\tau\right)$ for the Hamiltonian
$H_{\text{tqd}}\left(t\right)$ with respect to parameters
$\theta_{n}\left(t\right)$, where we will adopt it being independents. By
evaluating $\partial_{\theta_{n}}\Sigma\left(\tau\right)$, we obtain

$\displaystyle\partial_{\theta_{n}}\Sigma_{\text{tqd}}\left(\tau\right)=\frac{1}{2\tau}\int_{0}^{\tau}\frac{\partial_{\theta_{n}}\\{\text{Tr}[H_{\text{tqd}}^{2}\left(t\right)]\\}}{\mathchoice{{\hbox{$\displaystyle\sqrt{\text{Tr}\left[{H}_{\text{tqd}}^{2}\left(t\right)\right]\,}$}\lower
0.4pt\hbox{\vrule
height=8.63776pt,depth=-6.91023pt}}}{{\hbox{$\textstyle\sqrt{\text{Tr}\left[{H}_{\text{tqd}}^{2}\left(t\right)\right]\,}$}\lower
0.4pt\hbox{\vrule
height=8.63776pt,depth=-6.91023pt}}}{{\hbox{$\scriptstyle\sqrt{\text{Tr}\left[{H}_{\text{tqd}}^{2}\left(t\right)\right]\,}$}\lower
0.4pt\hbox{\vrule
height=6.0722pt,depth=-4.85779pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\text{Tr}\left[{H}_{\text{tqd}}^{2}\left(t\right)\right]\,}$}\lower
0.4pt\hbox{\vrule height=4.70554pt,depth=-3.76445pt}}}}\text{ }dt\text{ .}$
(380)

We then impose
$\partial_{\theta_{n}}\\{$Tr$[H_{\text{tqd}}^{2}\left(t\right)]\\}=0$ for all
time $t\in\left[0,\tau\right]$, which ensures
$\partial_{\theta_{n}}\Sigma_{\text{tqd}}\left(\tau\right)=0$. Thus, by using
Eq. (378), we write

$\displaystyle\partial_{\theta_{n}}\\{\text{Tr}[H_{\text{tqd}}^{2}\left(t\right)]\\}=2\theta_{n}\left(t\right)+2i\langle\dot{E}_{n}(t)|E_{n}(t)\rangle=0\text{
.}$ (381)

This implies

$\displaystyle\theta_{n}\left(t\right)=\theta^{\text{min}}_{n}\left(t\right)=-i\langle\dot{E}_{n}(t)|E_{n}(t)\rangle\text{
.}$ (382)

From the second derivative analysis, it follows that the choice for
$\theta_{n}\left(t\right)$ as in Eq. (382) necessarily minimizes the energy
cost, namely,
$\partial^{2}_{\theta_{n}}\Sigma_{\text{tqd}}\left(\tau\right)|_{\theta_{n}=\theta^{\text{min}}_{n}}>0$,
which concludes the proof.

Now, to proof the Theorem 4.2 we need to take the time derivative of the
Hamiltonian $H_{\text{SA}}(t)$, where we obtain

$\displaystyle\dot{H}_{\text{SA}}\left(t\right)=i\sum_{n=1}^{D}\frac{d}{dt}\left[\frac{{}}{{}}|\dot{E}_{n}(t)\rangle\langle
E_{n}(t)|+i\theta_{n}\left(t\right)|E_{n}(t)\rangle\langle
E_{n}(t)|\frac{{}}{{}}\right]\text{ .}$ (383)

Then, the matrix elements of $\dot{H}_{\text{SA}}\left(t\right)$ in the
eigenbasis $\left\\{|E_{m}(t)\rangle\right\\}$ of the Hamiltonian $H_{0}(t)$
read

$\displaystyle\langle
E_{k}(t)|\dot{H}_{\text{SA}}\left(t\right)|E_{m}(t)\rangle$
$\displaystyle=i\langle E_{k}(t)|\ddot{E}_{m}(t)\rangle+i\sum_{n=1}^{D}\langle
E_{k}(t)|\dot{E}_{n}(t)\rangle\langle\dot{E}_{n}(t)|E_{m}(t)\rangle$
$\displaystyle-\left[\dot{\theta}_{k}\left(t\right)\delta_{km}+\theta_{m}\left(t\right)\langle
E_{k}(t)|\dot{E}_{m}(t)\rangle+\theta_{k}\left(t\right)\langle\dot{E}_{k}(t)|E_{m}(t)\rangle\right].$
(384)

Now, in second term of the above equation we can use that $\langle
E_{k}(t)|\dot{E}_{n}(t)\rangle=-\langle\dot{E}_{k}(t)|E_{n}(t)\rangle$, to
write $\langle
E_{k}(t)|\dot{E}_{n}(t)\rangle\langle\dot{E}_{n}(t)|E_{m}(t)\rangle=\langle\dot{E}_{k}(t)|E_{n}(t)\rangle\langle
E_{n}(t)|\dot{E}_{m}(t)\rangle$ and thus

$\displaystyle\langle
E_{k}(t)|\dot{H}_{\text{SA}}\left(t\right)|E_{m}(t)\rangle$
$\displaystyle=i\frac{d}{d{t}}\left[\langle
E_{k}(t)|\dot{E}_{m}(t)\rangle\right]$
$\displaystyle-\left\\{\dot{\theta}_{k}\left(t\right)\delta_{km}+\left[\theta_{m}\left(t\right)-\theta_{k}\left(t\right)\right]\langle
E_{k}(t)|\dot{E}_{m}(t)\rangle\right\\}\text{ .}$ (385)

For $k=m$ in Eq. (385), we impose the vanishing of the diagonal elements of
$\dot{H}_{\text{SA}}(t)$, namely, $\langle
E_{k}(t)|\dot{H}_{\text{SA}}\left(t\right)|E_{k}(t)\rangle=0$. This yields

$\displaystyle\dot{\theta}_{m}\left(t\right)=i\frac{d}{dt}\left[\langle
E_{m}(t)|\dot{E}_{m}(t)\rangle\right]\text{ ,}$ (386)

On the other hand, for $k\neq m$ in Eq. (385), we now impose the vanishing of
the off-diagonal elements of $\dot{H}_{\text{SA}}(t)$, namely, $\langle
E_{k}(t)|\dot{H}_{\text{SA}}\left(t\right)|E_{m}(t)\rangle=0$ $(k\neq m)$.
This yields

$\displaystyle i\frac{d}{dt}\left[\langle
E_{k}(t)|\dot{E}_{m}(t)\rangle\right]=\left[\theta_{m}\left(t\right)-\theta_{k}\left(t\right)\right]\langle
E_{k}(t)|\dot{E}_{m}(t)\rangle\ \ \text{ for }k\neq m\text{ .}$ (387)

By taking $\langle E_{m}(t)|\dot{E}_{m}(t)\rangle\equiv c_{mm}$ in Eq. (386),
with $c_{mm}$ denoting by hypothesis complex constants, we get
$\theta_{m}\left(t\right)=\theta_{m}\left(0\right)\equiv\theta_{m}$, namely,
$\theta_{m}\left(t\right)$ is a constant function $\forall$ $m$. Moreover, by
using that $\langle E_{k}(t)|\dot{E}_{m}(t)\rangle\equiv c_{km}$ in Eq. (387),
with $c_{km}$ denoting nonvanishing complex constants, we obtain
$\theta_{k}=\theta_{m}$, $\forall$ $k,m$. If $c_{km}=0$, then $\theta_{k}$ and
$\theta_{m}$ are not necessarily equal, but Eq. (387) will also be satisfied
by this choice. Therefore, it follows that $\theta_{m}(t)$ can be simply taken
as

$\displaystyle\theta_{m}(t)=\theta_{m}=\theta\,\,\,\forall m\,,$ (388)

with $\theta$ a single real constant. This concludes the proof.

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