S(t,t_{0})=S(t)-S(t_{0})=-\text{Tr}\left[\rho(t)\ln[\rho(t)]\right]+\text{Tr}\left[\rho(t_{0})\ln[\rho(t_{0})]\right]\text{
, }$ (192)

that can be written in a differential way as

$dS(t)=-\left[\frac{d}{dt}\text{Tr}\left[\rho(t)\ln[\rho(t)]\right]\right]dt=-\text{Tr}[\dot{\rho}(t)\log\rho(t)]-\text{Tr}[\dot{\rho}(t)]\text{
, }$ (193)

and by using that $\text{Tr}[\dot{\rho}(t)]=0$, because
$\text{Tr}[\rho(t)]=1$, and $\dot{\rho}(t)=\pazocal{L}_{t}[\rho(t)]$ we get

$dS(t)=-\text{Tr}[\dot{\rho}(t)\log\rho(t)]dt=-\text{Tr}[\pazocal{L}_{t}[\rho(t)]\log\rho(t)]dt\text{
. }$ (194)

Now, let us to write

$\log\rho(t)=\frac{1}{D}\sum_{n=0}^{D_{-}}\varrho_{n}^{\log}(t)\sigma_{n}^{\dagger}\text{
, }$ (195)

so that we can define the vectors $\langle\langle\rho_{\log}(t)|$ associated
to $\log\rho(t)$ with components $\varrho_{n}^{\log}(t)$ obtained as
$\varrho_{n}^{\log}(t)=\text{Tr}[\sigma_{n}\log\rho(t)]$. Thus, we get

$dS(t)=-\frac{1}{D^{2}}\sum_{m=0}^{D_{-}}\sum_{n=0}^{D_{-}}\varrho_{m}(t)\varrho_{n}^{\log}(t)\text{Tr}[\pazocal{L}_{t}[\sigma_{m}]\sigma_{n}^{\dagger}]dt=-\frac{1}{D}\langle\langle\rho_{\log}(t)|\mathbbm{L}(t)|\rho(t)\rangle\rangle
dt\text{ . }$ (196)

Thus, from Eqs. (185), (191) and (196) we can study the thermodynamics of
adiabatic dynamics in a general framework. However, since work of an adiabatic
dynamics has been addressed in a number of applications, here we are
interested in study the implications of the adiabatic dynamics in the
expression of heat derived in Eqs. (185) and (196). To this end, let us
consider the most general case in which the system starts from a superposition
$|\rho(0)\rangle\rangle=\sum\nolimits_{\alpha,k_{\alpha}}r^{k_{\alpha}}_{\alpha}|\pazocal{D}^{k_{\alpha}}_{\alpha}(0)\rangle\rangle$,
associated with the initial arbitrary matrix density $\rho(0)$. Then, if we
take into account the situations where the system evolves adiabatically, we
use the evolution operator in Eqs. (135) and (134) to write the evolved as

$\displaystyle|\rho_{\text{ad}}(t)\rangle\rangle$
$\displaystyle=\pazocal{U}(t)|\rho(0)\rangle\rangle=\sum_{\beta}^{N}\sum_{k_{\beta}}^{N_{\beta}}\sum_{\alpha=1}^{N}\sum_{n_{\alpha},m_{\alpha}}^{N_{\alpha}}r^{k_{\beta}}_{\beta}\eta_{n_{\alpha}m_{\alpha}}(t)|\pazocal{D}_{\alpha}^{n_{\alpha}}(t)\rangle\rangle\langle\langle\pazocal{E}_{\alpha}^{m_{\alpha}}(0)|\pazocal{D}^{k_{\beta}}_{\beta}(0)\rangle\rangle$
$\displaystyle=\sum_{\beta}^{N}\sum_{n_{\beta},k_{\beta}}^{N_{\beta}}r^{k_{\beta}}_{\beta}\eta_{n_{\beta}k_{\beta}}(t)|\pazocal{D}_{\beta}^{n_{\beta}}(t)\rangle\rangle$
(197)

where we defined
$\eta_{n_{\alpha}m_{\alpha}}(t)=e^{\int_{0}^{t}\lambda_{\alpha}(\xi)d\xi}u_{n_{\alpha}m_{\alpha}}(t)$.
By using the above result, from Eq. (189) we write

$d\bar{}\hskip 1.60004ptQ_{\text{ad}}=\frac{1}{D}\langle\langle
h(t)|\mathbbm{L}(t)|\rho_{\text{ad}}(t)\rangle\rangle
dt=\frac{1}{D}\sum_{\beta}^{N}\sum_{n_{\beta},k_{\beta}}^{N_{\beta}}r^{k_{\beta}}_{\beta}\eta_{n_{\beta}k_{\beta}}(t)\langle\langle
h(t)|\mathbbm{L}(t)|\pazocal{D}_{\beta}^{n_{\beta}}(t)\rangle\rangle dt\text{
, }$ (198)

as the amount of heat exchanged between the system and its environment along
an adiabatic dynamics in open systems. A more useful expression can be found
using the quasi-eigenvector equation for $\mathbbm{L}(t)$ given in Eq. (30)
and reads as

$d\bar{}\hskip
1.60004ptQ_{\text{ad}}(t)=\frac{1}{D}\sum_{\beta}^{N}\sum_{n_{\beta},k_{\beta}}^{N_{\beta}}r^{k_{\beta}}_{\beta}\eta_{n_{\beta}k_{\beta}}(t)\left[\langle\langle
h(t)|\pazocal{D}_{\beta}^{n_{\beta}-1}(t)\rangle\rangle+\lambda_{\beta}(t)\langle\langle
h(t)|\pazocal{D}_{\beta}^{n_{\beta}}(t)\rangle\rangle\right]dt\text{ . }$
(199)

Moreover, let us consider the case where we have an one-dimensional Jordan
block decomposition of $\mathbbm{L}(t)$. In this case, we use the adiabatic
solution in Eq. (189) to write

$d\bar{}\hskip 1.60004ptQ_{\text{ad}}^{\text{1D}}(t)=\frac{1}{D}\langle\langle
h(t)|\mathbbm{L}(t)|\rho^{1\text{D}}_{\text{ad}}(t)\rangle\rangle
dt=\frac{1}{D}\sum_{\beta}^{N}r_{\beta}e^{\int_{0}^{t}\Lambda_{\beta}(t^{\prime})dt^{\prime}}\lambda_{\beta}(t)\langle\langle
h(t)|\pazocal{D}_{\beta}^{n_{\beta}}(t)\rangle\rangle dt\text{ . }$ (200)

The Eqs. (199) and (200) constitute the main result of this section, where it
is evident the possibility of heat exchange during an adiabatic dynamics in
open system. In addition, the Eq. (200) allows us to find a condition for
achieving an adiabatic process through an adiabatic dynamics. In fact, note
that if we start the dynamics from a superposition of eigenvectors
$\\{\pazocal{D}_{\beta}^{n_{\beta}}(0)\\}$ of $\mathbbm{L}(0)$ with
$\lambda_{\beta}(t)=0$ for all $t$, we get $d\bar{}\hskip
1.60004ptQ_{\text{ad}}^{\text{1D}}(t)=0$ for all $t$. In conclusion, we can
establish that an adiabatic dynamics in quantum mechanics is not in general
associated with an adiabatic process in quantum thermodynamics, with a
sufficient condition for thermal adiabaticity being the evolution within an
eigenstate set with vanishing eigenvalue of $\mathbbm{L}(t)$.

To end, from Eq. (196), we can write the von Neumann entropy variation as

$dS(t)=-\frac{1}{D}\langle\langle\rho^{\text{ad}}_{\log}(t)|\mathbbm{L}(t)|\rho^{\text{ad}}(t)\rangle\rangle
dt=-\frac{1}{D}\sum\nolimits_{i,k_{i}}c^{(k_{i})}_{i}e^{\int_{0}^{t}\tilde{\lambda}_{i,k_{i}}(t^{\prime})dt^{\prime}}\langle\langle\rho^{\text{ad}}_{\log}(t)|\mathbbm{L}(t)|\pazocal{D}^{(k_{i})}_{i}(t)\rangle\rangle
dt\text{ , }$

so that we can use the Eq. (30) to write

$dS(t)=\frac{1}{D}\sum\nolimits_{i,k_{i}}c^{(k_{i})}_{i}e^{\int_{0}^{t}\tilde{\lambda}_{i,k_{i}}(t^{\prime})dt^{\prime}}\Gamma_{i,k_{i}}(t)dt\text{
, }$ (201)

where
$\Gamma_{i,k_{i}}(t)=\langle\langle\rho^{\text{ad}}_{\log}(t)|\pazocal{D}^{(k_{i}-1)}_{i}(t)\rangle\rangle+\lambda_{i}(t)\langle\langle\rho^{\text{ad}}_{\log}(t)|\pazocal{D}^{(k_{i})}_{i}(t)\rangle\rangle$,
with $\langle\langle\rho^{\text{ad}}_{\log}(t)|$ standing for the adiabatic
evolved state associated with $\langle\langle\rho_{\log}(t)|$.

_Thermal adiabaticity for a qubit adiabatic dynamics –_ As an illustration of
how adiabatic dynamics is an adiabatic process, let us begin by considering a
two-level system initialized in a thermal equilibrium state
$\rho_{\text{th}}(0)$ for the Hamiltonian $H(0)$ at inverse temperature
$\beta\\!=\\!1/k_{\text{B}}T$, where $k_{\text{B}}$ and $T$ are the
Boltzmann’s constant and the absolute temperature, respectively. Now, let the
system be governed by a Lindblad equation, where the environment acts as a
dephasing channel in the energy eigenstate basis $\\{|E_{n}(t)\rangle\\}$ of
$H(t)$. Thus, we describe the coupling between the system and its reservoir
through

$\dot{\rho}(t)=\pazocal{L}[\rho(t)]=\frac{1}{i\hbar}[H(t),\rho(t)]+\gamma(t)\left[\Gamma^{\text{dp}}(t)\rho(t)\Gamma^{\text{dp}}(t)-\rho(t)\right]\text{
, }$ (202)

where $\Gamma^{\text{dp}}(t)=|E_{1}(t)\rangle\langle
E_{1}(t)|-|E_{0}(t)\rangle\langle E_{0}(t)|$, with $|E_{n}(t)\rangle$ being
the instantaneous eigenstates of $H(t)$, here we are assuming
$E_{1}(t)>E_{0}(t)$ for all $t$. In this case, the set of eigenvectors of
$\pazocal{L}[\bullet]$ can be obtained from set of operators
$P_{nm}(t)=|E_{n}(t)\rangle\langle E_{m}(t)|$, because we have

$\displaystyle\pazocal{L}[P_{nm}(t)]$
$\displaystyle=\frac{1}{i\hbar}[H(t),P_{nm}(t)]+\gamma(t)\left[\Gamma^{\text{dp}}(t)P_{nm}(t)\Gamma^{\text{dp}}(t)-P_{nm}(t)\right]$
$\displaystyle=\frac{1}{i\hbar}\left[E_{n}(t)-E_{m}(t)\right]P_{nm}(t)+\gamma(t)\left[\Gamma^{\text{dp}}(t)P_{nm}(t)\Gamma^{\text{dp}}(t)-P_{nm}(t)\right]P_{nm}(t)$
(203)

Now, by computing the second term in above equation we get

$\displaystyle\Gamma^{\text{dp}}(t)P_{nm}(t)\Gamma^{\text{dp}}(t)$
$\displaystyle=\left[|E_{1}(t)\rangle\langle E_{1}(t)|-|E_{0}(t)\rangle\langle
E_{0}(t)|\right]|E_{n}(t)\rangle\langle E_{m}(t)|\left[|E_{1}(t)\rangle\langle
E_{1}(t)|-|E_{0}(t)\rangle\langle E_{0}(t)|\right]$
$\displaystyle=\left[\delta_{1n}|E_{1}(t)\rangle-\delta_{0n}|E_{0}(t)\rangle\right]\left[\delta_{1m}\langle
E_{1}(t)|-\delta_{0m}\langle E_{0}(t)|\right]$
$\displaystyle=\sum_{k=0,1}\delta_{kn}\delta_{km}|E_{k}(t)\rangle\langle
E_{k}(t)|-\sum_{k=0,1}\delta_{kn}\delta_{\bar{k}m}|E_{k}(t)\rangle\langle
E_{\bar{k}}(t)|\text{ , }$ (204)

so that we can write

$\Gamma^{\text{dp}}(t)P_{nm}(t)\Gamma^{\text{dp}}(t)=\left\\{\begin{matrix}P_{nn}(t)&\text{for
}n=m\\\ -P_{nm}(t)&\text{for }n\neq m\end{matrix}\right.\text{ , }$ (205)

where we conclude that $P_{nm}(t)$ satisfies the eigenvalue equation
$\pazocal{L}[P_{nm}(t)]=\lambda_{nm}(t)P_{nm}(t)$ with

$\lambda_{nm}(t)=\left\\{\begin{matrix}0&\text{for }n=m\\\
\left[E_{n}(t)-E_{m}(t)\right]/i\hbar-2\gamma(t)&\text{for }n\neq
m\end{matrix}\right.\text{ . }$ (206)

Therefore, by writing $P_{nm}(t)$ in superoperator formalism the of
$|\pazocal{D}_{nm}(t)\rangle\rangle$, with components
$\pazocal{D}^{(i)}_{nm}(t)=\text{Tr}\left[P_{nm}(t)\sigma_{i}\right]$, the
eigenvalue equation for $\mathbbm{L}(t)$ can be written as

$\mathbbm{L}(t)|\pazocal{D}_{nm}(t)\rangle\rangle=\lambda_{nm}(t)|\pazocal{D}_{nm}(t)\rangle\rangle\text{
, }$ (207)

where $\lambda_{nm}(t)=E_{n}(t)-E_{m}(t)-2(1-\delta_{nm})\gamma(t)$. Since we
are considering the initial state as a thermal state, in the superoperator
formalism the initial state $\rho_{\text{th}}(0)$, we can write

$\rho_{\text{th}}(0)=\pazocal{Z}^{-1}(0)\sum_{n}e^{-\beta
E_{n}(0)}P_{nn}(0)\text{ \, \, }\Rightarrow\text{ \, \,
}|\rho_{\text{th}}(0)\rangle\rangle=\pazocal{Z}^{-1}(0)\sum_{n}e^{-\beta
E_{n}(0)}|\pazocal{D}_{nn}(0)\rangle\rangle\text{ , }$ (208)

where $\mathcal{Z}(0)=\text{Tr}\left[e^{-\beta H(0)}\right]$ is the partition
function of the system. Note that, since $|\rho_{\text{th}}(0)\rangle\rangle$
is given by a superposition of eigenvectors of $\mathbbm{L}(t)$ with
eigenvalue $\lambda_{nn}(t)=0$ for all $t$, we obtain from Eq. (200) that
$d\bar{}\hskip 1.60004ptQ_{\text{ad}}=0$ for all $t$ whenever the evolution is
adiabatic. Therefore, thermal adiabaticity is achieved for an arbitrary open-
system adiabatic dynamics subject to dephasing in the energy eigenbasis.
Hence, any internal energy variation for this situation should be identified
as work.

####  10.3 Experimental realization of heat exchange in adiabatic dynamics

Now, we will consider a case where the open system dynamics is adiabatic and
some amount of heat is exchanged between system and reservoir. The system used
here is the same as before, a two level system evolving under dephasing.
However, here we conveniently change the decohering basis and now the
decohering acts in computational basis $|0\rangle$ and $|1\rangle$, instead
the eigenstate basis of $H(t)$. Then the dynamics of the system is given by

$\dot{\rho}(t)=\pazocal{L}[\rho(t)]=\frac{1}{i\hbar}[H_{x},\rho(t)]+\gamma(t)\left[\sigma_{z}\rho(t)\sigma_{z}-\rho(t)\right]\text{
, }$ (209)

with $H_{x}=\hbar\omega\sigma_{x}$. Since we are interested in studying the
heat transfer during the adiabatic dynamics, we drive the system with a time-
independent Hamiltonian, so that $d\bar{}\hskip 1.60004ptW=0$ for all $t$ and
therefore any variation in internal energy of the system is due to heat flux.
Moreover, because we want to drive the system through an adiabatic dynamics,
the parameter $\gamma(t)$ needs to be time-dependent.

The system is initialized in the thermal of $H_{\text{x}}$ at inverse
temperature $\beta$ written as

$\rho(0)=\frac{1}{2}\left(\mathbbm{1}+\tanh[\beta\hbar\omega]\sigma_{x}\right)\text{
, }$ (210)

where rewrite the above state in superoperator formalism as the state
$|\rho(0)\rangle\rangle$ in the basis $\mathcal{O}_{\text{tls}}$ as

$|\rho(0)\rangle\rangle=|\sigma_{0}\rangle\rangle-\tanh[\beta\hbar\omega]|\sigma_{x}\rangle\rangle\text{
, }$ (211)

with the basis $|\sigma_{k}\rangle\rangle$ ($k=\\{0,x,y,z\\}$) defined in Eq.
(23). In this dynamics, the superoperator $\mathbbm{L}(t)$ associated with the
generator $\pazocal{L}[\bullet]$ reads

$\mathbbm{L}(t)=\begin{bmatrix}0&0&0&0\\\ 0&-2\gamma(t)&0&0\\\
0&0&-2\gamma(t)&-2\omega\\\ 0&0&2\omega&0\end{bmatrix}\text{ . }$ (212)

It is possible to show that the set
$\\{|\sigma_{0}\rangle\rangle,|\sigma_{x}\rangle\rangle\\}$ satisfies the
eigenvalue equation for $\mathbbm{L}(t)$ as

$\mathbbm{L}(t)|\sigma_{0}\rangle\rangle=0\text{ \ \ , \ \
}\mathbbm{L}(t)|\sigma_{x}\rangle\rangle=-2\gamma(t)|\sigma_{x}\rangle\rangle\text{
, }$ (213)

so, if the dynamics is adiabatic, we can write the evolved state as
$|\rho_{\text{ad}}(t)\rangle\rangle=r_{1}(t)|\sigma_{0}\rangle\rangle+r_{x}(t)|\sigma_{x}\rangle\rangle$,
where $r_{y}(t)=r_{y}(0)=0$ and $r_{z}(t)=r_{z}(0)=0$ because the coefficients
evolve independently form each other. Thus, from the adiabatic solution in
open quantum system given in Eq. (114), we obtain $r_{1}(t)=1$ and
$r_{x}(0)=-\tanh[\beta\hbar\omega]$, so that we can use
$\tilde{\lambda}_{1}=0$ and $\tilde{\lambda}_{x}=-2\gamma(t)$ to obtain

$|\rho_{\text{ad}}(t)\rangle\rangle=|\sigma_{0}\rangle\rangle-e^{-2\int_{0}^{t}\gamma(\xi)d\xi}\tanh[\beta\hbar\omega]|\sigma_{x}\rangle\rangle\text{
. }$ (214)

Notice that by rewriting the above equation in the standard operator
formalism, one gets

$\rho_{\text{ad}}(t)=\frac{1}{2}\left[\mathbbm{1}-e^{-2\int_{0}^{t}\gamma(\xi)d\xi}\tanh(\beta\hbar\omega)\sigma_{x}\right]\text{
, }$ (215)

moreover, by using this formalism, it is also possible to show that the
dephasing channel can be used as a thermalization process if we suitably
choose the parameter $\gamma(t)$ and the total evolution time
$\tau_{\text{dec}}$. In fact, we can define a new inverse temperature
$\beta_{\text{deph}}$ so that Eq. (214) behaves as thermal state, namely,

$|\rho(t)\rangle\rangle=|\sigma_{0}\rangle\rangle-\tanh[\beta_{\text{deph}}\hbar\omega]|\sigma_{x}\rangle\rangle\text{
, }$ (216)

where we immediately identify

$\beta_{\text{deph}}=\frac{1}{\hbar\omega}\text{arctanh}\left[e^{-2\int_{0}^{t}\gamma(\xi)d\xi}\tanh(\beta\hbar\omega)\right]\text{
. }$ (217)

Now, we use the Eq. (199) to compute the exchanged heat during an
infinitesimal time interval $dt$ as

$d\bar{}\hskip
1.60004ptQ_{\text{ad}}(t)=2\hbar\tanh(\beta\hbar\omega)\omega\gamma(t)e^{-2\int_{0}^{t}\gamma(\xi)d\xi}dt\text{
, }$ (218)

and therefore we get the total exchanged heat as a function of the total
evolution time $\tau_{\text{dec}}$ in which the system remains coupled to
reservoir as

$Q_{\text{ad}}(\tau_{\text{dec}})=\int_{0}^{\tau_{\text{dec}}}\left(\frac{d\bar{}\hskip
1.60004ptQ_{\text{ad}}(t)}{dt}\right)dt=\hbar\omega\tanh(\beta\hbar\omega)\left(1-e^{-2\bar{\gamma}\tau_{\text{dec}}}\right)\text{
, }$ (219)

where
$\bar{\gamma}=(1/\tau_{\text{dec}})\int_{0}^{\tau_{\text{dec}}}\gamma(\xi)d\xi$
is the average dephasing rate during $\tau_{\text{dec}}$. A first remark here
is that $Q_{\text{ad}}(\tau_{\text{dec}})>0$ for any value of $\bar{\gamma}>0$
and $\tau$, so this means that the reservoir here works as an artificial
thermal reservoir at inverse temperature
$\tilde{\beta}=\beta_{\text{deph}}<\beta$. We also considered a similar
analysis in an experimental simulation of a thermal engine in all optical
experimental setup [136]. In addition, due to the negative argument in the
exponential, the higher the mean-value of $\gamma(t)$ the faster the heat
exchange ends. Then, we can further compute the maximum exchanged heat from
Eq. (219), in the limit $\bar{\gamma}\tau_{\text{dec}}\gg 1$, as a quantity
independent of the environment parameters and given by
$Q_{\text{max}}=\hbar\omega\tanh(\beta\hbar\omega)$. This result can be also
obtained from Eqs. (183) and (215) by using the asymptotic steady state
$\rho_{\text{ss}}=\rho_{\text{ad}}(t\rightarrow\infty)\propto\mathbbm{1}$. It
would be worth highlighting that, for quantum thermal machines weakly coupled
to thermal reservoirs at different temperatures [134], the maximum heat
$Q_{\text{max}}$ is obtained with high-temperature hot reservoirs [137, 138,
139].

The experimental realization of the above results was done through the trapped
ion system already used previously, but in this case we need to consider an
additional challenger due to the parameter $\gamma(t)$ to be time-dependent.
To do that, it is used the same procedure as before with an additional element
able to provide a time-dependent decohering quantum channel with high
controllability. For further details, please see Appendix 7.A.4. In Fig. 12 we
show the experimental results for the heat exchange
$Q_{\text{ad}}(\tau_{\text{dec}})$ as a function of $\tau_{\text{dec}}$, where
we have chosen $\gamma(t)\\!=\\!\gamma_{0}(1+t/\tau_{\text{dec}})$, where here
the parameter $\tau_{\text{dec}}$ is experimentally controlled through the
time interval associated to the action of our decohering quantum channel. The
solid curves in Fig. 12 are computed from Eq. (219), while the experimental
points are directly computed through the variation of internal energy as
$Q(\tau_{\text{dec}})\\!=\\!U_{\text{fin}}-U_{\text{ini}}$, where
$U_{\text{fin(ini)}}\\!=\\!\text{Tr}[\rho_{\text{fin(ini)}}H_{x}]$. The
computation of $U_{\text{fin(ini)}}$ is directly obtained from quantum state
tomography of $\rho_{\text{fin(ini)}}$ for each value of $\tau_{\text{dec}}$.
Although the maximum exchanged heat is independent of $\gamma_{0}$, the
initial dephasing rate $\gamma_{0}$ affects the power for which the system
exchanges heat with the reservoir for a given evolution time
$\tau_{\text{dec}}$.

Figure 12: Heat $Q_{\text{ad}}(\tau_{\text{dec}})$ as a function of the total
evolution time $\tau_{\text{dec}}$ for several values of the parameter
$\gamma_{0}$. We use $\hbar\omega=82.662$ peV and $\beta^{-1}=17.238$ peV,
with the physical constants $\hbar\approx 6.578\cdot 10^{-16}$ eV$\cdot$s and
$k_{\text{B}}\approx 8.619\cdot 10^{-5}$ eV/K.

By completeness and to provide a more detailed view of this heat exchange, we
analyze the von Neumann entropy during the evolution. From Eq. (196), for the
particular case of an one-dimensional Jordan block decomposition, we write the
entropy variation in our dynamics as

$dS(t)=\frac{1}{2}\sum\nolimits_{i={0}}^{1}c_{i}e^{\int_{0}^{t}\tilde{\lambda}_{i}(t^{\prime})dt^{\prime}}\lambda_{i}(t)\langle\langle\rho^{\text{ad}}_{\log}(t)|\pazocal{D}_{i}(t)\rangle\rangle
dt\text{ , }$ (220)

where we used that
$\Gamma_{i}(t)=\lambda_{i}(t)\langle\langle\rho^{\text{ad}}_{\log}(t)|\pazocal{D}_{i}(t)\rangle\rangle$.
By computing $\langle\langle\rho^{\text{Ad}}_{\log}(t)|$, where we find

$\langle\langle\rho^{\text{Ad}}_{\log}(t)|=\log\left(\frac{1-g^{2}(t)}{4}\right)\langle\langle
1|-2\text{arctanh}[g(t)]\langle\langle x|\text{ , }$ (221)

with $g(t)\\!=\\!e^{-2\int_{0}^{t}\gamma(\xi)d\xi}\tanh(\beta\hbar\omega)$.
Then, from the set of adopted values for our parameters and the spectrum of
the Lindbladian, we conclude that

$dS(t)=4g(t)\gamma(t)\text{arctanh}[g(t)]dt\text{ . }$ (222)

Notice that the relation between heat and entropy can be obtained by rewriting
the exchanged heat $dQ$ in the interval $dt$ as $d\bar{}\hskip
1.60004ptQ_{\text{ad}}(t)=2\hbar\omega\gamma(t)g(t)dt$. In conclusion, the
energy variation can indeed be identified as heat exchanged along the
adiabatic dynamics. Indeed, by computing the thermodynamic relation between
$dS(t)$ and $d\bar{}\hskip 1.60004ptQ_{\text{ad}}(t)$ we get
$dS(t)=\beta_{\text{deph}}d\bar{}\hskip 1.60004ptQ_{\text{ad}}(t)$, where
$\beta_{\text{deph}}$ is the inverse temperature of the simulated thermal bath
shown in Eq. (217).

To guarantee that the amount of heat shown in Fig. 12 are associated with an
exchange heat in an adiabatic dynamics, we computed the fidelity
$\pazocal{F}(\tau_{\text{dec}})$ between the evolved state (solution of the
Eq. (209)) and the adiabatic solution given in Eq. (215). In Table 1 we show
the minimum value of the fidelity along the evolution, that is, we compute the
quantity
$\pazocal{F}_{\text{min}}=\min_{\tau_{\text{dec}}}\pazocal{F}(\tau_{\text{dec}})$.
From Table 1 it is possible to see that all of heat exchange behavior shown in
Fig. 12 are associated with an adiabatic dynamics in open system with high
fidelity.

Table 1: Minimum value of the experimental fidelity $\pazocal{F}_{\text{min}}=\min_{\tau_{\text{dec}}}\pazocal{F}(\tau_{\text{dec}})$ for each choice of $\gamma_{0}$. The maximum experimental error $\Delta\pazocal{F}_{\text{min}}$ for $\pazocal{F}_{\text{min}}$ is about $\Delta\pazocal{F}_{\text{min}}=0.13\%$ of $\pazocal{F}_{\text{min}}$. $\gamma_{0}$ | 314 Hz | 628 Hz | 1257 Hz | 3142 Hz | 6283 Hz
---|---|---|---|---|---
$\pazocal{F}_{\text{min}}$ | 0.9971(3) | 0.9965(4) | 0.9980(7) | 0.9952(8) | 0.9942(9)

Despite we have provided a specific open-system adiabatic evolution, we can
determine infinite classes of system-environment interactions exhibiting the
same amount of heat exchange $d\bar{}\hskip 1.60004ptQ$. In particular, there
are infinite engineered environments that are able to extract a maximum heat
amount $Q_{\text{max}}$. In fact, it is possible to determine infinite classes
of systems exhibiting the same amount of heat exchange $d\bar{}\hskip
1.60004ptQ$. This is provided in Theorem 1 below.

######  Theorem 3.1.

Let ${\mathcal{S}}$ be an open quantum system governed by a time-local master
equation in the form
$\dot{\rho}(t)=\pazocal{H}[\rho(t)]+\pazocal{R}_{t}[\rho(t)]$, where
$\pazocal{H}[\bullet]=(1/i\hbar)[H,\bullet]$ and
$\pazocal{R}_{t}[\bullet]=\sum\nolimits_{n}\gamma_{n}(t)[\Gamma_{n}(t)\bullet\Gamma^{\dagger}_{n}(t)-(1/2)\\{\Gamma^{\dagger}_{n}(t)\Gamma_{n}(t),\bullet\\}]$.
The Hamiltonian $H$ is taken as a constant operator so that no work is
realized by/on the system. Assume that the heat exchange between
${\mathcal{S}}$ and its reservoir during the quantum evolution is given by
$d\bar{}\hskip 1.60004ptQ$. Then, any unitarily related adiabatic dynamics
driven by
$\dot{\rho}^{\prime}(t)=\pazocal{H}^{\prime}[\rho^{\prime}(t)]+\pazocal{R}^{\prime}_{t}[\rho^{\prime}(t)]$,
where $\dot{\rho}^{\prime}(t)=U\dot{\rho}(t)U^{\dagger}$,
$\pazocal{H}^{\prime}[\bullet]=U\pazocal{H}[\bullet]U^{\dagger}$ and
$\pazocal{R}^{\prime}_{t}[\bullet]=U\pazocal{R}_{t}[\bullet]U^{\dagger}$, for
some constant unitary $U$, implies in an equivalent heat exchange
$d\bar{}\hskip 1.60004ptQ^{\prime}=d\bar{}\hskip 1.60004ptQ$.

  * Proof.

Let us consider that $\rho(t)$ is solution of

$\dot{\rho}(t)=\pazocal{H}[\rho(t)]+\pazocal{R}_{t}[\rho(t)]\text{ , }$ (223)

so, by multiplying both sides of the above equation by $U$ (on the left-hand-
side) and $U^{\dagger}$ (on the right-hand-side), we get

$\displaystyle U\dot{\rho}(t)U^{\dagger}$
$\displaystyle=U\pazocal{H}[\rho(t)]U^{\dagger}+U\pazocal{R}_{t}[\rho(t)]U^{\dagger}$
$\displaystyle=\frac{1}{i\hbar}U[H,\rho(t)]U^{\dagger}+\sum\nolimits_{n}\gamma_{n}(t)U\Gamma_{n}(t)\rho(t)\Gamma^{\dagger}_{n}(t)U^{\dagger}-\frac{1}{2}\sum\nolimits_{n}\gamma_{n}(t)U\\{\Gamma^{\dagger}_{n}(t)\Gamma_{n}(t),\rho(t)\\}]U^{\dagger}\text{
, }$ (224)

thus, by using the relations $[UAU^{\dagger},UBU^{\dagger}]=U[A,B]U^{\dagger}$
and $\\{UAU^{\dagger},UBU^{\dagger}\\}=U\\{A,B\\}U^{\dagger}$, we find

$\dot{\rho}^{\prime}(t)=\frac{1}{i\hbar}[UHU^{\dagger},\rho^{\prime}(t)]+\sum\nolimits_{n}\gamma_{n}(t)\Gamma^{\prime}_{n}(t)\rho^{\prime}(t)\Gamma^{\prime\dagger}_{n}(t)-\frac{1}{2}\sum\nolimits_{n}\gamma_{n}(t)\\{\Gamma^{\prime\dagger}_{n}(t)\Gamma^{\prime}_{n}(t),\rho^{\prime}(t)\\}\text{
, }$ (225)

where $\Gamma^{\prime}(t)=U\Gamma_{n}(t)U^{\dagger}$. In conclusion, we get
that $\rho^{\prime}(t)=U\rho(t)U^{\dagger}$ is a solution of

$\dot{\rho}^{\prime}(t)=\pazocal{H}^{\prime}[\rho^{\prime}(t)]+\pazocal{R}^{\prime}_{t}[\rho^{\prime}(t)]\text{
, }$ (226)

where

$\displaystyle\pazocal{H}^{\prime}[\bullet]$
$\displaystyle=\frac{1}{i\hbar}[UHU^{\dagger},\bullet]=U\pazocal{H}[\bullet]U^{\dagger},$
(227) $\displaystyle\pazocal{R}_{t}^{\prime}[\bullet]$
$\displaystyle=\sum\nolimits_{n}\gamma_{n}(t)[\Gamma^{\prime}_{n}(t)\rho^{\prime}(t)\Gamma^{\prime\dagger}_{n}(t)-\frac{1}{2}\\{\Gamma^{\prime\dagger}_{n}(t)\Gamma^{\prime}_{n}(t),\rho^{\prime}(t)\\}]=U\pazocal{R}_{t}[\bullet]U^{\dagger}\text{
. }$ (228)

Now, by taking into account that the Hamiltonian $H$ is a constant operator,
we have that no work is realized by/on the system. Then, by computing the
amount of heat extracted from the system in the prime dynamics during an
interval $t\in[0,\tau]$, we obtain

$Q^{\prime}=\text{Tr}[H^{\prime}\rho^{\prime}(\tau)]-\text{Tr}[H^{\prime}\rho^{\prime}(0)]\text{
, }$ (229)

where, by definition, we can use $\rho^{\prime}(t)=U\rho(t)U^{\dagger}$,
$\forall t\in[0,\tau]$. Hence

$Q^{\prime}=\text{Tr}[H^{\prime}U\rho(\tau)U^{\dagger}]-\text{Tr}[H^{\prime}U\rho(0)U^{\dagger}]=\text{Tr}[U^{\dagger}H^{\prime}U\rho(\tau)]-\text{Tr}[U^{\dagger}H^{\prime}U\rho(0)]=Q\text{
, }$ (230)

where we have used the cyclical property of the trace and that
$Q=\text{Tr}[H\rho(\tau)]-\text{Tr}[H\rho(0)]$. $\blacksquare$

As an example of application of the above theorem, let us consider a system-
reservoir interaction governed by
$\pazocal{R}^{\text{x}}_{t}[\bullet]=\gamma(t)\left[\sigma_{x}\bullet\sigma_{x}-\bullet\right]$
(bit-flip channel). We can then show that the results previously obtained for
dephasing can be reproduced if the quantum system is initially prepared in
thermal state of $H_{\text{y}}^{0}=\omega\sigma_{y}$. Such a result is clear
if we choose $U=R_{\text{x}}(\pi/2)R_{\text{z}}(\pi/2)$. Then, it follows that
$\pazocal{R}^{\text{x}}_{t}[\bullet]=U\pazocal{R}^{\text{z}}_{t}[\bullet]U^{\dagger}$
and $\pazocal{H}^{\prime}[\bullet]=U\pazocal{H}[\bullet]U^{\dagger}$, where
$R_{\text{z(x)}}(\theta)$ are rotation matrices with angle $\theta$ around
$z(x)$-axes for the case of a single qubit. Thus, the above theorem assures
that the maximum exchanged heat will be
$Q_{\text{max}}=\hbar\omega\tanh[\beta\hbar\omega]$.

###  11 Conclusions of this chapter

In this chapter we studied foundations on adiabatic dynamics of open system,
where we could to apply our results in context of quantum thermodynamics.
First, we have shown that, in completeness with the results present in
literature [25, 5], we have shown a class of quantum systems where the
adiabatic approximation for open systems holds in the asymptotic time limit
$t\rightarrow\infty$. However, such result is an arbitrarily valid consequence
of the one-dimensional Jordan decomposition of the Lindblad superoperator that
drives the system, the absence of level crossings, and the initialization of
the system as a superposition of only two eigenstates of such operator. As an
application, we studied both theoretical and experimental implementation of an
adiabatic version of the Deutsch algorithm, analyzing its run-time optimality
in open systems. In contrast with the general picture previously derived in
the literature [25, 5], we have shown that the adiabatic approximation for
open systems holds in the asymptotic time limit $t\rightarrow\infty$ in the
Deutsch algorithm. However, it is worth mentioning that, if we are interested
in maximizing the probability of measuring the target (pure) state as the
outcome of the algorithm, instead of the open system adiabatic mixed-state
density operator, then there is an optimal time window for the system
measurement (for previously related discussions on this topic, see also Refs.
[7, 5, 22, 140]). Concerning the experimental results, we have reported, to
the best of our knowledge, the first experimental investigation of the
adiabatic approximation in a fully controllable open system, where the total
evolution time and the decoherence rates can be freely set to verify the
quantitative validity conditions for the adiabatic behavior. By using a
hyperfine qubit encoded in ground-state energy levels of a trapped Ytterbium
ion, all of the theoretical predictions highlighted above have been
successfully realized.

As an application, we studied the thermodynamics of adiabatic evolutions in
context of open systems. From the general approach for adiabaticity in open
quantum systems discussed here, we provided a relationship between
adiabaticity in quantum mechanics and in quantum thermodynamics. In
particular, we derived a sufficient condition for which the adiabatic dynamics
in open quantum systems leads to adiabatic processes in thermodynamics. By
using a particular example of a single qubit undergoing an open-system
adiabatic evolution path, we have illustrated the existence of both adiabatic
and diabatic regimes in quantum thermodynamics, computing the associated heat
fluxes in the processes. As a further result, we also proved the existence of
an infinite family of decohering systems exhibiting the same maximum heat
exchange. From the experimental side, we have realized adiabatic open-system
evolutions using an Ytterbium trapped ion, with its hyperfine energy level
encoding a qubit (work substance). In particular, heat exchange and entropy
production can be optimized along the adiabatic path as a function of the
total evolution time. Our implementation exhibits high controllability,
opening perspectives for analyzing thermal machines (or refrigerators) in open
quantum systems under adiabatic evolutions. The associated effects of the
engineered reservoirs on the thermal efficiencies are left for future
research.

Then, the results presented in this chapter provide a framework to exploit the
experimental realization of adiabaticity in quantum computing under
decoherence, tackling features such as optimal run-time of an algorithm,
asymptotic time behavior, competition between adiabatic and relaxation time-
scales, outcome fidelities, thermodynamic processes via adiabatic dynamics,
among others.

##  Chapter 4 Transitionless Quantum Driving in closed systems

Adiabatic dynamics is a useful approach to achieve a number of quantum tasks.
However, as already previously highlighted, the constraints on the speed
evolution is a problem when we consider the undesired effects. As a strategy
to bypass these undesired effects in adiabatic dynamics, Mustafa Demirplak and
Stuart A. Rice proposed a special dynamics that allows us to mimic an
adiabatic dynamics without the requirement of slowly driving the system. The
problem solved by Demirplak and Rice in 2003 is how to speed up an adiabatic
dynamics [26]. To this end, Demirplak and Rice showed how to use an auxiliary
field to inhibit natural transitions between energy level of the system when
we implement fast evolutions. This additional field constitutes a counter
diabatic field paradigm, as mentioned in their revised paper, published two
years later [27]. In this chapter we will review the method developed by
Demirplak and Rice, revisited some years later by Michael Victor Berry [50].
As a contribution of this thesis to this field, we show how to generalize such
method and we highlight the benefits we can get from this generalized
approach. In recent years, the term “counter diabatic” has been replaced by
the terminology transitionless quantum driving (TQD) used by Berry [50]. Here
we will use the Berry’s terminology and the term “counter diabatic” will be
used whenever it is convenient for us. The results presented in this chapter
are associated with the following works:

$\bullet$ A. C. Santos and M. S. Sarandy, “Generalized shortcuts to
adiabaticity and enhanced robustness against decoherence”, J. Phys. A: Math.
Theor. 51, 025301 (2018).

$\bullet$ C.-K. Hu, J.-M. Cui, A. C. Santos, Y.-F. Huang, M. S. Sarandy, C.-F.
Li, and G.-C. Guo, “Experimental implementation of generalized transitionless
quantum driving”, Opt. Lett. 43, 3136 (2018).

$\bullet$ A. C. Santos, “Quantum gates by inverse engineering of a
Hamiltonian”, J. Phys. B: At. Mol. Opt. Phys. 51, 015501 (2018).

$\bullet$ A. C. Santos, A. Nicotina, A. M. Souza, R. S. Sarthour, I. S.
Oliveira, and M. S. Sarandy, “Optimizing NMR quantum information processing
via generalized transitionless quantum driving”, EPL (Europhysics Letters)
129, 30008 (2020).

###  12 Standard TQD in closed quantum systems

Consider $H_{0}(t)$ the driven reference (adiabatic) Hamiltonian of a quantum
system with dimension $D$, $|E_{n}(t)\rangle$ the set of instantaneous
eigenstates of $H(t)$ with eigenvalues $E_{n}(t)$. The solution for the
dynamics of the system that evolves through an adiabatic path (trajectory) is
given by

$|\psi_{\text{ad}}(t)\rangle=\sum_{n=1}^{D}c_{n}\exp\left[-\frac{i}{\hbar}\int_{0}^{t}E_{n}(t)+i\hbar\langle\dot{E}_{n}(t)|E_{n}(t)\rangle
dt\right]|E_{n}(t)\rangle\text{ , }$ (231)

with the coefficients $c_{i}$ defined by the initial state of the system
$|\psi(0)\rangle=\sum\nolimits_{n}^{D}c_{n}|E_{n}(0)\rangle$. By convenience
in the notation, from now on we will define the set of adiabatic phase
$\theta^{\text{ad}}_{n}(t)$ as quantal phase given by
$\theta^{\text{ad}}_{n}(t)=\theta^{\text{din}}_{n}(t)+\theta^{\text{geo}}_{n}(t)$,
where $\theta^{\text{din}}_{n}(t)$ and $\theta^{\text{geo}}_{n}(t)$ are the
dynamical and geometrical phases defined as [81]

$\theta^{\text{din}}_{n}(t)=-\frac{1}{\hbar}E_{n}(t)\text{ \ \ \ and \ \ \
}\theta^{\text{geo}}_{n}(t)=-i\langle\dot{E}_{n}(t)|E_{n}(t)\rangle\text{ . }$
(232)

In this way, it is straightforward to define the evolution operator
$U_{\text{ad}}(t)$ of an adiabatic dynamics as

$U_{\text{ad}}(t)=\sum_{n=1}^{D}e^{i\int_{0}^{t}\theta^{\text{ad}}_{n}(t)dt}|E_{n}(t)\rangle\langle
E_{n}(0)|\text{ , }$ (233)

because we can check that
$|\psi_{\text{ad}}(t)\rangle=U_{\text{ad}}(t)|\psi(0)\rangle$. From the above
operator $U_{\text{ad}}(t)$, Demirplak and Rice raised the question whether
there is another Hamiltonian (not the adiabatic) that shares of the same
evolution operator. Since we know the evolution operator of a given dynamics,
we can use an inverse engineering approach to determinate the Hamiltonian that
implements such dynamics. The Hamiltonian that mimics the adiabatic dynamics
can be obtained from inverse engineering using that

$\displaystyle H_{\text{tqd}}^{\text{std}}(t)$ $\displaystyle=-i\hbar
U_{\text{ad}}\left(t\right)\dot{U}_{\text{ad}}^{{\dagger}}\left(t\right)=\sum_{n=1}^{D}\sum_{m=1}^{D}e^{i\int_{0}^{t}\theta^{\text{ad}}_{n}(\xi)d\xi}|E_{n}(t)\rangle\langle
E_{n}(0)|E_{m}(0)\rangle\frac{d}{dt}\left[e^{-i\int_{0}^{t}\theta^{\text{ad}}_{n}(\xi)d\xi}\langle
E_{m}(t)|\right]$
$\displaystyle=H_{0}(t)+i\hbar\sum_{n}^{D}|\dot{E}_{n}(t)\rangle\langle
E_{n}(t)|+\langle\dot{E}_{n}(t)|E_{n}(t)\rangle|E_{n}(t)\rangle\langle
E_{n}(t)|\text{ , }$ (234)

where we already used the Eqs. (232) and
$H_{0}(t)=\sum_{n}E_{n}(t)|E_{n}(t)\rangle\langle E_{n}(t)|$, the spectral
decomposition of $H_{0}(t)$. A first point to be highlighted here is the
absence of any condition on the total evolution time, so a system driven by
the above Hamiltonian follows an adiabatic path framed by the reference
Hamiltonian $H_{0}(t)$ for any instant $t$. More rigorously, we showed that
naturally the speed of the evolution is constrained by the energy cost of the
implementation, with faster evolutions being more energy demanding [10].
Therefore, note that this approach requires our ability to include external
fields to the original field used to implement $H_{0}(t)$. These additional
fields work as inhibitors of diabatic transitions between energy levels of the
system. For this reason, these fields has been referred as counter diabatic
fields and the additional term in $H_{\text{tqd}}^{\text{std}}(t)$ as counter
diabatic Hamiltonian. Thus, we write
$H_{\text{tqd}}^{\text{std}}(t)=H_{0}(t)+H_{\text{cd}}(t)$, with

$H_{\text{cd}}(t)=i\hbar\sum_{n}^{D}|\dot{E}_{n}(t)\rangle\langle
E_{n}(t)|+\langle\dot{E}_{n}(t)|E_{n}(t)\rangle|E_{n}(t)\rangle\langle
E_{n}(t)|\text{ . }$ (235)

Here we are using the label “std” in $H_{\text{tqd}}^{\text{std}}(t)$ to
denote the Hamiltonian that implements a TQD as provided by the “standard”
model developed by Demirplak and Rice. It is convenient because we shall see
how to generalize this approach, in which this approach is recovered as
particular case of the generalized approach.

The standard TQD method has been investigated in a number of different
applications, where different discussions on the energetic cost of
implementing such evolution has been considered. For example, studies on
energetic cost of standard TQD were considered through the Frobenius norm of
the Hamiltonian [10, 33, 32], quantum speed limit inequalities [51] and
average thermodynamics work [141]. In all of these studies, standard TQD is
more costing than its adiabatic counterpart and we can say that speed up
adiabatic processes through this method demands additional energy resources.
However, in this section we will present how to provide more efficient
(energetically) TQD Hamiltonian by exploring the Gauge freedom of the phases
that accompanying the adiabatic and, consequently, the TQD evolution.

###  13 Generalized TQD in closed quantum systems

In some protocols of inverse engineering of a Hamiltonian [142, 31], the
starting point is the equation

$H(t)=-i\hbar U(t)\dot{U}^{\dagger}(t)\text{ , }$ (236)

with $U(t)$ being the operator that describes the desired evolution. Now,
observe that there are a number of adiabatic processes in which the quantal
phases collected along the evolution can be negligible, situations where we
are interested in the adiabatic trajectory associated to a single eigenstate
of $H_{0}(t)$. For this reason, we can use the quantal phases that
accompanying a TQD as a free parameter and define the generalized TQD
evolution operator given by

$U_{\text{tqd}}(t)=\sum_{n=1}^{D}e^{i\int_{0}^{t}\theta_{n}(\xi)d\xi}|E_{n}(t)\rangle\langle
E_{n}(0)|\text{ , }$ (237)

with $\theta_{n}(t)$ being arbitrary parameters, that can be adjusted to
minimize some physical quantity, for example. The TQD Hamiltonian considered
in previous section is a particular Hamiltonian that can be found from above
equation. In fact, by adopting the choice
$\theta_{n}(t)=\theta^{\text{ad}}_{n}(t)$, we get
$U_{\text{tqd}}(t)=U_{\text{ad}}(t)$. The operator $U_{\text{tqd}}(t)$
implements an generalized TQD, because it allows for a TQD with arbitrary
quantal phases, consequently we can find the generalized TQD Hamiltonian as

$\displaystyle H_{\text{tqd}}(t)$ $\displaystyle=-i\hbar
U_{\text{tqd}}(t)\dot{U}_{\text{tqd}}^{\dagger}(t)=\sum_{n=1}^{D}\sum_{m=1}^{D}e^{i\int_{0}^{t}\theta_{n}(\xi)d\xi}|E_{n}(t)\rangle\langle
E_{n}(0)|E_{m}(0)\rangle\frac{d}{dt}\left[e^{i\int_{0}^{t}\theta_{m}(\xi)d\xi}\langle
E_{m}(t)|\right]$
$\displaystyle=i\hbar\sum_{n=1}^{D}|\dot{E}_{n}(t)\rangle\langle
E_{n}(t)|+i\theta_{n}(t)|E_{n}(t)\rangle\langle E_{n}(t)|\text{ . }$ (238)

Now, if we identify the functions $\theta_{n}\left(t\right)$ as it has
originally been identified with the adiabatic phase
$\theta_{n}(t)=\theta^{\text{ad}}_{n}(t)$, which exactly mimics an adiabatic
evolution, then we can write $H_{\text{tqd}}(t)$ as
$H_{0}(t)+H_{\text{CD}}(t)=H_{\text{tqd}}^{\text{std}}(t)$. Now, it is
relevant to highlight here the absence of the original reference Hamiltonian
$H_{0}(t)$ in generalized TQD Hamiltonian $H_{\text{tqd}}(t)$. This result
makes clear that we do not need to know how to implement $H_{0}(t)$ if we want
to drive the system through an adiabatic trajectory of $H_{0}(t)$ via a TQD
method. Moreover, if we define the energy cost of TQD using the Frobenius
norm, as done in Refs. [10, 32, 33], one notes that
$||H_{\text{tqd}}(t)||\neq||H_{\text{tqd}}^{\text{std}}(t)||$. Therefore, a
question is raised here: Is the generalized TQD method energetically more
efficient than the standard one?

####  13.1 Energetically optimal transitionless quantum driving

As already mentioned, there is a number of situations for which an adiabatic
process do not need to be mimicked exactly, but only assure that the system is
kept in an instantaneous eigenstate of the original Hamiltonian $H_{0}(t)$
(independently of its associated quantum phase) [10, 32, 33, 64, 65, 66, 49,
67, 68, 42, 69, 70, 41]. This generalized dynamics in terms of arbitrary
phases $\theta_{n}(t)$ will be denoted as a transitionless evolution. Now, we
will show that $\theta_{n}(t)$ can be nontrivially optimized in transitionless
evolutions both in terms of energy cost and robustness against decoherence
effects. In this direction, we adopt as a measure of energy cost the average
Hilbert-Schmidt norm of the Hamiltonian throughout the evolution, which is
given by [45, 10, 51, 143]

$\Sigma\left(\tau\right)=\frac{1}{\tau}\int_{0}^{\tau}||H(t)||\text{
}dt=\frac{1}{\tau}\int_{0}^{\tau}\mathchoice{{\hbox{$\displaystyle\sqrt{\text{Tr}\left[{H}^{2}\left(t\right)\right]\,}$}\lower
0.4pt\hbox{\vrule
height=8.63776pt,depth=-6.91023pt}}}{{\hbox{$\textstyle\sqrt{\text{Tr}\left[{H}^{2}\left(t\right)\right]\,}$}\lower
0.4pt\hbox{\vrule
height=8.63776pt,depth=-6.91023pt}}}{{\hbox{$\scriptstyle\sqrt{\text{Tr}\left[{H}^{2}\left(t\right)\right]\,}$}\lower
0.4pt\hbox{\vrule
height=6.0722pt,depth=-4.85779pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\text{Tr}\left[{H}^{2}\left(t\right)\right]\,}$}\lower
0.4pt\hbox{\vrule height=4.70554pt,depth=-3.76445pt}}}\text{ }dt\text{ \ ,}$
(239)

where $\tau$ denotes the total evolution time. The energy cost
$\Sigma\left(\tau\right)$ aims at identifying changes in the energy coupling
constants and gap structure, which typically account for the effort of
speeding up adiabatic processes. It is a well-defined measure for finite-
dimensional Hamiltonians exhibiting non-degeneracies in their energy spectra,
so it is applicable, e.g., for generic systems composed of a finite number of
quantum bits (qubits) under magnetic or electric fields (see Refs. [143, 45]
for similar cost measures). More specially, the above energy quantifier can be
adequately applied to the Hamiltonians that describe the quantum systems
within the scope of this thesis. Note that Eq. (239) is non-invariant with
respect to a change of the zero energy offset, but we can adopt a fixed
reference frame where it can be used to quantify the energy cost involved in
attempts of accelerating the adiabatic path, either via an increasing of the
energy gap in the adiabatic approach or via a reduction of $\tau$ by adjusting
the relevant energy couplings in the counter-diabatic theory. In addition, as
$\tau$ can be set by the quantum speed limit [144], Eq. (239) allows us to
establish a trade-off between speed and energy cost for an arbitrary dynamics
[10, 51]. For instance, in NMR experimental setups, the quantity $||H(t)||$
represents how intense a magnetic field $\vec{B}(t)$ is expected to be in
order to control the speed of such a dynamics. In conclusion, the quantifier
defined in Eq. (239) will be used here to measure the energy cost expanded by
a generalized TQD with regards to its standard and adiabatic counterparts.

As a first result, we discuss how, for a fixed time $\tau$, the generalized
approach of TQD allows us to minimize the energy cost in a TQD by a suitable
choice of the arbitrary parameters $\theta_{n}\left(t\right)$. This
optimization can be analytically derived, which is established by Theorem 4.1
below (its derivation is provided in Appendix 9.A).

######  Theorem 4.1.

Consider a closed quantum system under adiabatic evolution governed by a
Hamiltonian $H_{0}\left(t\right)$. The energy cost to implement its
generalized transitionless counterpart, driven by the Hamiltonian
$H_{\text{tqd}}(t)$, can be minimized by setting

$\theta_{n}\left(t\right)=\theta^{\min}_{n}\left(t\right)=-i\langle\dot{E}_{n}(t)|E_{n}(t)\rangle\text{
.}$ (240)

In particular, for any evolution that satisfies the quantum parallel-transport
condition, given by $\langle\dot{E}_{n}(t)|E_{n}(t)\rangle=0$ [81], the energy
cost to implement a transitionless evolution is always optimized by choosing
$\theta_{n}^{\text{min}}\left(t\right)=0$. This approach is useful for
providing both realistic and energetically optimal Hamiltonians in several
physical scenarios. For example, by considering nuclear spins driven by a
magnetic field $\vec{B}(t)$ in a nuclear magnetic resonance setup, the energy
cost can be optimized by adjusting $\theta_{n}^{\text{min}}\left(t\right)$
such that the magnitude $B(t)$ of the magnetic field is reduced, since
$||H(t)||\propto B(t)$.

As a second result, it is possible to proof that the generalized TQD approach
can be used as a tool to yield time-independent Hamiltonians for
transitionless evolutions. In general, the Hamiltonian $H_{\text{tqd}}(t)$ has
its form constrained both by the choice of the phases
$\theta_{n}\left(t\right)$ and by eigenstates of the adiabatic Hamiltonian
$H_{0}(t)$. Thus, now we delineate under what conditions we can choose the set
$\\{\theta_{n}\left(t\right)\\}$ in order to obtain a time-independent
Hamiltonian for a transitionless evolution. To answer this question, we impose
$\dot{H}_{\text{tqd}}\left(t\right)=0$ considering arbitrary phases
$\theta_{n}\left(t\right)$. This leads to the Theorem 4.2 below (its
derivation is provided in Appendix 9.A).

######  Theorem 4.2.

Let $H_{0}\left(t\right)$ be a discrete quantum Hamiltonian, with
$\\{|E_{m}(t)\rangle\\}$ denoting its set of instantaneous eigenstates. If
$\\{|E_{m}(t)\rangle\\}$ satisfies $\langle
E_{k}(t)|\dot{E}_{m}(t)\rangle=c_{km}$, with $c_{km}$ complex constants
$\forall$ $k,m$, then a family of time-independent Hamiltonians
$H^{\\{\theta\\}}$ for generalized transitionless evolutions can be defined by
setting $\theta_{m}\left(t\right)=\theta$, with $\theta$ a single arbitrary
real constant $\forall m$.

In case where the hypothesis of the Theorem 4.2 are satisfied and the
parallel-transport condition is also satisfied for all eigenstates of
$H_{0}(t)$, we can get an energetically optimal time-independent TQD
Hamiltonian by setting
$\theta_{m}\left(t\right)=\theta_{m}^{\text{min}}\left(t\right)=0$, as we
shall see soon. Although we use the Frobenius norm of the Hamiltonian as
measurement of energy cost, we can define different quantifiers and physical
quantities to be optimized using these arbitrary quantal phases. Here we will
present three applications where different approaches to energy cost
optimization are considered.

####  13.2 Experimental implementation of Landau-Zener transitionless dynamics

As a first application, let us consider the dynamics of a two-level quantum
system, i.e., a qubit, evolving under the Landau-Zener Hamiltonian (here we
use the normalized time $s=t/\tau$)

$\displaystyle H_{0}^{\text{LZ}}(s)$
$\displaystyle=\hbar\Delta\sigma_{z}+\hbar\Omega_{0}(s)\sigma_{x}$ (241)

where $\Delta$ is a detuning, $\Omega(s)$ is the Rabi frequency and the time
$s$ is defined for the interval $[0,1]$. The adiabatic dynamic is obtained
from a very-slow continuum evolution for the Hamiltonian $H_{\text{LZ}}(t)$.
The above equation can be rewritten as

$\displaystyle
H_{0}^{\text{LZ}}\left(s\right)=\hbar\Delta\left[\sigma_{z}+\tan\vartheta\left(s\right)\sigma_{x}\right]\text{
, }$ (242)

with $\tan\vartheta\left(s\right)$ a dimensionless time-dependent parameter
associated with the Rabi frequency, with
$\vartheta\left(s\right)=\arctan[\Omega_{0}(s)/\Delta]$. This Hamiltonian
describes transitions in two-level systems exhibiting anti-crossings in its
energy spectrum [145]. In particular, it can be applied, e.g., to perform
adiabatic population transfer in a two-level system driven by a chirped field
[26] and to investigate molecular collision processes [146]. The instantaneous
ground $|E_{-}\left(s\right)\rangle$ and first excited
$|E_{+}\left(s\right)\rangle$ states of $H_{0}^{\text{LZ}}\left(s\right)$ are

$\displaystyle|E_{-}\left(s\right)\rangle$
$\displaystyle=-\sin\left[\frac{\vartheta\left(s\right)}{2}\right]|1\rangle+\cos\left[\frac{\vartheta\left(s\right)}{2}\right]|0\rangle\text{
,}$ (243a) $\displaystyle|E_{+}\left(s\right)\rangle$
$\displaystyle=\cos\left[\frac{\vartheta\left(s\right)}{2}\right]|1\rangle+\sin\left[\frac{\vartheta\left(s\right)}{2}\right]|0\rangle\text{
. }$ (243b)

The system is initialized in the ground state
$|E_{-}\left(0\right)\rangle=|0\rangle$ of $H_{0}^{\text{LZ}}\left(0\right)$.
By considering a unitary dynamics and a sufficiently large total evolution
time (adiabatic time), the qubit evolves to the instantaneous ground state
$|E_{-}\left(s\right)\rangle$ of $H_{0}^{\text{LZ}}\left(s\right)$. By
applying the standard TQD approach, it is possible to get the TQD Hamiltonian
$H_{\text{std}}^{\text{LZ}}(s)$ given by
$H_{\text{std}}^{\text{LZ}}(s)=H_{0}^{\text{LZ}}\left(s\right)+H_{\text{cd}}^{\text{LZ}}\left(s\right)$,
where

$\displaystyle
H_{\text{cd}}^{\text{LZ}}\left(s\right)=\frac{1}{\tau}i\hbar\sum_{k=\pm}|d_{s}E_{k}\left(s\right)\rangle\langle
E_{k}\left(s\right)|=\frac{\hbar
d_{s}\vartheta\left(s\right)}{2\tau}\sigma_{y}\text{ , }$ (244)

because the eigenstates of the Hamiltonian $H_{0}^{\text{LZ}}\left(s\right)$
the parallel-transport condition.

Now, let us discuss the generalized transitionless dynamics theory for the
Landau-Zener model. For optimal energy cost, Eq. (240) establishes
$\theta_{n}\left(t\right)=\langle
d_{s}E_{\pm}\left(s\right)|E_{\pm}\left(s\right)\rangle=0$ for the states in
Eqs. (243). Therefore, the optimal Hamiltonian
${H}_{\text{opt}}^{\text{LZ}}\left(s\right)$ is given by

$\displaystyle
H_{\text{opt}}^{\text{LZ}}\left(s\right)=H_{\text{cd}}^{\text{LZ}}\left(s\right)\text{
. }$ (245)

From Eq. (244), we can see that $H_{0}^{\text{LZ}}\left(s\right)$ satisfies
the hypotheses of Theorem 4.2 if, and only if, we choose the linear
interpolation $\vartheta(s)=\vartheta_{0}s$. Thus, we adopt this choice for
simplicity and, consequently, we have
$H_{\text{opt}}^{\text{LZ}}\left(s\right)=(\vartheta_{0}/2\tau)\sigma_{y}$. We
observe that a complete (avoided) level crossing picture for the Landau-Zener
model is described by varying the parameter $\tan\vartheta\left(s\right)$ from
$-\infty$ to $+\infty$. Here, we are taking a narrower range for
$\tan\vartheta\left(s\right)$, which simplifies the description of the
transitionless dynamics for the model.

Again, the experimental realization was done using the Ytterbium trapped ion
setup shown in Fig. (10). We use it because we aim to investigate both energy
cost and robustness against decoherence of the optimal TQD scheme. In our
experiment we used both resonant and off-resonant microwaves fields to
implement adiabatic and TQD dynamics. The adiabatic Hamiltonian dynamics was
performed by using a non-resonant microwave, where the time-dependent
effective Rabi frequency is given by
$\Omega_{\text{eff}}(s)=[\Omega_{0}^{2}(s)+\Delta^{2}]^{1/2}$. This is also
known as “generalized” Rabi frequency, i.e, Rabi frequency with detuning [147,
148, 149]. To drive the system by using the standard TQD, we use an
independent microwave to simulate the counter-diabatic term
$H_{\text{cd}}(s)$. This additional field is a resonant microwave
($\Delta=0$), whose Rabi frequency can be obtained from $H_{\text{cd}}(s)$ in
Eq. (244) as $\Omega_{0\text{cd}}(s)=d_{s}\vartheta\left(s\right)/2\tau$. On
the other hand, different from standard TQD Hamiltonian, the optimal dynamics
driven by $H_{\text{opt}}(s)$ could be implemented using a single resonant
microwave with Rabi frequency $\Omega_{0\text{cd}}(s)$, where we have turned-
off the non-resonant field used for simulate $H_{0}(s)$.

In order to quantify the energy resources employed in the quantum evolution,
we study the intensity of the fields used to perform the adiabatic dynamics as
well as standard and optimal TQD. The field intensity is associated with the
Rabi frequency through the relation $I(s)=\Gamma\Omega_{0}^{2}(s)$, where
$\Gamma$ is a constant that depends on the microwave amplifier. Considering
the whole evolution time, we can define the average intensity field as

$\displaystyle\bar{I}(\tau)=\frac{1}{\tau}\int_{0}^{\tau}I(t)dt=\Gamma\int_{0}^{1}|\Omega(s)|^{2}ds\text{
, }$ (246)

where we use the parametrization $s=t/\tau$. Thus, from Eqs. (241), (244) and
(245), we get the field intensity for the adiabatic, standard and optimal TQD
Hamiltonian, respectively, as

$\displaystyle\bar{I}_{0}(\tau)$
$\displaystyle=\Gamma\int\nolimits_{0}^{1}|\Omega_{\text{0}}(s)|^{2}ds=\Delta^{2}\Gamma\int\nolimits_{0}^{1}\tan^{2}\vartheta\left(s\right)ds\text{
, }$ (247a) $\displaystyle\bar{I}_{\text{std}}(\tau)$
$\displaystyle=\bar{I}_{0}(\tau)+\bar{I}_{\text{opt}}(\tau)=\Delta^{2}\Gamma\int\nolimits_{0}^{1}\tan^{2}\vartheta\left(s\right)ds+\frac{\Gamma}{4\tau^{2}}\int\nolimits_{0}^{1}|d_{s}\vartheta\left(s\right)|^{2}ds\text{
, }$ (247b) $\displaystyle\bar{I}_{\text{opt}}(\tau)$
$\displaystyle=\Gamma\int\nolimits_{0}^{1}|\Omega_{\text{CD}}(s)|^{2}ds=\frac{\Gamma}{4\tau^{2}}\int\nolimits_{0}^{1}|d_{s}\vartheta\left(s\right)|^{2}ds\text{
. }$ (247c)

Notice that for the standard TQD we find
$\bar{I}_{\text{std}}(\tau)=\bar{I}_{0}(\tau)+\bar{I}_{\text{opt}}(\tau)$,
since the standard TQD field is composed by both the adiabatic and the optimal
TQD contributions. The parameter $\Gamma$ is a characteristic parameter that
depends on the device used to amplify and modulate the driving microwave
acting on the system. Then, in order to give a description of the energy cost
independent of any auxiliary device, we take the relative field intensities
expressed in unities of the adiabatic intensity $\bar{I}_{0}$. By doing this,
it is possible to disregard the constant $\Gamma$. In fact, we define
$\pazocal{I}_{\text{std}}(\tau)=\bar{I}_{\text{std}}(\tau)/\bar{I}_{0}$ and
$\pazocal{I}_{\text{opt}}(\tau)=\bar{I}_{\text{opt}}(\tau)/\bar{I}_{0}$, and
adopt the normalization $\pazocal{I}_{0}(\tau)=1$.

From Eqs. (247), it is possible to identify a value of $\tau_{\text{B}}$, for
the total evolution time, in which the intensity fields for implementing
optimal TQD becomes less intense than the adiabatic intensity. In fact,
firstly note that $\bar{I}_{\text{std}}(\tau)\geq\bar{I}_{\text{opt}}(\tau)$,
but on the other side, we can see that the value of $\tau$ implies into
$\bar{I}_{\text{opt}}(\tau)\geq\bar{I}_{0}(\tau)$ or
$\bar{I}_{\text{opt}}(\tau)\leq\bar{I}_{0}(\tau)$. So, here we impose the
desired situation in which $\bar{I}_{\text{opt}}(\tau)\leq\bar{I}_{0}(\tau)$
to obtain the relation

$\displaystyle\frac{\Gamma}{4\tau^{2}}\int\nolimits_{0}^{1}|d_{s}\vartheta\left(s\right)|^{2}ds\leq\Delta^{2}\Gamma\int\nolimits_{0}^{1}\tan^{2}\vartheta\left(s\right)ds\text{
, }$ (248)

therefore we conclude that, in this situation, $\tau$ should satisfy

$\displaystyle\tau\geq\frac{1}{|\Delta|}\mathchoice{{\hbox{$\displaystyle\sqrt{\frac{\int\nolimits_{0}^{1}|d_{s}\vartheta\left(s\right)|^{2}ds}{\int\nolimits_{0}^{1}\tan^{2}\vartheta\left(s\right)ds}\,}$}\lower
0.4pt\hbox{\vrule
height=17.42888pt,depth=-13.94316pt}}}{{\hbox{$\textstyle\sqrt{\frac{\int\nolimits_{0}^{1}|d_{s}\vartheta\left(s\right)|^{2}ds}{\int\nolimits_{0}^{1}\tan^{2}\vartheta\left(s\right)ds}\,}$}\lower
0.4pt\hbox{\vrule
height=12.23886pt,depth=-9.79114pt}}}{{\hbox{$\scriptstyle\sqrt{\frac{\int\nolimits_{0}^{1}|d_{s}\vartheta\left(s\right)|^{2}ds}{\int\nolimits_{0}^{1}\tan^{2}\vartheta\left(s\right)ds}\,}$}\lower
0.4pt\hbox{\vrule
height=9.35274pt,depth=-7.48222pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\frac{\int\nolimits_{0}^{1}|d_{s}\vartheta\left(s\right)|^{2}ds}{\int\nolimits_{0}^{1}\tan^{2}\vartheta\left(s\right)ds}\,}$}\lower
0.4pt\hbox{\vrule height=9.35274pt,depth=-7.48222pt}}}\text{ . }$ (249)

This result shows the existence of a boundary total evolution time
$\tau_{\text{B}}$ given by

$\displaystyle\tau_{\text{B}}=\frac{1}{|\Delta|}\mathchoice{{\hbox{$\displaystyle\sqrt{\frac{\int\nolimits_{0}^{1}|\vartheta^{\prime}\left(s\right)|^{2}ds}{\int\nolimits_{0}^{1}\tan^{2}\vartheta\left(s\right)ds}\,}$}\lower
0.4pt\hbox{\vrule
height=17.42888pt,depth=-13.94316pt}}}{{\hbox{$\textstyle\sqrt{\frac{\int\nolimits_{0}^{1}|\vartheta^{\prime}\left(s\right)|^{2}ds}{\int\nolimits_{0}^{1}\tan^{2}\vartheta\left(s\right)ds}\,}$}\lower
0.4pt\hbox{\vrule
height=12.23886pt,depth=-9.79114pt}}}{{\hbox{$\scriptstyle\sqrt{\frac{\int\nolimits_{0}^{1}|\vartheta^{\prime}\left(s\right)|^{2}ds}{\int\nolimits_{0}^{1}\tan^{2}\vartheta\left(s\right)ds}\,}$}\lower
0.4pt\hbox{\vrule
height=9.35274pt,depth=-7.48222pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\frac{\int\nolimits_{0}^{1}|\vartheta^{\prime}\left(s\right)|^{2}ds}{\int\nolimits_{0}^{1}\tan^{2}\vartheta\left(s\right)ds}\,}$}\lower
0.4pt\hbox{\vrule height=9.35274pt,depth=-7.48222pt}}}\text{ , }$ (250)

so that if $\tau>\tau_{\text{B}}$ we get
$\bar{I}_{\text{Opt}}(\tau)<\bar{I}_{\text{Ad}}(\tau)$, otherwise we find
$\bar{I}_{\text{Opt}}(\tau)>\bar{I}_{\text{Ad}}(\tau)$. In case where
$\tau=\tau_{\text{B}}$ we have identical field intensities. It would be worth
highlighting that $\tau_{\text{B}}$ does not depend on the $\Gamma$. Notice
that, after $\tau_{\text{B}}$, the shortcut to adiabaticity defined by the
optimal TQD can be implemented by spending less energy resources, as measured
by the field intensity, than the adiabatic approach.

Figure 13: (a) The calculated relative field intensity $\pazocal{I}(\tau)$ for
standard TQD $\pazocal{I}_{\text{std}}(\tau)$ and optimal TQD
$\pazocal{I}_{\text{opt}}(\tau)$ as function of the total evolution time
$\tau$, where the horizontal black line represents $\pazocal{I}_{0}(\tau)$.
(b) Fidelity for unitary dynamics (continuum lines) and non-unitary one
(dashed lines) under dephasing. The symbols and lines represent experimental
data and theoretical results, respectively. In both figures the dashed gray
vertical line denotes the boundary time $\tau_{\text{B}}\approx 0.052$ ms,
obtained from Eq. (250), between the regions
$\bar{I}_{\text{opt}}(\tau)>\bar{I}_{0}(\tau)$ (left hand side) and
$\bar{I}_{\text{opt}}(\tau)<\bar{I}_{0}(\tau)$ (right hand side). Here we set
$\vartheta\left(s\right)=\pi s/3$, $\Delta=2\pi\times 1$ KHz and
$\Omega_{0}\left(s\right)=\Delta\tan(\pi s/3)$ is used in our experiment. The
decohering rate was kept as $\gamma=2.5$ KHz for all non-unitary dynamics. The
first gray vertical line denotes the boundary time $\tau_{\text{B}}\approx
0.052$ ms obtained from Eq. (250).

It is important to mention here that the energy cost as computed by the
Frobenius norm of the Hamiltonian does not represents a real energy cost. In
fact, it is possible to show that the norm of the Hamiltonian
$H_{0}^{\text{LZ}}\left(s\right)$ depends on the parameter $\Delta$, but since
of $\Delta$ can changed without change the microwave intensity, the norm of
$H_{0}^{\text{LZ}}\left(s\right)$ cannot be used as a realistic cost of the
dynamics in this particular case. However, if we consider this analysis, the
Theorem 4.1 guarantees that
$\Sigma_{\text{std}}(\tau)>\Sigma_{\text{opt}}(\tau)$ for every $\tau$. Thus,
because we spend less energy with the optimal TQD scheme, it is convenient to
study the impact of this optimal TQD on the robustness of the dynamics against
decoherence. To do a direct comparison between adiabatic dynamics, standard
and optimal TQD, we implemented each dynamics evolving under a phase damping
channel described by the non-unitary contribution given in Eq. (148).

In the experiment we consider a constant dephasing rate as where $\gamma=2.5$
KHz is the dephasing rate and a linear interpolation for the function
$\vartheta\left(s\right)=\vartheta_{0}s$ (where we get a time-independent
optimal TQD Hamiltonian), with $\vartheta_{0}=\pi/3$. Therefore, we have
$\Omega_{0}\left(s\right)=\Delta\tan(\pi s/3)$, where we pick the detuning
parameter $\Delta$ as $2\pi\times 2$ KHz. As previously, here we quantify the
robustness of the protocols from the fidelity defined in Eq. (165). However,
since we have a pure state as our target state $|\psi_{\text{tg}}(s)\rangle$,
it is possible to proof that the Eq. (165) becomes [150]

$\displaystyle\pazocal{F}(\tau)=\mathchoice{{\hbox{$\displaystyle\sqrt{\langle\psi_{\text{tg}}(s)|\rho\left(s\right)|\psi_{\text{tg}}(s)\rangle\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\textstyle\sqrt{\langle\psi_{\text{tg}}(s)|\rho\left(s\right)|\psi_{\text{tg}}(s)\rangle\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\scriptstyle\sqrt{\langle\psi_{\text{tg}}(s)|\rho\left(s\right)|\psi_{\text{tg}}(s)\rangle\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\langle\psi_{\text{tg}}(s)|\rho\left(s\right)|\psi_{\text{tg}}(s)\rangle\,}$}\lower
0.4pt\hbox{\vrule height=7.5pt,depth=-6.00003pt}}}\text{ , }$ (251)

with $\rho\left(s\right)$ being solution of Schrödinger equation, obtained by
numerical calculations, and $|\psi_{\text{tg}}(s)\rangle$ given by Eq. (243a).
The result for the fidelity $\pazocal{F}(\tau)$ as a function of $\tau$ for
both free-decohering and decohering dynamics are shown in Fig. 13b.

The continuum lines, with full symbols denoting the associated experimental
data, in Fig. 13b denotes the values of $\pazocal{F}(\tau)$ under a unitary
dynamics for each protocol and the corresponding experimental data points.
This is possible because the the maximum total evolution time is $\tau=10$ ms,
while the coherence time is of order of $200$ ms for the qubit used here. As
expected, the black curve shows that the adiabatic behavior is achieved for
long total evolution time $\tau\gg\tau_{0}\approx 0.021$ ms, with $\tau_{0}$
computed from $\tau_{0}=\max_{s\in[0,1]}|\langle
E_{-}(s)|d_{s}H_{0}(s)|E_{+}(s)\rangle/g^{2}(s)|$, where
$g(s)=E_{+}(s)-E_{-}(s)$ is the gap between fundamental $|E_{-}(s)\rangle$ and
excited $|E_{+}(s)\rangle$ energy levels [21]. It is important highlight here
that, for fast evolutions ($\tau<\tau_{0}$), both optimal and traditional TQD
provide a high fidelity protocol, while adiabatic dynamics fails. Naturally,
this high performance is accompanied by costly fields used for implementing
the TQD protocols, as previously shown in Fig. 13b. On the other hand, it is
important to highlight that for total evolution time $\tau>\tau_{\text{B}}$,
where the optimal TQD field is smaller than both adiabatic and traditional TQD
ones, the high performance of optimal TQD is kept.

By looking now at the effect of the dephasing channel (dashed curves, with
empty symbols denoting their respective experimental data) from Fig. 13, we
can see that optimal TQD exhibits the highest robustness for every $\tau$,
with its advantage increasing as $\tau$ increases. For fast evolution times,
this high performance is again associated with costly fields in comparison
with the adiabatic fields. However, for large evolution times, while the
traditional TQD fidelity converges to adiabatic fidelity (as expected, because
the counter-diabatic Hamiltonian becomes less effective than the adiabatic one
[33]), the fidelity behavior of optimal TQD is much better than both adiabatic
dynamics and traditional TQD, with less energy resources spent in the process.
This is a remarkable result, since we can achieve both better fidelities with
less energy fields involved. Table 2 summarizes fidelity and intensity fields
for the TQD models considered here. We highlight that
$\pazocal{F}_{\text{opt}}(\tau)\gtrapprox\pazocal{F}_{\text{std}}(\tau)$ and
there is a range of $\tau$, where we get
$\pazocal{F}_{\text{opt}}(\tau)>\pazocal{F}_{0}(\tau)$ even when
$\bar{I}_{\text{opt}}(\tau)\ll\bar{I}_{0}(\tau)$. In conclusion, the
generalized TQD approach allows us for getting, at least to this case, time-
independent TQD Hamiltoniansthat are more robust and less energetically
costing than its adiabatic and standard counterparts.

Table 2: Intensity fields and fidelity relations for both standard and optimal TQD. Model | $\tau<\tau_{\text{B}}$ | $\tau=\tau_{\text{B}}$ | $\tau>\tau_{\text{B}}$
---|---|---|---
Standard |  | $\pazocal{F}_{\text{std}}(\tau)>\pazocal{F}_{0}(\tau)$
---
$\bar{I}_{\text{std}}(\tau)>\bar{I}_{0}(\tau)$
| $\pazocal{F}_{\text{std}}(\tau)>\pazocal{F}_{0}(\tau)$
---
$\bar{I}_{\text{std}}(\tau)>\bar{I}_{0}(\tau)$
| $\pazocal{F}_{\text{std}}(\tau)\gtrapprox\pazocal{F}_{0}(\tau)$
---
$\bar{I}_{\text{std}}(\tau)\gtrapprox\bar{I}_{0}(\tau)$
Optimal |  | $\pazocal{F}_{\text{opt}}(\tau)>\pazocal{F}_{0}(\tau)$
---
$\bar{I}_{\text{opt}}(\tau)>\bar{I}_{0}(\tau)$
| $\pazocal{F}_{\text{opt}}(\tau)>\pazocal{F}_{0}(\tau)$
---
$\bar{I}_{\text{opt}}(\tau)=\bar{I}_{0}(\tau)$
| $\pazocal{F}_{\text{opt}}(\tau)>\pazocal{F}_{0}(\tau)$
---
$\bar{I}_{\text{opt}}(\tau)<\bar{I}_{0}(\tau)$

###  14 Generalized TQD for a single-spin in a rotating magnetic field

As a second application, we study optimal transitionless evolution of a single
spin-$\frac{1}{2}$ particle in presence of a rotating magnetic field [72],
whose Hamiltonian $H_{0}(t)$ is given by the Eq. (34). Here we do not assume
any restriction on the parameters $\omega_{0}$ and $\omega_{1}$, as done
previously. As we know, the set of eigenvectors for is given in Eqs. (35),
then one can show that the generalized TQD Hamiltonian reads

$\displaystyle H^{\text{gen}}_{\text{tqd}}(t)$
$\displaystyle=\frac{\hbar}{2}\theta(t)\sin\alpha(\sigma_{x}\cos\omega
t+\sigma_{y}\sin\omega
t)+\frac{\hbar}{2}\left[\omega+\theta(t)\cos\alpha\right]\sigma_{z}\text{ , }$
(252)

where $\theta(t)=\theta_{0}(t)-\theta_{1}(t)$ and
$\alpha=\arctan[\omega_{1}/\omega_{0}]$, with $\theta_{0}(t)$ and
$\theta_{1}(t)$ being the generalized quantal phases (arbitrary). Thus, note
that a transitionless quantum driving is not obtained by changing the angular
frequency of the RF field, but rather by changing the intensity of the
magnetic fields. In fact, as in Hamiltonian $H^{\text{nmr}}_{0}(t)$, we can
write the magnetic field $\vec{B}^{\text{gen}}_{\text{tqd}}(t)$ associated
with the Hamiltonian
$H^{\text{gen}}_{\text{tqd}}(t)=-\gamma\vec{S}\cdot\vec{B}^{\text{gen}}_{\text{tqd}}(t)$
as

$\displaystyle\vec{B}^{\text{gen}}_{\text{tqd}}(t)=\left[B_{\omega}+B_{\theta}(t)\cos\alpha\right]\hat{z}+B_{\theta}(t)\sin\alpha(\cos\omega
t\hat{x}+\sin\omega t\hat{y})\text{ , }$ (253)

where $B_{\theta}(t)=-\theta(t)/\gamma$ is associated with the free parameter
$\theta(t)$ and $B_{\omega}=-\omega/\gamma$. Therefore, the generalized
transitionless theory allows us to use the phases $\theta_{n}(t)$ ($n=0,1$) to
minimize the magnetic field required to implement the desired dynamics. In
some cases, such a field is fixed and no free parameter can be used to
optimize the field intensity. For example, if we set the parameters
$\theta_{n}(t)$ in order to obtain the exact adiabatic dynamics with adiabatic
phases $\theta^{\text{ad}}_{n}(t)$, we find the standard shortcut to
adiabaticity, which is provided by the Hamiltonian in Eq. (234), with the
counter-diabatic term $H_{\text{cd}}(t)$ reading

$\displaystyle
H_{\text{cd}}(t)=\frac{\hbar\omega}{2}\left[\sin^{2}\alpha\sigma_{z}-\frac{1}{2}\sin
2\alpha(\sigma_{x}\cos\omega t+\sigma_{y}\sin\omega t)\right]\text{ , }$ (254)

leading to the standard quantum driving Hamiltonian in the form
$H^{\text{std}}_{\text{tqd}}(t)=-\gamma\vec{S}\cdot\vec{B}^{\text{std}}_{\text{tqd}}(t)$,
with the magnetic field $\vec{B}^{\text{std}}_{\text{tqd}}(t)$ to implement
$H^{\text{std}}_{\text{tqd}}(t)$ reading

$\displaystyle\vec{B}^{\text{std}}_{\text{tqd}}(t)$
$\displaystyle=\left[B_{xy}-\frac{1}{2}B_{\omega}\sin
2\alpha\right](\cos\omega t\hat{x}+\sin\omega
t\hat{y})+\left[B_{z}+B_{\omega}\sin^{2}\alpha\right]\hat{z}\text{ , }$ (255)

where $B_{\omega}$ is the additional magnetic field. By analyzing the energy
resources to implement the desired evolution, we can find the optimal protocol
for the transitionless dynamics in terms of the magnetic field intensities
required by the Hamiltonian. In particular, it is possible to show that the
optimal field is obtained by setting
$\theta_{n}(t)=i\langle\dot{E}_{n}(t)|E_{n}(t)\rangle$, as established in
Theorem 4.1. Therefore, the optimal transitionless counterpart reads

$\displaystyle
H^{\text{opt}}_{\text{tqd}}(t)=\frac{\hbar\omega}{2}\left[\sin^{2}\alpha\sigma_{z}-\frac{1}{2}\sin
2\alpha(\sigma_{x}\cos\omega t+\sigma_{y}\sin\omega t)\right]\text{ , }$ (256)

that can be written as
$H^{\text{opt}}_{\text{tqd}}(t)=-\gamma\vec{S}\cdot\vec{B}^{\text{opt}}_{\text{tqd}}(t)$
with the optimal magnetic $\vec{B}^{\text{opt}}_{\text{tqd}}(t)$ identified as

$\displaystyle\vec{B}^{\text{opt}}_{\text{tqd}}(t)=B_{\omega}\sin^{2}\alpha\hat{z}-\frac{1}{2}B_{\omega}\sin
2\alpha(\cos\omega t\hat{x}+\sin\omega t\hat{y})\text{ . }$ (257)

We can see that the norms of the fields satisfy
$||\vec{B}^{\text{std}}_{\text{tqd}}(t)||>||\vec{B}_{0}(t)||$ and
$||\vec{B}^{\text{std}}_{\text{tqd}}(t)||>||\vec{B}^{\text{opt}}_{\text{tqd}}(t)||$
for any choice of the set of parameters
$\Omega=\\{\omega,\omega_{0},\omega_{1}\\}$. This means that a fast evolution
based on the standard shortcut exactly mimicking the adiabatic phase, such as
given by Eq. (234), always requires more energy resources than the original
adiabatic dynamics. On the other hand, the relation between
$||\vec{B}^{\text{opt}}_{\text{tqd}}(t)||$ and $||\vec{B}_{0}(t)||$ depends on
the parameters $\omega$, $\omega_{0}$ and $\omega_{1}$ used in
$||\vec{B}_{0}(t)||$. In particular, this means that the optimal shortcut to
adiabaticity, such as given by Eq. (256), can be faster while spending even
less resources than the original adiabatic dynamics. More specifically, this
trade-off can be expressed through the norm relation

$\displaystyle\frac{||\vec{B}_{0}(t)||}{||\vec{B}^{\text{opt}}_{\text{tqd}}(t)||}=\frac{B_{1}^{2}+B_{0}^{2}}{B_{1}B_{\omega}}=\frac{\omega_{1}^{2}+\omega_{0}^{2}}{\omega_{1}\omega}\text{
. }$ (258)

As we can see, for the case $\omega_{1}=\omega_{0}=\omega$, we get
$||\vec{B}_{0}(t)||=2||\vec{B}^{\text{opt}}_{\text{tqd}}(t)||$. This shows the
advantage of the optimal TQD approach in comparison with its adiabatic
counterpart, whose implementation requires a higher magnetic field. It is
worth mentioning that, since the experimental implementation of the driven
Hamiltonian in NMR is obtained in the rotating frame, the dynamics here is
implemented by a phase modulated pulse with constant amplitude applied off
resonance. This approach is similar to the adiabatic pulse technique where
large swept-frequency offsets are used to make the effective magnetic field,
observed in the rotating frame, approximately collinear with the spin
magnetization [151]. In our experiment no swept-frequency has been used. The
energy cost is well defined from the transverse RF field used to drive the
system, since the $Z$ component of the Hamiltonian is implemented by a
frequency shift, with no real cost associated with it.

####  14.1 Experimental implementation

In order to demonstrate the advantage of the TQD approach as a tool in quantum
control, we have performed an experimental realization using a two-qubit NMR
system, namely, the 1H and 13C spin$-1/2$ nuclei in the Chloroform molecule,
with up and down spin states denoted by $|1\rangle$ and $|0\rangle$,
respectively [152]. The experiment has been realized at room temperature in a
Varian 500 MHz spectrometer. More specifically, we have compared the dynamics
of the system driven by fields $\vec{B}_{0}(t)$,
$\vec{B}^{\text{std}}_{\text{tqd}}(t)$ and
$\vec{B}^{\text{opt}}_{\text{tqd}}(t)$. Since we need a single qubit, the 1H
spin has been driven by phase modulated magnetic fields, while the 13C spin
has been decoupled during the experiment. In the rotating frame, the
Hamiltonian associated with the magnetic pulse is written as

$\displaystyle
H_{\text{rf}}=\frac{\hbar}{2}\left[\omega_{0}\sigma_{z}+\omega_{1}(\sigma_{x}\cos\phi+\sigma_{y}\sin\phi)\right]\text{
. }$ (259)

By changing the amplitude of the pulses, the offset frequency and using time
phase modulation, one can reproduce the desired dynamics. The evolution has
been implemented in an on-resonance condition
$\omega_{z}\\!=\\!\omega_{xy}\\!=\\!\omega\\!=\\!2\pi\times 200$Hz. This means
that, even though the evolution obeys the adiabatic condition, it can lead to
population transfer, resulting in a violation of the expected adiabatic
evolution as a resonance effect [52]. From the values set for $\omega$,
$\omega_{z}$ and $\omega_{xy}$, Eq. (258) implies that
$||\vec{B}_{0}(t)||=2||\vec{B}^{\text{opt}}_{\text{tqd}}(t)||$, so that
$||\vec{B}^{\text{opt}}_{\text{tqd}}(t)||\\!<\\!||\vec{B}_{0}(t)||\\!<\\!||\vec{B}^{\text{std}}_{\text{tqd}}(t)||$.
Notice that the optimal TQD indeed spends less energy resources as measured by
the strength of the magnetic field applied. The system is then initially
prepared in the ground state of $H_{0}(0)$, given by
$|\psi(0)\rangle=\cos\alpha|0\rangle-\sin\alpha|1\rangle$, and evolved
according to the desired Hamiltonian. The quantum state experimentally
obtained is determined via quantum state tomography [153] and compared to the
theoretically evaluated instantaneous ground state of $H_{0}(t)$. This is
performed by computing the fidelity $\mathcal{F}(t)$ between the instantaneous
ground state of $H_{0}(t)$, represented by the density operator
$\rho_{\text{gs}}(t)$, and the dynamically evolved quantum state driven by the
Hamiltonians $H_{0}(t)$, $H^{\text{opt}}_{\text{tqd}}(t)$, and
$H^{\text{std}}_{\text{tqd}}(t)$, represented by the density operator
$\rho(t)$. The fidelity $\mathcal{F}(t)$ is defined here as the relative
purity given by [154, 155, 156]

$\mathcal{F}(t)=\frac{\left|\text{Tr}[\rho_{\text{gs}}(t)\rho(t)]\right|}{\mathchoice{{\hbox{$\displaystyle\sqrt{\text{Tr}[\rho_{\text{gs}}^{2}(t)]\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\textstyle\sqrt{\text{Tr}[\rho_{\text{gs}}^{2}(t)]\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\scriptstyle\sqrt{\text{Tr}[\rho_{\text{gs}}^{2}(t)]\,}$}\lower
0.4pt\hbox{\vrule
height=5.25pt,depth=-4.20003pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\text{Tr}[\rho_{\text{gs}}^{2}(t)]\,}$}\lower
0.4pt\hbox{\vrule
height=3.75pt,depth=-3.00002pt}}}\mathchoice{{\hbox{$\displaystyle\sqrt{\text{Tr}[\rho^{2}(t)]\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\textstyle\sqrt{\text{Tr}[\rho^{2}(t)]\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\scriptstyle\sqrt{\text{Tr}[\rho^{2}(t)]\,}$}\lower
0.4pt\hbox{\vrule
height=5.25pt,depth=-4.20003pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\text{Tr}[\rho^{2}(t)]\,}$}\lower
0.4pt\hbox{\vrule height=3.75pt,depth=-3.00002pt}}}}\text{ . }$ (260)

The theoretical and experimental results are shown in Fig. 14 for the three
distinct dynamics. Since the experiment is performed on-resonance, the
evolution driven by the Hamiltonian $H_{0}(t)$ is found to be nonadiabatic,
oscillating as a function of time. On the other hand, the shortcuts given by
$H^{\text{opt}}_{\text{tqd}}(t)$, and $H^{\text{std}}_{\text{tqd}}(t)$ keep a
non-transitional evolution, being immune to the resonance effect.

Figure 14: Fidelity $\mathcal{F}(t)$ between the instantaneous ground state of
$H_{0}(t)$ and the dynamically evolved quantum states driven by the
Hamiltonians $H_{0}(t)$, $H^{\text{opt}}_{\text{tqd}}(t)$, and
$H^{\text{std}}_{\text{tqd}}(t)$. The curves denote the theoretical
predictions and the symbols are the experimental values. The standard
$H_{0}(t)$ implementation is more sensitive to RF miscalibration, so that we
have less tolerance to field inhomogeneity compared to the other cases. For
this reason, we may have a fluctuation between experimental and theoretical
results in some points.

The fitting of the experimental data has been performed in an open quantum
system scenario, which is accomplished by adjusting the parameters of the
Lindblad equation whose non-unitary contribution is given by Eq. (144). The
NMR system considered in this work is mainly affected by dephasing, which
occurs due to the inhomogeneity of the static magnetic field. This variation
causes the spins of all the molecules to slowly desynchronize and, therefore,
lose coherence across the sample. In this case, we have a single Lindblad
operator $\Gamma_{1}(t)=\gamma_{0}\sigma_{z}$, where the decohering rate
$\gamma_{0}$ is given by
$\gamma_{0}=\mathchoice{{\hbox{$\displaystyle\sqrt{1/T_{2}\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\textstyle\sqrt{1/T_{2}\,}$}\lower
0.4pt\hbox{\vrule
height=7.5pt,depth=-6.00003pt}}}{{\hbox{$\scriptstyle\sqrt{1/T_{2}\,}$}\lower
0.4pt\hbox{\vrule
height=5.25pt,depth=-4.20003pt}}}{{\hbox{$\scriptscriptstyle\sqrt{1/T_{2}\,}$}\lower
0.4pt\hbox{\vrule height=3.75pt,depth=-3.00002pt}}}$, with $T_{2}$ denoting
the dephasing relaxation time of the system. Other non-unitary effects, such
as generalized amplitude damping characterized by a relaxation scale $T_{1}$,
are also present, but they are negligible for the time ranges considered in
the experiment. Indeed, in our realization, we have $T_{2}=0.25$ s and
$T_{1}=5.11$ s. Notice then that $T_{1}$ is two orders of magnitude larger
than the total time of evolution, so that generalized amplitude damping does
not exhibit significant effect. Hence, by considering dephasing as a main
effect, the non-unitary part of the dynamics is dictated by the Eq. (148). As
we showed previously, due to the resonance configuration [52, 61], the
adiabatic dynamics for $H_{0}(t)$ does not hold in general, being achieved
just for some particular instants of time. Notice that the oscillation in the
adiabatic fidelity is damped as a function of time. This is a by-product of
dephasing in the system. Moreover, there are also errors induced by the
implementation of RF pulses. Those errors are about $3\%$ per pulse.
Remarkably, shortcuts to adiabaticity show considerable robustness against the
decoherence and unitary errors.

As shown in Fig. 14, fidelity stays close to one throughout the evolution,
which means that the states of the evolution are, indeed, kept as
instantaneous eigenstates of $H_{0}(t)$. Therefore, as expected, the adiabatic
behavior of the system is drastically affected by the resonance phenomenon. On
the other hand, the standard TQD and the optimal TQD are both immune to the
resonance destructive effect. In particular, we have shown that the optimal
TQD approach provides a transitionless Hamiltonian that can be implemented
with low intensity fields in comparison with the fields used to implement the
adiabatic and the standard TQD Hamiltonians.

###  15 Quantum gates by Controlled TQD

In this section we show a useful application of TQD, namely application to
quantum information processing, where we use TQD as a shortcuts to
adiabaticity to speed up adiabatic quantum gates. More specifically, it has
been applied to perform universal quantum computation (QC) via either counter-
diabatic controlled evolutions [10] or counter-diabatic quantum teleportation
[32]. As hybrid models, these approaches provide a convenient digital
architecture for physical realizations while potentially keeping both the
generality and some inherently robustness of analog implementations.
Experimentally, digitized implementations of quantum annealing processes have
been recently provided [8], with controlled quantum gates adiabatically
realized with high fidelity via superconducting qubits [157]. In this Section,
by focusing on controlled evolutions, we will now show that counter-diabatic
QC can be more robust against decoherence than its adiabatic counterpart as
long as the gate runtime is suitably determined within a range of evolution
times.

####  15.1 Quantum gates by adiabatic controlled evolutions

In adiabatic protocols, we need to start the evolution at some eigenstate of
the Hamiltonian $H_{0}(0)$, and at the end of the evolution ($t=\tau$) we
should get the respective updated eigenstate of $H_{0}(\tau)$. A important
point to highlight here is that the input state of the computation should not
depend on the $H_{0}(0)$, and vice-versa. It is a strong requirement because
we want to design a universal model of computation, so the input state needs
to be considered as an unknown state. To this end, we consider a bipartite
system composed by a target subsystem $\pazocal{T}$ and an auxiliary subsystem
$\pazocal{A}$, whose individual Hilbert spaces $\pazocal{H}{T}$ and
$\pazocal{H}{A}$ have dimensions $d{T}$ and $d{A}$, respectively. The
auxiliary subsystem $\pazocal{A}$ will be driven by a family of time-dependent
Hamiltonians $\\{\mathcal{H}_{k}\left(s\right)\\}$, with $0\leq k\leq d{T}-1$.
The target subsystem will be driven by a complete set $\left\\{P_{k}\right\\}$
of orthogonal projectors over $\pazocal{T}$, which satisfy
$P_{k}P_{m}=\delta_{km}P_{k}$ and $\sum_{k}P_{k}=\mathbbm{1}$. In a controlled
adiabatic dynamics, the composite system $\pazocal{T}\pazocal{A}$ will be
governed by a Hamiltonian in the form [9]

$H_{0}\left(s\right)=\sum\nolimits_{k}P_{k}\otimes\mathcal{H}_{k}\left(s\right)\text{
, }$ (261)

with
$\mathcal{H}_{k}\left(s\right)=g\left(s\right)\mathcal{H}_{k}^{\left(f\right)}+f\left(s\right)\mathcal{H}^{\left(b\right)}$,
where $\mathcal{H}^{(b)}$ is the beginning Hamiltonian,
$\mathcal{H}^{(f)}_{k}$ is the contribution $k$ to the final Hamiltonian, and
the time-dependent functions $f(s)$ and $g(s)$ satisfy the boundary conditions
$f\left(0\right)=g\left(1\right)=1$ and $g\left(0\right)=f\left(1\right)=0$.

Now, we prepare $\pazocal{T}\pazocal{A}$ in the initial state
$|\Psi_{\text{init}}\rangle=|\psi\rangle\otimes|\varepsilon_{b}\rangle$, where
$|\psi\rangle$ is an arbitrary state of $\pazocal{T}$ and
$|\varepsilon_{b}\rangle$ is the ground state of
$\mathcal{H}^{\left(b\right)}$. Then $|\Psi_{\text{init}}\rangle$ is the
ground state of the initial Hamiltonian
$\mathbbm{1}\otimes\mathcal{H}^{\left(b\right)}$. By applying the adiabatic
theorem [3, 21], a sufficiently slowing-varying evolution of $H_{0}(t)$ will
drive the system (up to a phase) to the final state

$\left|\Psi_{\text{final}}\right\rangle=\sum\nolimits_{k}P_{k}|\psi\rangle\otimes\left|\varepsilon_{k}\right\rangle\text{
, }$ (262)

where $\left|\varepsilon_{k}\right\rangle$ is the ground state of
$\mathcal{H}_{k}^{\left(f\right)}$ [9]. Note that an arbitrary projection
$P_{k}$ over the unknown state $|\psi\rangle$ can be yielded by performing a
convenient measurement over $\pazocal{A}$. Therefore, by adjusting the set of
projectors $P_{k}$ and the Hamiltonians $\mathcal{H}_{k}\left(s\right)$, it is
possible to control the output state $|\psi_{\text{out}}\rangle$ of
$\pazocal{T}$, so that single and $N$-qubit gates can be implemented [10]. In
fact,

Let us begin by considering $\pazocal{T}$ as a single qubit and a single-qubit
gate as an arbitrary rotation of angle $\phi$ around a direction $\hat{n}$
over the Bloch sphere. Under this consideration, the Hamiltonian that
adiabatically implements such a single-qubit gate for an arbitrary input state
$|\psi\rangle=a|0\rangle+b|1\rangle$, with $a,b\in\mathbbm{C}$, is given by
[9]

$H_{0}^{\text{sg}}\left(s\right)=P_{+}\otimes\mathcal{H}_{0}\left(s\right)+P_{-}\otimes\mathcal{H}_{\phi}\left(s\right)\text{
,}$ (263)

where $\\{P_{\pm}\\}$ is a complete set of orthogonal projectors over the
Hilbert space of the target qubit. The projectors can be parametrized as
$P_{\pm}=(\mathbbm{1}\pm\hat{n}\cdot\vec{\sigma})/2$, with $\hat{n}$
associated with the direction of the target qubit on the Bloch sphere. In Eq.
(263), each Hamiltonian $\mathcal{H}_{\xi}\left(s\right)$ ($\xi=\\{0,\phi\\}$)
acts on $\pazocal{A}$, and is given by [9]

$\mathcal{H}_{\xi}\left(s\right)=-\omega\\{\sigma_{z}\cos(\varphi_{0}s)+\sin(\varphi_{0}s)[\sigma_{x}\cos\xi+\sigma_{y}\sin\xi]\\}\text{
, }$ (264)

with $\varphi_{0}$ denoting an arbitrary parameter that sets the success
probability of obtaining the desired state at the end of the evolution. This
parameter plays a role in the energy performance of counter-diabatic QC, with
probabilistic counter-diabatic QC ($\varphi_{0}\neq\pi$) being energetically
more favorable than its deterministic ($\varphi_{0}=\pi$) counterpart [33].

The projectors $\\{P_{\pm}\\}$ may be written in terms of two basis vectors
$\\{|n_{\pm}\rangle\\}$ in the Bloch sphere as
$\\{P_{\pm}\\}=|n_{\pm}\rangle\langle n_{\pm}|$, where

$\displaystyle|n_{+}\rangle$
$\displaystyle=\cos(\varepsilon/2)|0\rangle+e^{i\delta}\sin(\varepsilon/2)|1\rangle\text{
,}$ (265) $\displaystyle|n_{-}\rangle$
$\displaystyle=-\sin(\varepsilon/2)|0\rangle+e^{i\delta}\cos(\varepsilon/2)|1\rangle\text{
.}$ (266)

Thus, a quantum gate is encoded as a rotation of $\phi$ around the vector
$|n_{+}\rangle$ (as we shall illustrate in Fig. 15a). Now, by expressing the
state $|\psi\rangle$ in the basis $\\{|n_{\pm}\rangle\\}$, we write
$|\psi\rangle=\alpha|n_{+}\rangle+\beta|n_{-}\rangle$, with
$\left|\hat{n}_{\pm}\right\rangle$ being a state in the direction $\hat{n}$
and $\alpha,\beta\in\mathbbm{C}$. We then prepare the system in the initial
state $|\Psi(0)\rangle=|\psi\rangle|0\rangle$. Then, assuming an adiabatic
dynamics, the evolved state $|\Psi(s)\rangle$ is given by the superposition

$|\Psi(s)\rangle=\alpha|n_{+}\rangle|E_{-,0}(s)\rangle+\beta|n_{-}\rangle|E_{-,\phi}(s)\rangle=\cos\left(\frac{\varphi_{0}s}{2}\right)|\psi\rangle|0\rangle+\sin\left(\frac{\varphi_{0}s}{2}\right)|\psi_{\text{rot}}\rangle|1\rangle\text{
,}$ (267)

with
$|\psi_{\text{rot}}\rangle=\alpha|n_{+}\rangle+e^{i\phi}\beta|n_{-}\rangle$
being the rotated desired state and the ground $|E_{-,\xi}(s)\rangle$ and
first excited $|E_{+,\xi}(s)\rangle$ states of
$\mathcal{H}_{\xi}\left(s\right)$ given by

$\displaystyle|E_{-,\xi}(s)\rangle$
$\displaystyle=\cos(\varphi_{0}s/2)|0\rangle+e^{i\xi}\sin(\varphi_{0}s/2)|1\rangle\text{
,}$ (268) $\displaystyle|E_{+,\xi}(s)\rangle$
$\displaystyle=-\sin(\varphi_{0}s/2)|0\rangle+e^{i\xi}\cos(\varphi_{0}s/2)|1\rangle\text{
.}$ (269)

We observe that, due to the dynamics of the auxiliary qubit through two
adiabatic paths, there are quantum (adiabatic) phases
$\theta^{\text{ad}}_{0}(s)$ and $\theta^{\text{ad}}_{\phi}(s)$ accompanying
the evolutions associated with $|E_{-,0}(s)\rangle$ and
$|E_{-,\phi}(s)\rangle$, respectively. Then, relative phases should in
principle be considered in Eq. (267). However, as shown in the Ref. [9], such
phases satisfy $\theta^{\text{ad}}_{0}(s)=\theta^{\text{ad}}_{\phi}(s)$. Thus,
they factorize as a global phase of the state $|\Psi(s)\rangle$. At the end of
the evolution, a measurement on the auxiliary qubit yields the rotated state
with probability $\sin^{2}\left(\varphi_{0}/2\right)$ and the input state with
probability $\cos^{2}\left(\varphi_{0}/2\right)$. The computation process is
therefore probabilistic, which succeeds if the auxiliary qubit ends up in the
state $|1\rangle$. Otherwise, the target system automatically returns to the
input state and we simply restart the protocol. In the adiabatic scenario, the
parameter $\varphi_{0}$ can then be adjusted in order to obtain the optimal
fidelity $1$ by taking the limit $\varphi_{0}\rightarrow\pi$, implying in a
deterministic computation.

This model can be easily adapted to implement controlled single-qubit gates
(two-qubit gates). To this end, the target system has to be increased from one
qubit to two qubits, where one of these qubits will be used as control and the
second one as target. Again, we need to consider the auxiliary qubit in order
to provide the universality of the protocol. Here, we adopt that the single-
qubit gate acts on the target register if the state of the control register is
$|1\rangle$. With this convention, the Hamiltonian that implements a
controlled single-qubit gate is given by

$H_{0}^{\text{cg}}\left(s\right)=(\mathbbm{1}-P_{1,-})\otimes\mathcal{H}_{0}\left(s\right)+P_{1,-}\otimes\mathcal{H}_{\phi}\left(s\right)\text{
,}$ (270)

where now the set the orthogonal projectors is given by
$P_{k,\pm}=|k\rangle\langle k|\otimes|n_{\pm}\rangle\langle n_{\pm}|$, where
$|k\rangle$ denotes the computational basis. The input state of the target
system is now written as
$|\psi_{2}\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle$, with
$a,b,c,d\in\mathbbm{C}$ and $|nm\rangle=|n\rangle|m\rangle$ denoting the
control and target register, respectively. By rewriting $|\psi_{2}\rangle$ in
terms of the basis $|n_{\pm}\rangle$, we have
$|\psi_{2}\rangle=\alpha|0n_{+}\rangle+\beta|0n_{-}\rangle+\gamma|1n_{+}\rangle+\delta|1n_{-}\rangle$,
with $\alpha,\beta,\gamma,\delta\in\mathbbm{C}$. Therefore, by assuming
adiabatic evolution, the system evolves from the state
$|\Psi_{2}(0)\rangle=|\psi_{2}\rangle|0\rangle$, to the instantaneous state

$\displaystyle|\Psi(s)\rangle$
$\displaystyle=\alpha|0n_{+}\rangle|E_{-,0}(s)\rangle+\beta|0n_{-}\rangle|E_{-,0}(s)\rangle+\gamma|1n_{+}\rangle|E_{-,0}(s)\rangle+\delta|1n_{-}\rangle|E_{-,\phi}(s)\rangle$
$\displaystyle=\cos\left(\frac{\varphi_{0}s}{2}\right)|\psi_{2}\rangle|0\rangle+\sin\left(\frac{\varphi_{0}s}{2}\right)|\psi_{2\text{rot}}\rangle|1\rangle\text{
, }$ (271)

with
$|\psi_{2\text{rot}}\rangle=\alpha|0n_{+}\rangle+\beta|0n_{-}\rangle+\gamma|1n_{+}\rangle+e^{i\phi}\delta|1n_{-}\rangle$
being the rotated desired state. We then see that the final state
$|\Psi(1)\rangle$ allows for a probabilistic interpretation for the evolution
and, consequently, the computation protocol can again be taken as
probabilistic ($\varphi_{0}\neq\pi$) or deterministic ($\varphi_{0}=\pi$).

####  15.2 Quantum gates by standard and optimal TQD controlled evolutions

As shown in Ref. [10], the standard TQD Hamiltonian that implements the
shortcut to controlled adiabatic evolutions reads

$H_{\text{std}}\left(s\right)=\sum_{k}P_{k}\otimes\mathcal{H}_{k}^{\text{std}}\left(s\right)\text{
, }$ (272)

where each Hamiltonian $\mathcal{H}_{k}^{\text{std}}\left(s\right)$ is the
corresponding standard TQD Hamiltonian associated with the Hamiltonian
$\mathcal{H}_{k}\left(t\right)$ given in Eq. (261). Thus, we write
$\mathcal{H}_{k}^{\text{std}}\left(s\right)=\mathcal{H}_{k}\left(s\right)+\mathcal{H}_{k}^{\text{cd}}(s)$,
with $\mathcal{H}_{k}^{\text{cd}}(s)$ the corresponding counter-diabatic term
of $\mathcal{H}_{k}\left(s\right)$. For the single and two-qubit gates
previously discussed, the counter-diabatic Hamiltonians
$\mathcal{H}_{\xi}^{\text{cd}}(s)$ associated with the Hamiltonians
$\mathcal{H}_{\xi}\left(s\right)$ in Eq. (264) are time-independent and given
by ($\xi=\\{0,\phi\\}$)

$\mathcal{H}_{\xi}^{\text{cd}}(s)=\mathcal{H}_{\xi}^{\text{cd}}=\hbar\frac{\vartheta_{0}}{2\tau}\left[\sigma_{y}\cos\xi-\sigma_{x}\sin\xi\right]\text{
. }$ (273)

Therefore, one note that the cost of performing standard TQD requires the
knowledge of the eigenvalues and eigenstates of
$\mathcal{H}_{k}\left(t\right)$. Moreover, we can show that, for an arbitrary
$n$-controlled quantum gate, we get time-independent counter-diabatic
Hamiltonians [10]. Therefore, the Eq. (273) shows that the implementation of
the shortcut can be achieved with a very simple assistant Hamiltonian in the
quantum dynamics. Then, the standard TQD Hamiltonian that implements a single-
qubit gate reads

$H_{\text{std}}^{\text{sg}}\left(s\right)=P_{+}\otimes\mathcal{H}_{0}^{\text{std}}\left(s\right)+P_{-}\otimes\mathcal{H}_{\phi}^{\text{std}}\left(s\right)\text{
, }$ (274)

with

$\mathcal{H}_{\xi}^{\text{std}}\left(s\right)=\mathcal{H}_{\xi}\left(s\right)+\hbar\frac{\vartheta_{0}}{2\tau}\left[\sigma_{y}\cos\xi-\sigma_{x}\sin\xi\right]\text{
. }$ (275)

Similarly, we derive a general TQD Hamiltonian in which the quantal phases are
considered as arbitrary. Remarkably, the Hamiltonian satisfy the conditions of
the Theorem 4.2 and the parallel transport condition. Therefore, the optimal
TQD reads

$H_{\text{opt}}^{\text{sg}}\left(s\right)=P_{+}\otimes\mathcal{H}_{0}^{\text{opt}}\left(s\right)+P_{-}\otimes\mathcal{H}_{\phi}^{\text{opt}}\left(s\right)\text{
, }$ (276)

in which each optimal Hamiltonian
$\mathcal{H}_{\xi}^{\text{opt}}\left(s\right)$ is given by
$\mathcal{H}_{\xi}^{\text{opt}}\left(s\right)=\mathcal{H}_{\xi}^{\text{opt}}=\mathcal{H}_{\xi}^{\text{cd}}$.
Thus, the application of generalized TQD theory to quantum computation allows
us to get a optimal costing time-independent TQD Hamiltonian
$H^{\text{sg}}_{\text{opt}}$ given by

$H_{\text{opt}}^{\text{sg}}=P_{+}\otimes\mathcal{H}_{0}^{\text{cd}}+P_{-}\otimes\mathcal{H}_{\phi}^{\text{cd}}\text{
, }$ (277)

where $\mathcal{H}_{k}^{\text{cd}}$ is given by the Eq. (273). To end, by
using the same discussion as above, the complete set of TQD Hamiltonian to be
used as to achieve an universal scheme of quantum computation is obtained as

$\displaystyle H_{\text{std}}^{\text{cg}}\left(s\right)$
$\displaystyle=(\mathbbm{1}-P_{1,-})\otimes\mathcal{H}_{0}^{\text{std}}\left(s\right)+P_{1,-}\otimes\mathcal{H}_{\phi}^{\text{std}}\left(s\right)\text{
, }$ (278) $\displaystyle H_{\text{opt}}^{\text{cg}}$
$\displaystyle=(\mathbbm{1}-P_{1,-})\otimes\mathcal{H}_{0}^{\text{opt}}+P_{1,-}\otimes\mathcal{H}_{\phi}^{\text{opt}}\text{
, }$ (279)

with the projector $P_{1,-}$ defined as in Eq. (270), and the Hamiltonians
$\mathcal{H}_{\xi}^{\text{std}}\left(s\right)$ and
$\mathcal{H}_{\xi}^{\text{opt}}$ given as in Eqs. (273) and (275),
respectively.

As a important remark, it is evident that a number of strategies can be used
to implement quantum gates. For example, inverse engineering of a Hamiltonian
is an interesting strategy to design a class of Hamiltonians to quantum
computation [31], where it is shown how arbitrary single and controlled
quantum gates are implementable through very simple time-independent
Hamiltonians, and to speed up state engineering [158], as well. Here we are
using a different approach to achieve the same result as in Refs. [31, 158].
However, among the infinite curves connecting the input to the output state,
here we are interested in that dynamics where the path followed by the system
in the Hilbert space is the adiabatic path. For this reason, these number of
additional controls (and fields), auxiliary qubits and measurements
(sometimes) are required.

The performance of optimal TQD approach in comparative with its adiabatic and
standard counterpart is studied in Ref. [71], where different scenarios of
energy resource and decohering effects were considered. As main result, in any
case of decoherence and energetic resource, there always exist dynamical
regimes (short evolution times) for which the optimal transitionless
evolutions are more robust and therefore a preferred approach in a decohering
physical environment. This has been shown for both the dephasing and GAD
channels acting on the eigenstate bases. It is also possible to show the
advantage in other bases, such as the computational basis. The general picture
is that the gain will typically occur during some finite time range,
disappearing in the limit of long evolution times. These results are
encouraging for the generalized transitionless approach in the open-system
realm as long as local Hamiltonians are possible to be designed. In the
specific case of quantum gate Hamiltonians, this approach can be applied, e.g.
to derive robust local building blocks for analog implementations of quantum
circuits (see, e.g., Refs. [157, 8]).

####  15.3 Optimal TQD quantum computation in NMR experimental setup

The experimental realization of quantum gates is implemented through the
Chloroform molecule by taking the 13C nucleus as the target qubit and the 1H
nucleus as the auxiliary subsystem. In our experiment [72], we encode the
computational states $|0\rangle$ and $|1\rangle$ into the spin states
$|\\!\\!\downarrow\rangle$ and $|\\!\\!\uparrow\rangle$, respectively. The
schematic representation of the quantum dynamics is illustrated in Fig. 15b,
where the composite system is initially prepared at $t=0$ and measured in
$t=\tau$. The decohering time scales in our system are $T_{1\text{C}}=7.33$ s
and $T_{2\text{C}}=4.99$ s for 13C nucleus and $T_{1\text{H}}=14.52$ s and
$T_{2\text{H}}=0.77$ s for 1H nucleus.

Figure 15: (a) Representation of a single-qubit gate as an arbitrary rotation
implemented in the Bloch sphere. (b) Schematic representation of the quantum
dynamics for the Chloroform molecule, with the target and ancilla qubits
encoded in the Carbon and Hydrogen nuclei, respectively.

We have considered the experimental implementation of the single-qubit phase
gate $Z$, which yields the Pauli matrix $\sigma^{z}$ applied to the input
qubit, which is prepared in an arbitrary initial state along a direction
$\hat{r}$ in the Bloch sphere. To this gate, the adiabatic, standard TQD, and
optimal TQD Hamiltonians are given by (the subscript “C” and “H” denotes
operation on Carbon and Hydrogen nucleus, respectively)

$\displaystyle H^{\text{ph}}_{0}(t)$
$\displaystyle=-2\pi\nu\left[\cos\left(\frac{\pi
t}{\tau}\right)\mathbbm{1}^{\text{C}}\otimes\sigma_{z}^{\text{H}}+\sin\left(\frac{\pi
t}{\tau}\right)\sigma_{z}^{\text{C}}\otimes\sigma_{x}^{\text{H}}\right]\text{
, }$ (280a) $\displaystyle H^{\text{ph}}_{\text{std}}(t)$
$\displaystyle=H^{\text{ph}}_{0}(t)+\frac{\pi}{2\tau}\sigma_{z}^{\text{C}}\otimes\sigma_{y}^{\text{H}}\text{
, }$ (280b) $\displaystyle H^{\text{ph}}_{\text{opt}}$
$\displaystyle=\frac{\pi}{2\tau}\sigma_{z}^{\text{C}}\otimes\sigma_{y}^{\text{H}}\text{
, }$ (280c)

which are obtained from Eqs. (270), (274), and (277), respectively, with
$\omega=2\pi\nu$ and $\nu$ denoting a real frequency in the experiment. It is
worth highlighting that a study on the energy cost to implement adiabatic and
TQD Hamiltonians has been previously discussed in Ref. [71] from an operator
norm approach. Here, we are interested in analyzing the cost from an
alternative point of view, where we associate fixed energy amounts for each
pulse in a pulse sequence. Therefore, this includes the effective energy spent
to implement each pulse of magnetic field, while disregarding the free
evolution contributions to the quantum dynamics.

Particularly, in our experimental implementation of the phase gate $Z$, we
have set $\nu=35$ Hz. We can see that, differently from the standard TQD and
adiabatic Hamiltonians, the optimal TQD protocol provides a time-independent
Hamiltonian to realize the phase gate $Z$. In Fig. 16a, we present the pulse
sequences used to implement each Hamiltonian in Eqs. (280) for any input
state. As an illustration, the initial state of the target qubit considered in
our experiment has been taken as
$|\psi(0)\rangle=|+\rangle^{\text{C}}\otimes|0\rangle^{\text{H}}$, with
$|+\rangle=(1/\mathchoice{{\hbox{$\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\,}$}\lower
0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}})(|0\rangle+|1\rangle)$.
Each pulse sequence implements the correct dynamics up to a rotation around
the $Z$-axis over the auxiliary qubit. Since the final state of the auxiliary
qubit is $|1\rangle$, the circuits provide the correct output up to a global
phase. The pulse composition is described in details in Appendix 7.B.1.

Figure 16: (a) Adiabatic, standard TQD, and optimal TQD protocol pulses. Two-
qubit blocks labeled with $\Delta t_{j,n}$ represent a free evolution of the
chloroform molecule during a time interval $\Delta t_{j,n}$, while single-
qubit blocks are rotations $(\vartheta)_{\varphi_{n}}$ in Hilbert space (See
Appendix 7.B.1 for more details). (b) Fidelity for the $Z$ phase gate
implementation over the initial input state
$|+\rangle=(1/\mathchoice{{\hbox{$\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\,}$}\lower
0.4pt\hbox{\vrule height=3.22221pt,depth=-2.57779pt}}})(|0\rangle+|1\rangle)$
encoded in the Carbon nucleus, with the Hydrogen nucleus representing the
auxiliary qubit in the initial state $|0\rangle$. Fidelity $\mathcal{F}(\tau)$
is computed between the instantaneous ground state of $H^{\text{ph}}_{0}(t)$
and the dynamically evolved quantum states driven by the Hamiltonians
$H^{\text{ph}}_{0}(t)$, $H^{\text{ph}}_{\text{std}}(t)$, and
$H^{\text{ph}}_{\text{opt}}(t)$. Solid curves and discrete symbols represent
theoretical predictions and experimental results, respectively.

The Hamiltonians in Eq. (280a) and (280b) are time-dependent and do not
commute at different instants of time, i.e.
$[H^{\text{ph}}_{0}(t_{1}),H^{\text{ph}}_{0}(t_{2})]\neq 0$ and
$[H^{\text{ph}}_{\text{std}}(t_{1}),H^{\text{ph}}_{\text{std}}(t_{2})]\neq 0$
for some $t_{1}\neq t_{2}$. Then, their pulse sequences are required to
implement the Dyson series for the corresponding unitaries [150]. Thus, as
shown in Figs. 16a-I and 16a-II, the $N$ repetitions are associated to
a“trotterization” of the Dyson series for the adiabatic and standard TQD
protocols, respectively, with the implementation of the Dyson series being
exact in the limit $N\rightarrow\infty$. The repeated application of the
sequence can lead to the accumilation to experimental sistematic errors, in
order to avoid the rrors we have employed NMR composite pulses [159]. On the
other hand, the optimal TQD Hamiltonian $H^{\text{ph}}_{\text{opt}}$ can be
implemented by using a very short pulse sequence, as shown in Fig 16a-III.

An immediate discussion arising from the analysis above is the amount of
energy required to implement the protocol (the pulse sequence associated with
the Dyson series). In order to conveniently study the energy cost of
implementing the optimal TQD protocol, we need to analyze the energy cost of
implementing the sequence of pulses able to reproduce the Dyson series for
each dynamics. Consequently, from the pulse sequence presented in Fig 16a, the
energy cost evolving the optimal TQD is (at least) $N$ times less than both
that associated with the adiabatic and the standard TQD protocols. In fact, if
we consider that each pulse in Fig. 16a [given by a rotation
$(\vartheta)_{\varphi_{n}}$] is associated with an energy cost $E_{0}$, the
overall energy cost for implementing the optimal TQD is
$E^{\text{opt}}_{\text{tqd}}=3E_{0}$ (disregarding the energy cost of free
evolutions), while the energy cost of the adiabatic and the standard TQD are
given by $E_{\text{ad}}=2(N+1)E_{0}$ and
$E^{\text{std}}_{\text{tqd}}=5NE_{0}$, respectively. From this analysis, it
follows that generalized TQD can be used to provide Hamiltonians exhibiting
optimal pulse sequence.

We also consider the $\mathcal{F}(t)$ between the instantaneous ground state
of $H_{Z}(t)$ and the dynamically evolved quantum states driven by the
Hamiltonians $H_{Z}(t)$, $H^{\text{opt}}_{\text{tqd}}(t)$, and
$H^{\text{std}}_{\text{tqd}}(t)$. This is shown in Fig. 16b. Notice that, as
expected for the Hamiltonian $H_{Z}(t)$, the fidelity improves as the
evolution time is increased. In contrast, $H^{\text{opt}}_{\text{tqd}}(t)$ and
$H^{\text{std}}_{\text{tqd}}(t)$ are capable of achieving high fidelities for
any evolution time, since they are designed to mimic the adiabatic evolution
for arbitrary $\tau$. Notice also that, for the time scale considered in the
experiment, which goes up to $\tau=12$ ms, decoherence has little effect,
since this $\tau$ is still much smaller than the dephasing relaxation time
scale $T_{2}$.

###  16 Conclusions of this chapter

In this chapter we have developed a generalized minimal resource demanding
counter-diabatic theory, which is able to yield efficient shortcuts to
adiabaticity via fast transitionless evolutions. To this end, we explore the
gauge freedom of the quantal phases that accompany the evolution dynamics, so
that we can fix them in order to minimize some physical quantity of our
interest. In particular, we have focused on minimum energy and pulse costs for
implement a optimal TQD, comparing it with its associated adiabatic dynamics
and traditional TQD counterparts. The performance of the theory has been
experimentally investigated in two different experimental setups. Firstly, by
using the trapped Ytterbium ion qubit the result are: i) The optimal version
of generalized TQD has been shown to be a useful protocol for obtaining the
optimal shortcut to adiabaticity. While adiabatic and traditional TQD require
time-dependent quantum control, optimal TQD can be experimentally realized by
using time-independent fields. In addition, the necessity of auxiliary fields
in traditional TQD is not a requirement for implementing TQD via its optimal
version. ii) Optimal TQD is an energetically optimal protocol of shortcut to
adiabaticity. By considering the average intensity fields as a measure of
energy cost for implementing the protocols discussed here, we were able to
show that the optimal version of generalized TQD may be energetically less
demanding. This result is kept also for alternatives definitions of energy
cost, e.g. taking into account the detuning contribution. iii) By simulating
an environment-system coupling associated with the dephasing channel, we have
shown that optimal TQD can be more robust than the adiabatic dynamics and the
traditional TQD while at the same time spending less energy resources for a
finite time range. Once the Landau-Zener Hamiltonian can be implemented in
other physical systems, e.g. nuclear magnetic resonance [150] and two-level
systems driven by a chirped field [26], it is reasonable to think that the
results obtained here can be also realized in other experimental
architectures.

As a second experimental verification, we investigated the generalized
approach for TQD in an NMR setup. As a first application in this architecture,
we considered the adiabatic dynamics of a single spin$-1/2$ particle in a
resonant time-dependent rotating magnetic field. As expected, the adiabatic
behavior of the system is drastically affected by the resonance phenomenon. On
the other hand, the standard TQD and the optimal TQD are both immune to the
resonance destructive effect. In particular, we have shown that the optimal
TQD approach provides a transitionless Hamiltonian that can be implemented
with low intensity fields in comparison with the fields used to implement the
adiabatic and the standard TQD Hamiltonians. The second application was done
through the study of the adiabatic and counter-diabatic implementations of
single-qubit quantum gates in NMR. From the generalized approach for TQD, we
have addressed the problem of the feasibility of the shortcuts to
adiabaticity, as provided by TQD protocols, in the context of quantum
computation via controlled evolutions [9, 10]. By using the generalized TQD
Hamiltonian [71], we have presented the optimal solution in terms of pulse
sequence and resources to implement fast quantum gates through counter-
diabatic quantum computation with high fidelity. The energy-optimal protocol
presented here is potentially useful for speeding up digitized adiabatic
quantum computing [73]. Our study explicitly illustrates the performance of
generalized TQD in terms of both energy resources and optimal pulse sequence.
Since digitized adiabatic quantum computing requires the Trotterization of the
adiabatic dynamics, our protocol could be useful in reducing the number of
steps used in digital adiabatic quantum processes. In addition, such process
is independent of the experimental approach used to digitize the adiabatic
quantum evolution.

These results are encouraging for the generalized transitionless approach in
the open-system realm as long as local Hamiltonians are possible to be
designed. In the specific case of quantum gate Hamiltonians, this approach can
be applied, e.g. to derive robust local building blocks for analog
implementations of quantum circuits (see, e.g., Refs. [157, 8]).

##  Chapter 5 Overview and future perspectives

In this thesis we presented general results on adiabatic dynamics in closed
and open systems, and shortcuts to adiabaticity by transitionless quantum
driving. In the context of adiabatic dynamics in closed system we presented
potentially useful theorems that can be viewed as a validation mechanism for
the well-known adiabaticity conditions, where changes of frame in Schrödinger
equation are required. In particular, for the closed system case, we have both
theoretically and experimentally shown that several relevant ACs, which
include the traditional AC, are neither sufficient nor necessary to describe
the adiabatic behavior of a qubit in an oscillating field considered here.
This result leads us to the conclusion that we do not typically have a fully
applicable AC. However, from an adequate choice of the reference frame used to
describe the system dynamics, sufficiency and necessity of the ACs are
fundamentally obtained under certain conditions through the non-inertial frame
map. As a potential application of adiabatic dynamics, we provided a way to
enhance the charging performance of quantum devices able to store energy
(quantum batteries). We propose the notion of adiabatic QBs, where the
adiabatic behavior takes place in the non-inertial frame and it allows for a
stable charging process of the battery, since in inertial frame the dynamics
is not adiabatic. To this aim, we propose the notion of a new quantum
observable that quantifies the energy transfer rate from the quantum battery
to a consumption hub, the power operator. Applications of this operator have
demonstrated its usefulness in quantum battery context. For open systems, our
theoretical results give complementary contributions to the results known in
the literature. The validity conditions used to predict the adiabatic behavior
of a quantum system evolving through non-unitary dynamics are reviewed. We
have shown that the adiabatic approximation for open systems holds in the
asymptotic time limit $t\rightarrow\infty$ as an arbitrarily valid consequence
of the one-dimensional Jordan decomposition of the Lindblad superoperator, the
absence of level crossings, and the initialization of the system as a
superposition of only two eigenstates of Lindbladian that drives the system.
Our main result in adiabaticity in open system refers to the application in
quantum thermodynamics, where we can provide some enhanced understanding on
the thermal adiabatic process and dynamical adiabatic evolutions. To end, as
an original contribution to the TQD theory, we have developed a generalized
minimal energy demanding counter-diabatic theory, which is able to yield
efficient shortcuts to adiabaticity via fast transitionless evolutions.
Moreover, we have investigated the robustness of adiabatic dynamics, standard
and optimal TQD under decoherence. There always exist dynamical regimes for
which generalized transitionless evolutions are more robust and therefore a
preferred approach in a decohering setup.

### Future perspectives

As future perspectives, there are a number of open questions to be addressed.

1) The study introduced here on adiabatic dynamics allows us to realize
further future research as detailed below.

1-a) The consideration and influence of different reference frames in the
adiabatic dynamics of the system lead us to the fundamental question of the
existence of a preferable non-inertial frame. In addition, our development
takes into account a single resonant field acting on the system, so that a
more general approach for an arbitrary number of fields is an open question.

1-b) The experimental implementation of the two-cell adiabatic quantum battery
proposed here is a desired task to be taken in consideration as future
research. The charging and discharging process of the quantum battery proposed
in this thesis will be investigated in a transmon qubit superconducting system
in collaboration with experimental groups. The questions to be experimentally
addressed here refers to stable charging/discharging of quantum batteries
based on Bell states and its robustness against decohering effects.

1-c) When we use the terminology “quantum” in quantum devices, it is required
some detailed discussion on what quantum is in such devices. For example, the
quantum advantage of quantum batteries may be associated with quantum
entanglement [86, 87, 88, 89, 90] and it becomes evident in multi-cell quantum
batteries. Therefore, the scalability of the model proposed here is of great
interest, where we can verify the real role of entanglement for the power of
the charging/discharging process of the battery. Moreover, since the
experimental verification of the expected quantum advantage has not been
addressed so far, collaboration with experimental groups will be considered in
future research.

2) As for the generalized TQD theory, we investigated its performance in
minimizing the energy cost of fast evolutions, and the pulse sequence required
to implement optimal TQD. However, we can think about a number of applications
that can be benefited from these particular evolutions. For example,
applications of generalized TQD to holonomic quantum computing [160],
superadiabatic charging of quantum batteries through minimal intensity fields
[62, 63] and optimal superadiabatic quantum thermal engines [46, 161].

2-a) In the general theory of optimal control [162], one studies how to find a
set of fields $\\{\epsilon_{n}(t)\\}$ (pulses, for example) that drives the
system along a path $|\psi(t)\rangle$ in such way that we can maximize the
quantity $\mathcal{O}[\psi]=\langle\psi(t)|\pazocal{O}|\psi(t)\rangle$, for
some predefined observable $\pazocal{O}$ [163]. Then, how can we use the
generalized approach of TQD in a general approach of optimal control? What is
the general formalism to find optimal quantal phases from an arbitrary
observable $\pazocal{O}$?

2-b) The standard TQD has been applied to enhance performance of quantum
thermal engines concerning the adiabatic version of such heat engines [46,
161, 164]. However, an open question here is whether this enhancement is due
to the additional fields which acts on the system. As future research we will
address this question by using the generalized TQD and providing a scenario
where TQD and adiabatic dynamics spend a same amount of energy. Moreover, it
is known that if we speed up a thermodynamics cycle in order to enhance the
power, some amount of energy is thrown away as a dissipated work due to
diabatic transitions [165, 166]. In particular, it was shown that shortcuts to
adiabaticity can be a good strategy to deal with this problem [34]. The open
question is whether there is some optimal TQD Hamiltonian able to minimize
undesired entropy production along some superadiabatic thermal cycle.

2-c) Given the notion of adiabatic quantum battery and the role of TQD in
quantum thermodynamic tasks, the proposal of superadiabatic quantum batteries
based on TQD is an exciting possibility. Although the adiabatic version of
such devices allows for a stable charging process, the average power is
limited by the total evolution time imposed by the adiabatic theorem. Then,
standard and generalized TQD positively contribute to design a high power and
stable energy transference process.

2-d) The extension for open systems of the standard version of TQD, as
proposed by Demirplak and Rice, was proposed by Vacanti et al [167]. However,
different from closed system TQD, the open system TQD requires our ability of
controlling both fields and decohering environment to drive the system along
of an open system adiabatic path. Therefore, the extension of the generalized
approach of TQD for open system is highly desirable as topic for future
research.

##  Chapter 6 Change of frame in quantum mechanics

Here we present the calculations related with a change of frame in quantum
mechanics, where we explore resonance phenomena and present auxiliary
calculations on the validity conditions of adiabaticity from different
reference frame representations.

##  Appendix 6.A Spin 1/2 particle in a rotating magnetic field

Let us start from below Hamiltonian

$\displaystyle
H(t)=\frac{\hbar\omega_{0}}{2}\sigma_{z}+\frac{\hbar\omega_{1}}{2}\left[\cos(\omega
t)\sigma_{x}+\sin(\omega t)\sigma_{y}\right]\text{ , }$ (281)

which describes the spin-$\frac{1}{2}$ particle in presence of a rotating
magnetic field. The above dynamics is considered concerning the laboratory
inertial frame. In such frame, the unitary system dynamics is governed by Von
Neumann’s equation given by

$\displaystyle\dot{\rho}(t)=\frac{1}{i\hbar}[H(t),\rho(t)]\text{ . }$ (282)

Due the oscillatory behavior of the transverse field, it is convenient to
rewrite the system dynamics in rotating frame provided by the time-dependent
unitary transformation $R(t)=e^{i\frac{\omega}{2}t\sigma_{z}}$. By doing that,
we can implement a rotation in above equation as

$\displaystyle
R(t)\dot{\rho}(t)R^{\dagger}(t)=\frac{1}{i\hbar}R(t)[H(t),\rho(t)]R^{\dagger}(t)=\frac{1}{i\hbar}[R(t)H(t)R^{\dagger}(t),R(t)\rho(t)R^{\dagger}(t)]\text{
. }$ (283)

where we can define the system rotated state as
$\rho_{\text{R}}(t)=R(t)\rho(t)R^{\dagger}(t)$, so that

$\displaystyle R(t)\dot{\rho}(t)R^{\dagger}(t)$
$\displaystyle=\dot{\rho}_{\text{R}}(t)-\dot{R}(t)\overbrace{R^{\dagger}(t)R(t)}^{\mathbbm{1}}\rho(t)R^{\dagger}(t)-R(t)\rho(t)\overbrace{R^{\dagger}(t)R(t)}^{\mathbbm{1}}\dot{R}^{\dagger}(t)$
$\displaystyle=\dot{\rho}_{\text{R}}(t)-\dot{R}(t)R^{\dagger}(t)\rho_{\text{R}}(t)-\rho_{\text{R}}(t)R(t)\dot{R}^{\dagger}(t)\text{
, }$ (284)

in which we can use $\dot{R}(t)R^{\dagger}(t)=-R(t)\dot{R}^{\dagger}(t)$ to
get

$\displaystyle
R(t)\dot{\rho}(t)R^{\dagger}(t)=\dot{\rho}_{\text{R}}(t)-[\dot{R}(t)R^{\dagger}(t),\rho_{\text{R}}(t)]\text{
. }$ (285)

Thus, we have

$\displaystyle\dot{\rho}_{\text{R}}(t)$
$\displaystyle=\frac{1}{i\hbar}[R(t)H(t)R^{\dagger}(t),\rho_{\text{R}}(t)]+[\dot{R}(t)R^{\dagger}(t),\rho_{\text{R}}(t)]$
$\displaystyle=\frac{1}{i\hbar}[R(t)H(t)R^{\dagger}(t),\rho_{\text{R}}(t)]+\frac{i\hbar}{i\hbar}[\dot{R}(t)R^{\dagger}(t),\rho_{\text{R}}(t)]\text{
, }$ (286)

and we conclude that the dynamics of the system in rotating frame will be
given by

$\displaystyle\dot{\rho}_{\text{R}}(t)=\frac{1}{i\hbar}[H_{\text{R}}(t),\rho_{\text{R}}(t)]\text{
, }$ (287)

where the new Hamiltonian in rotating frame is
$H_{\text{R}}(t)=R(t)H(t)R^{\dagger}(t)+i\hbar\dot{R}(t)R^{\dagger}(t)$. By
computing the new Hamiltonian

$\displaystyle H_{\text{R}}(t)$
$\displaystyle=\frac{\hbar\omega_{0}}{2}e^{i\frac{\omega}{2}t\sigma_{z}}\sigma_{z}e^{-i\frac{\omega}{2}t\sigma_{z}}+\frac{\hbar\omega_{1}}{2}e^{i\frac{\omega}{2}t\sigma_{z}}\left[\cos(\omega
t)\sigma_{x}+\sin(\omega
t)\sigma_{y}\right]e^{-i\frac{\omega}{2}t\sigma_{z}}+i\hbar\frac{d}{dt}\left[e^{i\frac{\omega}{2}t\sigma_{z}}\right]e^{-i\frac{\omega}{2}t\sigma_{z}}$
$\displaystyle=\frac{\hbar\omega_{0}}{2}\sigma_{z}+\frac{\hbar\omega_{1}}{2}\left[\cos(\omega
t)e^{i\frac{\omega}{2}t\sigma_{z}}\sigma_{x}e^{-i\frac{\omega}{2}t\sigma_{z}}+\sin(\omega
t)e^{i\frac{\omega}{2}t\sigma_{z}}\sigma_{y}e^{-i\frac{\omega}{2}t\sigma_{z}}\right]-\hbar\frac{\omega}{2}\sigma_{z}\text{
, }$ (288)

now we can use that

$\displaystyle
e^{i\frac{\omega}{2}t\sigma_{z}}\sigma_{x}e^{-i\frac{\omega}{2}t\sigma_{z}}$
$\displaystyle=\cos\left(\omega t\right)\sigma_{x}-\sin\left(\omega
t\right)\sigma_{y}\text{ , }$ (289) $\displaystyle
e^{i\frac{\omega}{2}t\sigma_{z}}\sigma_{y}e^{-i\frac{\omega}{2}t\sigma_{z}}$
$\displaystyle=\cos\left(\omega t\right)\sigma_{y}+\sin\left(\omega
t\right)\sigma_{x}\text{ , }$ (290)

to get the time-independent Hamiltonian

$\displaystyle H_{\text{R}}(t)=H_{\text{R}}$
$\displaystyle=\frac{\hbar\omega_{0}-\omega}{2}\sigma_{z}+\frac{\hbar\omega_{1}}{2}\sigma_{x}\text{
. }$ (291)

Thus, the solution of the dynamics in rotating frame reads as

$\displaystyle\rho_{\text{R}}(t)=\exp\left[-\frac{i}{\hbar}H_{\text{R}}t\right]\rho(0)\exp\left[\frac{i}{\hbar}H_{\text{R}}t\right]\text{
, }$ (292)

so that in non-rotating one we get

$\displaystyle\rho(t)=R(t)\rho_{\text{R}}(t)R^{\dagger}(t)=e^{i\frac{\omega}{2}t\sigma_{z}}e^{-\frac{i}{\hbar}H_{\text{R}}t}\rho(0)e^{\frac{i}{\hbar}H_{\text{R}}t}e^{-i\frac{\omega}{2}t\sigma_{z}}\text{
. }$ (293)

##  Appendix 6.B Resonance in a two-level system driven by oscillating field

Let us consider a Landau-Zener type Hamiltonian given by

$\displaystyle
H(t)=\frac{\hbar\omega_{0}}{2}\sigma_{z}+\frac{\hbar\omega_{\text{x}}}{2}\sin(\omega
t)\sigma_{x}\text{ .}$ (294)

We can rewrite Eq. (294) as

$\displaystyle
H(t)=\frac{\hbar\omega_{0}}{2}\left[\sigma_{z}+\tan\theta\sin(\omega
t)\sigma_{x}\right]\text{ , }$ (295)

where $\theta=\arctan[\omega_{\text{x}}/\omega_{0}]$. It is not obvious that
the Hamiltonian $H(t)$ in Eq. (295) exhibits a resonant behavior. In order to
see this fact, let us define an time-dependent oscillating frame
$R(t)=e^{-i\frac{\omega}{2}t\sigma_{z}}$. In this oscillating frame, the
Hamiltonian is given by

$\displaystyle
H_{\text{R}}(t)=R(t)H(t)R^{\dagger}(t)+iR(t)\dot{R}^{\dagger}(t)=\hbar\frac{\omega_{0}-\omega}{2}\sigma_{z}+\hbar\frac{\sin(\omega
t)\tan\theta}{2}\vec{\omega}_{xy}(t)\cdot\vec{\sigma}_{xy}\text{ , }$ (296)

with $\vec{\omega}_{xy}(t)=\omega_{0}[\cos(\omega t)\hat{x}-\sin(\omega
t)\hat{y}]$ and $\vec{\sigma}_{xy}=\sigma_{x}\hat{x}+\sigma_{y}\hat{y}$.

##  Appendix 6.C Proof of the theorems

Let us consider two Hamiltonians, an inertial frame Hamiltonian $H(t)$ and its
non-inertial counterpart $H_{\pazocal{O}}(t)$, which are related by a time-
dependent unitary $\pazocal{O}(t)$. The dynamics associated with Hamiltonians
$H(t)$ and $H_{\pazocal{O}}(t)$ are given by

$\displaystyle\dot{\rho}(t)$
$\displaystyle=\frac{1}{i\hbar}[H(t),\rho(t)]\text{ , }$ (297)
$\displaystyle\dot{\rho}_{\pazocal{O}}(t)$
$\displaystyle=\frac{1}{i\hbar}[H_{\pazocal{O}}(t),\rho_{\pazocal{O}}(t)]\text{
, }$ (298)

where
$H_{\pazocal{O}}(t)=\pazocal{O}(t)H(t)\pazocal{O}^{\dagger}(t)+i\hbar\dot{\pazocal{O}}(t)\pazocal{O}^{\dagger}(t)$
and $\rho_{\pazocal{O}}(t)=\pazocal{O}(t)\rho(t)\pazocal{O}^{\dagger}(t)$.
Then, the connection between the evolved states $|\psi(t)\rangle$ and
$|\psi_{\pazocal{O}}(t)\rangle$ in inertial and non-inertial frames,
respectively, is given by
$|\psi_{\pazocal{O}}(t)\rangle=\pazocal{O}(t)|\psi(t)\rangle$, $\forall
t\in[t_{0},\tau]$. By considering the initial state in inertial frame given by
a single eigenstate of $H(t)$, namely
$|\psi(t_{0})\rangle=|E_{k}(t_{0})\rangle$, the adiabatic dynamics in this
frame is written as

$\displaystyle|\psi(t)\rangle=e^{i\int_{t_{0}}^{t}\theta_{k}(\xi)d\xi}|E_{k}(t)\rangle\text{
, }$ (299)

where $\theta_{k}(t)=-E_{k}(t)/\hbar+i\langle E_{k}(t)|(d/dt)|E_{k}(t)\rangle$
is the adiabatic phase composed by the dynamical and geometrical phase,
respectively. On the other hand, an adiabatic behavior is obtained in non-
inertial frame if and only if

$\displaystyle|\langle
E^{\mathcal{O}}_{m}(t)|\psi_{\pazocal{O}}(t)\rangle|=|\langle
E^{\pazocal{O}}_{m}(t_{0})|\psi_{\pazocal{O}}(t_{0})\rangle|\text{ , }\forall
m\,,\,\forall t\in[t_{0},\tau]\text{ . }$ (300)

Therefore, we can write

$\displaystyle|\langle E^{\pazocal{O}}_{m}(t)|\psi_{\pazocal{O}}(t)\rangle|$
$\displaystyle=|\langle
E^{\pazocal{O}}_{m}(t_{0})|\psi_{\pazocal{O}}(t_{0})\rangle|\text{ , }$
$\displaystyle|\langle E^{\pazocal{O}}_{m}(t)|\pazocal{O}(t)|\psi(t)\rangle|$
$\displaystyle=|\langle
E^{\pazocal{O}}_{m}(t_{0})|\pazocal{O}(t_{0})|\psi(t_{0})\rangle|\text{ , }$
$\displaystyle|\langle E^{\pazocal{O}}_{m}(t)|\pazocal{O}(t)|E_{k}(t)\rangle|$
$\displaystyle=|\langle
E^{\pazocal{O}}_{m}(t_{0})|\pazocal{O}(t_{0})|E_{k}(t_{0})\rangle|\text{ . }$
(301)

Thus, Eq. (301) establishes a necessary and sufficient condition to obtain an
adiabatic evolution in the non-inertial frame, assuming an adiabatic evolution
in the original frame. To conclude our proof, let us consider the converse
case, where the system starts in a eigenstate of
$|E^{\mathcal{O}}_{m}(t_{0})\rangle$ in non-inertial frame. If the dynamics is
adiabatic we write

$\displaystyle|\psi_{\pazocal{O}}(t)\rangle=e^{i\int_{t_{0}}^{t}\theta^{\pazocal{O}}_{m}(\xi)d\xi}|E^{\pazocal{O}}_{m}(t)\rangle\text{
, }$ (302)

where $\theta^{\pazocal{O}}_{m}(t)$ is the adiabatic phase collected in this
frame. The dynamics will be adiabatic in the inertial frame if and only if

$\displaystyle|\langle E_{m}(t)|\psi(t)\rangle|=|\langle
E_{m}(t_{0})|\psi(t_{0})\rangle|\text{ , }\forall m\,,\,\forall
t\in[t_{0},\tau]\text{ . }$ (303)

Therefore, by using the same procedure as before, we get the condition

$\displaystyle|\langle
E_{k}(t)|\pazocal{O}^{\dagger}(t)|E^{\pazocal{O}}_{m}(t)\rangle|$
$\displaystyle=|\langle
E_{k}(t_{0})|\pazocal{O}^{\dagger}(t_{0})|E^{\pazocal{O}}_{m}(t_{0})\rangle|\text{
, }$ (304)

which is equivalent to Eq. (301). This ends the proof of the Theorem 2.1. Now,
let us proof the second theorem discussed in this thesis.

To this end, we need to consider a time-independent Hamiltonian
$H_{\mathcal{O}}$ in the non-inertial frame, so that its evolution operator
can be written as
$U_{\pazocal{O}}(t,t_{0})=e^{-\frac{i}{\hbar}H_{\pazocal{O}}(t-t_{0})}$. Thus,
we can write the dynamics in non-inertial frame as

$\displaystyle|\psi_{\pazocal{O}}(t)\rangle=e^{-\frac{i}{\hbar}H_{\pazocal{O}}(t-t_{0})}|\psi_{\pazocal{O}}(t_{0})\rangle\text{
. }$ (305)

Moreover, assuming adiabatic dynamics in the inertial frame, we get

$\displaystyle|\langle E_{k}(t)|\psi(t)\rangle|=|\langle
E_{k}(t_{0})|\psi(t_{0})\rangle|\text{ . }$ (306)

By using the relationship between inertial and non-inertial frames as
$|\psi(t)\rangle=\pazocal{O}^{\dagger}(t)|\psi_{\pazocal{O}}(t)\rangle$, we
can write

$\displaystyle|\langle
E_{k}(t)|\pazocal{O}^{\dagger}(t)e^{-\frac{i}{\hbar}H_{\pazocal{O}}(t-t_{0})}|\psi_{\pazocal{O}}(t_{0})\rangle|=|\langle
E_{k}(t_{0})|\psi(t_{0})\rangle|\text{ , }$ (307)

where we have used the Eq. (305). Now, by taking
$|\psi_{\pazocal{O}}(t_{0})\rangle=\pazocal{O}(t_{0})|\psi(t_{0})\rangle$, we
obtain

$\displaystyle|\langle
E_{k}(t)|\pazocal{O}^{\dagger}(t)e^{-\frac{i}{\hbar}H_{\pazocal{O}}(t-t_{0})}\pazocal{O}(t_{0})|\psi(t_{0})\rangle|=|\langle
E_{k}(t_{0})|\psi(t_{0})\rangle|\text{ . }$ (308)

Thus, by inserting the initial state
$|\psi(t_{0})\rangle=|E_{n}(t_{0})\rangle$ in Eq. (308), we get

$\displaystyle|\langle
E_{k}(t)|\pazocal{O}^{\dagger}(t)e^{-\frac{i}{\hbar}H_{\pazocal{O}}(t-t_{0})}\pazocal{O}(t_{0})|E_{n}(t_{0})\rangle|=|\langle
E_{k}(t_{0})|E_{n}(t_{0})\rangle|\text{ . }$ (309)

This concludes the proof of Theorem 2.2.

####  6.C.1 Application to the oscillating Hamiltonian

In order to show how the Theorem 2.1 allows us to predict the adiabatic
behavior of the system driven by the Hamiltonian in Eq. (294), let us first
consider a generic system under action of a single time-dependent oscillating
field with characteristic frequency $\omega$, whose Hamiltonian reads

$\displaystyle
H(\omega,t)=\hbar\omega_{0}H_{0}+\hbar\omega_{\text{T}}H_{\text{T}}(\omega,t)\text{
, }$ (310)

where we consider the transverse term
$\hbar\omega_{\text{T}}H_{\text{T}}(\omega,t)$ as a perturbation, so that
$||\omega_{0}H_{0}||\gg||\omega_{\text{T}}H_{\text{T}}(\omega,t)||$, $\forall
t\in[0,\tau]$. In this case, the eigenstates $|E_{n}(t)\rangle$ and energies
$E_{n}(t)$ of $H(\omega,t)$ can be obtained as perturbation of eigenstates
$|E_{n}^{0}\rangle$ and energies $E_{n}^{0}$ of $\hbar\omega_{0}H_{0}$ as (up
to a normalization coefficient)

$\displaystyle|E_{n}(t)\rangle$
$\displaystyle=|E_{n}^{0}\rangle+\pazocal{O}\left(||\hbar\omega_{\text{T}}H_{\text{T}}(\omega,t)||\right)\text{
, }$ (311) $\displaystyle E_{n}(t)$
$\displaystyle=E_{n}^{0}+\pazocal{O}\left(||\hbar\omega_{\text{T}}H_{\text{T}}(\omega,t)||\right)\text{
. }$ (312)

On the other hand, in the non-inertial frame, we have
$H_{\pazocal{O}}(t)=\pazocal{O}(t)H(t)\pazocal{O}^{\dagger}(t)+i\hbar\dot{\pazocal{O}}(t)\pazocal{O}^{\dagger}(t)$,
which yields

$\displaystyle