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# Robust Transitionless Quantum Driving: A Concatenated Approach

Zhi-Cheng Shi  Fujian Key Laboratory of Quantum Information and Quantum Optics
(Fuzhou University), Fuzhou 350108, China  Department of Physics, Fuzhou
University, Fuzhou 350108, China  Cheng Zhang  Fujian Key Laboratory of
Quantum Information and Quantum Optics (Fuzhou University), Fuzhou 350108,
China  Department of Physics, Fuzhou University, Fuzhou 350108, China  Li-Tuo
Shen  Fujian Key Laboratory of Quantum Information and Quantum Optics (Fuzhou
University), Fuzhou 350108, China  Department of Physics, Fuzhou University,
Fuzhou 350108, China  Jie Song  Department of Physics, Harbin Institute of
Technology, Harbin 150001, China  Yan Xia  Fujian Key Laboratory of Quantum
Information and Quantum Optics (Fuzhou University), Fuzhou 350108, China
Department of Physics, Fuzhou University, Fuzhou 350108, China  X. X. Yi
Center for Quantum Sciences and School of Physics, Northeast Normal
University, Changchun 130024, China

###### Abstract

We propose a concatenated approach for implementing transitionless quantum
driving regardless of adiabatic conditions while being robustness with respect
to all kinds of systematic errors induced by pulse duration, pulse amplitude,
detunings, and Stark shift etc. The current approach is particularly efficient
for all time-dependent pulses with arbitrary shape, and only the phase
differences between pulses is required to properly modulate. The simple
physical implementation without the help of pulse shaping techniques or extra
pulses makes this approach quite universal and provides a different avenue for
robust quantum control by the time-dependent Hamiltonian.

_Introduction._ The adiabatic passage (AP) Allen and Eberly (1975); Oreg  _et
al._ (1984); Král  _et al._ (2007) plays an important role in quantum optics
Scully and Zubairy (1997) and quantum computation Nielsen and Chuang (2000);
Vandersypen and Chuang (2005); Albash and Lidar (2018), due to the
insensitivity to deviations in physical parameters. In the adiabatic process,
the system always evolves along an eigenstate connecting the initial state
with the target state. One typical example of AP is the stimulated Raman
adiabatic passage (STIRAP) Kuklinski  _et al._ (1989); Shore  _et al._ (1992);
Vitanov  _et al._ (2017); Fedoseev  _et al._ (2021); Liu  _et al._ (2019);
Vitanov (2020), a popular way for population inversion in three-level systems.
Regardless of its robustness, the common characteristic of AP and its
variations (e.g., rapid adiabatic passage Rickes  _et al._ (2000); Oberst  _et
al._ (2007); Ding  _et al._ (2020) and fractional-STIRAP Marte  _et al._
(1991); Vitanov  _et al._ (1999)) is incompleteness of population transfer and
slow change of parameters over time.

To overcome this shortcoming, a technique called shortcuts to adiabaticity
(STA) Demirplak and Rice (2003, 2005); Berry (2009); Guéry-Odelin  _et al._
(2019); Chen  _et al._ (2010a, b); Ibáñez  _et al._ (2012); Vitanov and
Drewsen (2019); Kölbl  _et al._ (2019); Mostafavi  _et al._ (2019); Chen  _et
al._ (2021) has emerged, which can directly cancel nonadiabatic transitions by
introducing extra counterdiabatic fields. This technique allows the system
perfectly evolving along the eigenstate of the original Hamiltonian at a very
fast rate, but suffers a great loss on the robustness of AP since it requires
to exactly know system parameters in advance. Besides, the pulse shaping
technique is demanded to tailor the original waveform, and some
counterdiabatic fields sometimes are prohibited from a physical point of view
so that its realization is usually challenging.

As another alternative, the composite adiabatic passage (CAP) technique
Torosov  _et al._ (2011), a combination of the best of both composite pulses
Levitt and Freeman (1979); Wimperis (1994); Genov  _et al._ (2014); Wang  _et
al._ (2014); Calderon-Vargas and Kestner (2017); Morton  _et al._ (2005);
Dridi  _et al._ (2020); Ivanov  _et al._ (2022); Torosov  _et al._ (2020);
Gevorgyan and Vitanov (2021); Shi  _et al._ (2022); Torosov and Vitanov (2022)
and adiabatic passage Zlatanov and Vitanov (2017, 2020), has been proposed to
achieve complete population inversion in two-level systems, while its
extension version named composite STIRAP Torosov and Vitanov (2013); Bruns
_et al._ (2018) achieves the same objective in three-level systems. The main
advantage of CAP over STA is the retained robustness of AP and the simplicity
of implementation, because one just properly modulates the relative phases
between different pulses. Nevertheless, some fundamental issues still need to
be addressed in CAP, e.g., the unaccessible of obtaining universal rotation
operations (or arbitrary superposition states) and the requirement of longer
duration compared with traditional adiabatic passages.

In this Letter, through carefully designing the phase differences, we propose
a dynamical mechanism for perfect transitionless quantum driving in a robust
manner, while simultaneously perform high precision quantum operations even
_without_ knowing the magnitudes of various errors in system parameters. The
current approach is constructed by multiply concatenating the Hamiltonians
with the same arbitrary pulse shape but different well-designed constant
phases. In particular, it is unnecessary to satisfy the adiabatic condition,
and thus the total duration does not have to be long.

Figure 1: Sketch diagram of the concatenated approach. Different colors
represent different uses, and
$\bm{\vartheta}_{1}=\bm{\vartheta}_{1}^{\prime}=\bm{\vartheta}_{1}^{\prime\prime}=0$.
In the blue box (the first hierarchy), the phase differences
$\bm{\theta}_{mn}$ in the propagator
$U_{2}(\bm{\theta}_{1}^{\prime})=U_{1}(\bm{\theta}_{N_{1}})\cdots
U_{1}(\bm{\theta}_{1})$ are applied to achieve perfect transitionless quantum
driving. The only difference between the different blue boxes is the phase
shift $\bm{\vartheta}_{n}$. By continuing to concatenate the propagators
$U_{2}(\bm{\theta}_{n}^{\prime})$ in the green box (the second hierarchy),
i.e.,
$U_{3}(\bm{\theta}_{1}^{\prime\prime})=U_{2}(\bm{\theta}_{N_{2}}^{\prime})\cdots
U_{2}(\bm{\theta}^{\prime}_{1})$, we design the phase differences
$\bm{\theta}_{mn}^{\prime}$ for the robustness against the nonadiabatic
transition induced by various errors. Similarly, the phase differences
$\bm{\theta}_{mn}^{\prime\prime}$ in the orange box (the third hierarchy) are
devoted to performing high precision quantum operations, where
$U_{4}(\bm{\theta}_{1}^{\prime\prime\prime})=U_{3}(\bm{\theta}_{N_{3}}^{\prime\prime})\cdots
U_{3}(\bm{\theta}^{\prime\prime}_{1})$. Certainly, the propagators
$U_{4}(\bm{\theta}_{n}^{\prime\prime\prime})$ can be further concatenated to
realize other uses. In this way, we achieve transitionless quantum driving in
a robust manner even in the presence of various systematic errors.

_Triviality of constant phases._ Consider a quantum system with near-neighbor
interactions, and the general form of the time-dependent Hamiltonian is
($\hbar=1$)

$\displaystyle H(t)=\sum\lambda_{j,j}(t)|j\rangle\langle
j|+\lambda_{j,j+1}(t)|j\rangle\langle j\\!+\\!1|+\mathrm{H.c.},~{}$ (1)

where $\lambda_{j,j}(t)$ denote level energies, and $\lambda_{j,j+1}(t)$ are
the coupling strengths of near-neighbor levels. When introducing extra
constant phases to the coupling strengths, the Hamiltonian becomes

$\displaystyle H(t,\bm{\theta})$ $\displaystyle=$
$\displaystyle\sum\lambda_{j,j}(t)|j\rangle\langle
j|+\lambda_{j,j+1}(t)e^{i\theta_{j}}|j\rangle\langle j\\!+\\!1|+\mathrm{H.c.}$
(2) $\displaystyle=$
$\displaystyle\sum\lambda_{j,j}(t)|\tilde{j}\rangle\langle\tilde{j}|+\lambda_{j,j+1}(t)|\tilde{j}\rangle\langle\widetilde{j\\!+\\!1}|+\mathrm{H.c.},$
(3)

where $\bm{\theta}=(\theta_{1},\theta_{2},\dots)$ represents a vector
parameterizing different constant phases in external fields. Through making
appropriate transformations to the original basis:
$|\tilde{j}\rangle=\exp(-{i\sum_{k=1}^{k=j-1}\theta_{k}})|j\rangle$, the form
of the Hamiltonian with different constant phases is the same as the previous
one. Namely, the addition of constant phases simply means that we are choosing
a set of new basis and the expression of the propagator remains unchanged.
Therefore, the physical property of the Hamiltonian (2) is still trivial after
introducing arbitrary constant phases.

_Nontriviality of phase differences._ Even though the choice of constant
phases is arbitrary, _phase differences_ are of nontrivial physical
significance, and can be used for implementing perfect transitionless quantum
driving in a robust manner. To make it more clear, we demonstrate the detailed
design workflow in Fig. 1, where the propagator corresponding to the
Hamiltonian $H(t,\bm{\theta}_{n})$ reads
$U_{1}(\bm{\theta}_{n})=\mathbf{T}\exp{\left[-i\int
H(t,\bm{\theta}_{n})t\right]}$ with $\mathbf{T}$ being a time-ordering
operator. As shown in the blue box, we initially concatenate $N_{1}$
Hamiltonians $H(t,\bm{\theta}_{n})$ with the same time-varying shape but
different $\bm{\theta}_{n}$ to create a composite propagator
$U_{2}(\bm{\theta}_{1}^{\prime})=U_{1}(\bm{\theta}_{N_{1}})\cdots
U_{1}(\bm{\theta}_{1})$, where the phase differences
$\bm{\theta}_{mn}=\bm{\theta}_{m}-\bm{\theta}_{n}$ are properly modulated to
achieve perfect transitionless quantum driving.

When the quantum system exhibits various errors, transitionless quantum
driving becomes imperfect and thus produces nonadiabatic transitions. To solve
it, we continue to concatenating ${N_{2}}$ propagators
$U_{2}(\bm{\theta}_{n}^{\prime})$ to produce a new one
$U_{3}(\bm{\theta}_{1}^{\prime\prime})=U_{2}(\bm{\theta}_{N_{2}}^{\prime})\cdots
U_{2}(\bm{\theta}^{\prime}_{1})$, where $U_{2}(\bm{\theta}_{n}^{\prime})$ are
generated by adding different constant phase shifts $\bm{\vartheta}_{n}$ to
$N_{1}$ Hamiltonians $H(t,\bm{\theta}_{n})$. Through altering the phase
differences
$\bm{\theta}^{\prime}_{mn}=\bm{\theta}^{\prime}_{m}-\bm{\theta}^{\prime}_{n}$,
the propagator $U_{3}(\bm{\theta}_{1}^{\prime\prime})$ would sharply suppress
the nonadiabatic transition induced by various errors.

Note that the propagator $U_{3}(\bm{\theta}_{1}^{\prime\prime})$ becomes
approximately diagonalized after the first two concatenations, but there still
exists phase errors on diagonal elements and those errors would reduce the
precision of quantum operations. Thus, we require to further concatenate
${N_{3}}$ propagators $U_{3}(\bm{\theta}_{n}^{\prime\prime})$ to obtain the
new one
$U_{4}(\bm{\theta}_{1}^{\prime\prime\prime})=U_{3}(\bm{\theta}_{N_{3}}^{\prime\prime})\cdots
U_{3}(\bm{\theta}^{\prime\prime}_{1})$. Similarly, the phase differences
$\bm{\theta}^{\prime\prime}_{mn}=\bm{\theta}^{\prime\prime}_{m}-\bm{\theta}^{\prime\prime}_{n}$
are finely tuned to promote the accuracy of quantum operations. Indeed, the
propagators $U_{4}(\bm{\theta}_{n}^{\prime\prime\prime})$ can be also
concatenated for more other uses if necessary. Through this concatenated
approach, we obtain a sequence for implementing robust quantum operations
under transitionless quantum driving.

_Transitionless quantum driving._ To illustrate this, let us consider a three-
level $\Lambda$ system driven by a Stokes and pump (SP) pulse pair, while two-
photon resonance is kept. The form of the Hamiltonian reads

$\displaystyle\mathrm{H}(t)$ $\displaystyle\\!=\\!$
$\displaystyle\Delta(t)\sigma_{ee}\\!+\\!\left[\Omega_{p}(t)e^{i\theta_{p}}\sigma_{ge}\\!+\\!\Omega_{s}(t)e^{i\theta_{s}}\sigma_{fe}\\!+\\!\mathrm{H.c.}\right]\\!,~{}~{}~{}$
(4)

where $\sigma_{kl}=|k\rangle\langle l|$ ($k,l=g,f,e$), and the
$|g\rangle(|f\rangle)\leftrightarrow|e\rangle$ transition is driven by the
pump (Stokes) pulse with the coupling strength $\Omega_{p}(t)$
[$\Omega_{s}(t)$], the phase $\theta_{p}$ ($\theta_{s}$), and the detuning
$\Delta(t)$. The instantaneous eigenstates of $H(t)$ are
$|d(t)\rangle=\cos\phi(t)\exp({-i\theta_{sp}})|g\rangle-\sin\phi(t)|f\rangle,~{}|E_{+}(t)\rangle=\sin\varphi(t)\exp({i\theta_{s}})|b(t)\rangle+\cos\varphi(t)|e\rangle,~{}\mathrm{and}~{}|E_{-}(t)\rangle=\cos\varphi(t)\exp{(i\theta_{s})}|b(t)\rangle-\sin\varphi(t)|e\rangle$,
with $\theta_{sp}=\theta_{s}-\theta_{p}$,
$\Omega(t)=\sqrt{\Omega_{p}(t)^{2}+\Omega_{s}(t)^{2}}$, $\tan
2\varphi(t)=2\Omega(t)/\Delta(t)$, $\tan\phi(t)=\Omega_{p}(t)/\Omega_{s}(t)$,
and the bright state
$|b(t)\rangle=\sin{\phi(t)}|g\rangle+\cos{\phi(t)}\exp{(i\theta_{sp})}|f\rangle$.

In STIRAP, there is a time delay in the SP pulse pair. Because of this
asynchrony, $\phi(t)$ is a variable quantity over time, inevitably leading to
nonadiabatic transitions between distinct eigenstates. Here, we demand that
the SP pulse pair must be synchronized [i.e., $\phi(t)$ is constant] to make
the dark state $|d(t)\rangle$ completely decouple to other adiabatic states
sup . To extract a freely adjustable phase (e.g., $\theta_{s}$), the
eigenstates $|E_{+}(t)\rangle$ and $|E_{-}(t)\rangle$ are reassociated to form
a set of new basis $\\{|d(t)\rangle,|b(t)\rangle,|e\rangle\\}$, and we adopt
this set of basis to reveal the dynamical mechanism of transitionless quantum
driving.

Figure 2: Fidelity $F$ of the target state
$|\Psi_{T}\rangle=1/\sqrt{2}(|g\rangle+|f\rangle)$ (top panels) and population
$P_{e}$ of the state $|e\rangle$ (bottom panels) vs pulse amplitude error
$\delta_{\Omega_{s}}$ and detuning error $\delta_{\Delta}$ for different
$\mathcal{R}_{\frac{\pi}{4}}(N_{1}\times N_{2},N_{3})$ sequences. Here,
$F=|\langle\Psi_{T}|\Psi(t)\rangle|^{2}$, and
$\mathcal{R}_{\alpha}(N_{1}\times N_{2},N_{3})$ refers to performing a
rotation operation around the axis $\bm{n}$ by the angle $\alpha$ in the Bloch
sphere to obtain the target state $\cos\alpha|g\rangle+\sin\alpha|f\rangle$,
where the initial state is $|g\rangle$ and the total number of SP pulse pairs
is $N_{1}\times N_{2}\times N_{3}$ with $N_{1}=2$. The $n^{\mathrm{th}}$
Stokes pulse has a Gaussian shape
$\Omega_{s}(t)=A\Omega_{0}\exp[(t-3n\tau)^{2}/\tau^{2}]$ with the duration
$T=6\tau$.

In the synchronization of the SP pulse pair, the general expression for the
system propagator in the basis $\\{|d(t)\rangle,|b(t)\rangle,|e\rangle\\}$ is
given by

$\displaystyle U_{1}(\bm{\theta})$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}1&0&0\\\\[2.15277pt]
0&se^{i\alpha}&re^{-i\theta_{s}}\\\\[2.15277pt]
0&-r^{*}e^{i\theta_{s}}&se^{-i\alpha}\end{array}\right],$ (8)

where $r$ and $\alpha$ are determined by the parameters of the SP pulse pair
and $s=\sqrt{1-|r|^{2}}$. Definitely, the appearance of $r$ in Eq. (8) is the
result of the coupling between the states $|b(t)\rangle$ and $|e\rangle$ in
this system. To completely nullify this nonadiabatic transition, we just have
to concatenate two SP pulse pairs with well-designed phase differences, and
the propagator reads
$U_{2}(\bm{\theta}_{1}^{\prime})=U_{1}(\bm{\theta}_{2})U_{1}(\bm{\theta}_{1})$,
where $U_{1}(\bm{\theta}_{n_{1}})$ is the propagator for the
$n_{1}^{\mathrm{th}}$ SP pulse pair with
$\bm{\theta}_{n_{1}}=(\theta_{s,n_{1}},\theta_{p,n_{1}})$, and
$\theta_{s(p),{n_{1}}}$ refers to the phase of the $n_{1}^{\mathrm{th}}$
Stokes (pump) pulse, ${n_{1}}=1,2$. When the phase differences satisfy sup

$\displaystyle\\!\\!{\theta}_{21}=\theta_{s,2}-\theta_{s,1}=\pi-2\alpha,~{}~{}\theta_{s,n_{1}}-\theta_{p,n_{1}}=\theta_{sp},$
(9)

the propagator becomes diagonalizable, i.e.,

$\displaystyle U_{2}(\bm{\theta}_{1}^{\prime})$ $\displaystyle=$
$\displaystyle|d(t)\rangle\langle d(t)|\\!+\\!e^{i2\alpha}|b(t)\rangle\langle
b(t)|\\!+\\!e^{-i2\alpha}|e\rangle\langle e|.~{}~{}~{}~{}$ (10)

According to this propagator, we can perfectly steer the system evolution
along the dark/bright state without any transitions to the excited state
$|e\rangle$. Therefore, two SP pulse pairs are sufficient to achieve perfect
transitionless quantum driving, where the shape of the Stokes (pump) pulse can
be arbitrary and does not require to satisfy the adiabatic condition.

When returning to the original representation, the propagator (10) in the
subspace spanned by $\\{|g\rangle,|f\rangle\\}$ becomes
$\mathcal{U}=\exp\left({-i\alpha\bm{n}\cdot\bm{\sigma}}\right)$ with
$\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ being the Pauli operators. It
actually represents a rotation operation around the axis
$\bm{n}=(-\sin\phi(t)\cos\theta_{sp},\sin\phi(t)\sin\theta_{sp},\cos\phi(t))$
by the angle $\alpha$ in the Bloch sphere. As a result, we obtain a universal
quantum gate, its evolution along the dark/bright state without any
transitions. In particular, different rotation operations [i.e., reflecting in
different $\alpha$, $\phi(t)$, and $\theta_{sp}$] can be freely modulated by
the coupling strength $\Omega_{s}(t)$, the detuning $\Delta(t)$, or the phase
$\theta_{s}$ in this system.

_Robustness._ In reality, the system always exhibits various errors in
physical parameters (e.g., pulse duration, pulse amplitude, detunings, and
more), resulting in two unfavorable effects: the generation of nonadiabatic
transitions and the imprecision of quantum control. Actually, both effects can
be largely avoided by further concatenating the propagators with well-designed
constant phases. Next, we elaborate on this point.

We can see in Eq. (8) that systematic errors mainly cause the deviation of two
quantities: $r$ and $\alpha$. Fortunately, the concatenated dynamical
mechanism for transitionless quantum driving is inherently immune to the
deviation in the quantity $r$, because the design of phase differences is
independent of $r$ sup . As a result, the system evolution is completely
unaffected by the deviation in the quantity $r$. For the deviation in the
quantity $\alpha$, it makes the phase difference satisfying Eq. (9) slightly
different, resulting in the generation of the nonadiabatic transition. To
reduce this transition, we can concatenate $N_{2}$ propagators with different
constant phases:
$U_{3}({\theta}_{1}^{\prime\prime})=U_{2}(\bm{\theta}_{N_{2}}^{\prime})\cdots
U_{2}(\bm{\theta}^{\prime}_{1})$, where
$\bm{\theta}_{n_{2}}^{\prime}=(\theta_{s,{n_{2}}},\theta_{p,{n_{2}}})$, and
$\theta_{s(p),{n_{2}}}$ refers to the phase of the $(2n_{2}-1)^{\mathrm{th}}$
Stokes (pump) pulse in this situation, $n_{2}=1,\dots,N_{2}$. For
$N_{2}=2^{M}$, $M=1,2,\dots$, when the phase differences satisfy sup

$\displaystyle\theta^{\prime}_{2^{M}n_{2}-2^{M-1}+1,2^{M}n_{2}-2^{M}+1}=\pi-2^{M+1}\alpha,$
(11)

where $\theta_{m,n}^{\prime}=\theta_{s,m}-\theta_{s,n}$, the sequence is
accurate to the $(M+1)^{\mathrm{th}}$ order deviation. For other pulse
numbers, i.e., $N_{2}\neq 2^{M}$, we can adopt the concatenated method or
numerical method to obtain the phase differences sup .

Upon the nonadiabatic transition induced by the deviation in the quantity
$\alpha$ being dramatically suppressed, the propagator
$U_{3}({\theta}_{1}^{\prime\prime})$ can be approximately written as a
diagonalizable form:
$U_{3}({\theta}^{\prime\prime}_{1})\approx|d(t)\rangle\langle
d(t)|+\exp[{i2\beta(1+\delta_{\beta})}]|b(t)\rangle\langle
b(t)|+\exp[{-i2\beta^{\prime}(1+\delta_{\beta^{\prime}})}]|e\rangle\langle
e|$. It is worth mentioning that the deviation $\delta_{\beta}$ cannot be
completely eliminated in the second concatenation Wang  _et al._ (2012), and
thus introduces the fault of quantum operations. To decline its influence, we
need to return back to the original basis $\\{|g\rangle,|f\rangle\\}$, and
then further concatenate $N_{3}$ propagators with distinct phase differences:
$U_{4}({\theta}_{1}^{\prime\prime\prime})=U_{3}({\theta}_{N_{3}}^{\prime\prime})\cdots
U_{3}({\theta}^{\prime\prime}_{1})$, where
$U_{3}({\theta}^{\prime\prime}_{n_{3}})\approx\exp\left[{-{i}\beta(1+\delta_{\beta})\bm{n}\cdot\bm{\sigma}}\right]$
and ${\theta}_{n_{3}}^{\prime\prime}$ refers to the phase differences
$\theta_{sp,{n_{3}}}$ between the $(4n_{3}-1)^{\mathrm{th}}$ Stokes pulse and
pump pulse, $n_{3}=1,\dots,N_{3}$. Here, the pulse number $N_{3}(N_{3}\geq 3)$
can be arbitrarily selected. When $N_{3}$ is small, e.g., $N_{3}=3$, the
analytical expression of phase differences is given by sup

$\displaystyle\theta_{21}^{\prime\prime}$ $\displaystyle=$ $\displaystyle
2\arctan\pm\left(\sqrt{1-P_{f}^{2}}\pm\sqrt{2P_{f}-P_{f}^{2}}\right),$ (12a)
$\displaystyle\theta_{32}^{\prime\prime}$ $\displaystyle=$ $\displaystyle
2\arctan\pm\left(\sqrt{1-P_{f}^{2}}\mp\sqrt{2P_{f}-P_{f}^{2}}\right),$ (12b)

where
$\theta_{mn}^{\prime\prime}=\theta_{m}^{\prime\prime}-\theta_{n}^{\prime\prime}$,
and $P_{f}$ represents the population of the state $|f\rangle$. As for a
longer sequence (i.e., $N_{3}>3$), the performance of the robustness against
$\delta_{\beta}$ becomes better since more adjustable phases are contained,
and it is instructive to adopt numerical calculations to obtain the solutions.

In Fig. 2, we demonstrate the robust performance of the quantum operation in
the presence of two systematic errors by different
$\mathcal{R}_{\frac{\pi}{4}}(N_{1}\times N_{2},N_{3})$ sequences, where
$\mathcal{R}_{\frac{\pi}{4}}(N_{1}\times N_{2},N_{3})$ represents the rotation
around the axis $\bm{n}$ with the angle ${{\pi}/{4}}$, and the total pulse
number $N=N_{1}\times N_{2}\times N_{3}$ with $N_{1}=2$. For simplicity, the
shape of each Stokes pulse is chosen as Guassian here, while other shapes can
still work sup .

As shown in Figs. 2(a) and 2(b), a concatenation of two SP pulse pairs to
achieve perfect transitionless quantum driving possesses preliminary
robustness against the pulse amplitude and detuning errors. Whereas, it does
not work very well on large systematic errors. When the SP pulse pairs are
continuously recombined for a second time [cf. $N_{2}=4$ in Figs. 2(c) and
2(d)], the robust performance of transitionless quantum driving is
dramatically enhanced so as to obtain an extremely low leakage population of
the state $|e\rangle$. Note that this recombination makes little contribution
to promoting the precision of rotation operations, since the fidelity is
sensitive to systematic errors yet. Therefore, we need to execute the third
concatenation of SP pulse pairs. It is shown in Figs. 2(e)–2(j) that the high-
fidelity region gradually enlarges as $N_{3}$ increases, while the ability of
leakage suppression is still reserved, which demonstrates that the deviation
$\delta_{\beta}$ can be favorably compensated by properly designing the phase
difference of each pulse pair. Certainly, when concurrently increasing $N_{2}$
and $N_{3}$, both population leakage and operation precision are efficiently
improved.

Figure 3:  Generation of the maximum coherent state $|\Psi_{T}\rangle$ and the
corresponding phase waveform of the Stokes pulse for the
$\mathcal{R}_{\frac{\pi}{4}}(4,3)$ sequence. The top (middle) panel represents
the population evolution of the state $|f\rangle$ ($|e\rangle$) in the
absence/presence of the errors $\delta_{\Omega_{s}}$ and $\delta_{\Delta}$.
After twelve SP pulse pairs with well-designed phase differences, the system
is primely driven to the state $|\Psi_{T}\rangle$ even though it exhibits
significant pulse amplitude and detuning errors; see the dashed and dotted
curves. In the bottom panel, the phase differences $\theta_{21}$ between the
$2n^{\mathrm{th}}$ and $(2n-1)^{\mathrm{th}}$ SP pulse pairs are used for
perfect transitionless quantum driving; the phase differences
$\theta_{21}^{\prime}$ between the $(4n-1)^{\mathrm{th}}$ and
$(4n-3)^{\mathrm{th}}$ SP pulse pairs are devoted to suppress the nonadiabatic
transition induced by the pulse amplitude and detuning errors; the phase
differences $\theta_{(n+1)n}^{\prime\prime}$ between the
$(4n+1)^{\mathrm{th}}$ and $(4n-3)^{\mathrm{th}}$ SP pulse pairs are employed
for improving the fidelity of the Hadamard gate with a specific phase. All
phase differences $\theta_{mn}$, $\theta_{mn}^{\prime}$, and
$\theta_{mn}^{\prime\prime}$ are accessible by properly modulating the phases
$\theta_{s}$ and $\theta_{p}$ of SP pulse pairs.

To see more clearly, we demonstrate in Fig. 3 the population evolution of two
states $|f\rangle$ and $|e\rangle$ and the phase waveform of the Stokes pulse
by the $\mathcal{R}_{\frac{\pi}{4}}(4,3)$ sequence. For a single SP pulse
pair, the system exhibits nonadiabatic transitions, i.e., the leakage to the
excited state $|e\rangle$ (see the blue curve at $t=T$), because the adiabatic
condition is broken. By concatenating two SP pulse pairs and properly
adjusting the phase difference $\theta_{21}$, the transition to the excited
state is completely eliminated, as shown by the blue curve at $t=2T$. When the
systematic errors are large, the evolution path seriously deviates from the
original one, as shown by the dashed and dotted curves. We then concatenate
four SP pulse pairs and modulate the phase difference $\theta_{21}^{\prime}$
to reduce the unfavorable influence on the transition of the excited state
induced by two systematic errors. In this circumstance, the nonadiabatic
transition is sharply suppressed, but the fidelity of the rotation operation
is not much improved; see the curves at $t=4T$. Through further concatenating
three groups of four SP pulse pairs (twelve in total) and controlling the
phase difference $\theta_{21}^{\prime\prime}$ and
$\theta_{32}^{\prime\prime}$, the system evolution is strictly restricted in
the subspace $\\{|g\rangle,|f\rangle\\}$, and the rotation operation becomes
remarkable error tolerant.

_Conclusion._ We have developed a concatenated approach of constructing a
dynamical mechanism for perfect transitionless quantum driving and improving
the robustness with respect to various systematic errors simultaneously. It is
particularly significant that the pulse shape can be arbitrary while the
quantum system does not have to satisfy the adiabatic condition. By properly
designing the phase differences between different Hamiltonians, the unwanted
nonadiabatic transitions are sharply suppressed even in the presence of all
kinds of errors. At the same time, quantum operations with very high fidelity
are naturally obtained. Of course, this concatenated procedure can still be
carried on for other uses. The simplicity, flexibility, and versatility of the
concatenated approach opens a promising avenue for realizing high precision
control in quantum information processing.

###### Acknowledgements.

This work is supported by the Natural Science Foundation of Fujian Province
under Grant No. 2021J01575, the Natural Science Funds for Distinguished Young
Scholar of Fujian Province under Grant No. 2020J06011, and the Project from
Fuzhou University under Grant No. JG202001-2.

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###  S0.1 Supplementary Material for “Robust Transitionless Quantum Driving: A
Concatenated Approach”

##  S1 SI. Dark state decoupling from other eigenstates

In this section, we give a detailed derivation process that the Stokes and
pump (SP) pulse pair must be synchronized for the dark state decoupling from
other eigenstates. The system of interest is a three-level $\Lambda$ structure
and driven by a Stokes and pump (SP) pulse pair under the two-photon resonance
condition. The form of the Hamiltonian reads ($\hbar=1$)

$\displaystyle\mathrm{H}(t)$ $\displaystyle\\!=\\!$
$\displaystyle\Delta(t)|e\rangle\langle
e|\\!+\\!\Omega_{p}(t)e^{i\theta_{p}}|g\rangle\langle
e|\\!+\\!\Omega_{s}(t)e^{i\theta_{s}}|f\rangle\langle e|\\!+\\!\mathrm{H.c.},$
(S1)

where the transition $|g\rangle(|f\rangle)\leftrightarrow|e\rangle$ is driven
by the pump (Stokes) pulse with the coupling strength $\Omega_{p}(t)$
[$\Omega_{s}(t)$], the phase $\theta_{p}$ ($\theta_{s}$), and the detuning
$\Delta(t)$. For this Hamiltonian, we easily obtain the eigenenergies

$\displaystyle E_{0}(t)$ $\displaystyle=$ $\displaystyle 0,$ (S2a)
$\displaystyle E_{+}(t)$ $\displaystyle=$
$\displaystyle\frac{\Delta(t)}{2}+\sqrt{\frac{\Delta(t)^{2}}{4}+\Omega(t)^{2}}=\Omega(t)\cot\varphi(t),$
(S2b) $\displaystyle E_{-}(t)$ $\displaystyle=$
$\displaystyle\frac{\Delta(t)}{2}-\sqrt{\frac{\Delta(t)^{2}}{4}+\Omega(t)^{2}}=-\Omega(t)\tan\varphi(t),$
(S2c)

and the corresponding eigenstates

$\displaystyle|E_{0}(t)\rangle$ $\displaystyle=$
$\displaystyle|d(t)\rangle=\cos\phi(t)e^{-i\theta_{sp}}|g\rangle-\sin\phi(t)|f\rangle,$
(S3a) $\displaystyle|E_{+}(t)\rangle$ $\displaystyle=$
$\displaystyle\sin\phi(t)\sin\varphi(t)e^{i\theta_{p}}|g\rangle+\cos\phi(t)\sin\varphi(t)e^{i\theta_{s}}|f\rangle+\cos\varphi(t)|e\rangle=\sin\varphi(t)e^{i\theta_{s}}|b(t)\rangle+\cos\varphi(t)|e\rangle,~{}~{}~{}~{}~{}$
(S3b) $\displaystyle|E_{-}(t)\rangle$ $\displaystyle=$
$\displaystyle\sin\phi(t)\cos\varphi(t)e^{i\theta_{p}}|g\rangle+\cos\phi(t)\cos\varphi(t)e^{i\theta_{s}}|f\rangle-\sin\varphi(t)|e\rangle=\cos\varphi(t)e^{i\theta_{s}}|b(t)\rangle-\sin\varphi(t)|e\rangle,$
(S3c)

where $\Omega(t)=\sqrt{\Omega_{p}(t)^{2}\\!+\\!\Omega_{s}(t)^{2}}$, $\tan
2\varphi(t)=2\Omega(t)/\Delta(t)$, $\tan\phi(t)=\Omega_{p}(t)/\Omega_{s}(t)$,
$\theta_{sp}=\theta_{s}\\!-\\!\theta_{p}$ is the phase difference between the
Stokes and pump pulse, and
$|b(t)\rangle=\sin{\phi(t)}|g\rangle+\cos{\phi(t)}e^{i\theta_{sp}}|f\rangle$
represents the bright state.

The transition mechanism of the dark state is very easily revealed in the
adiabatic representation spanned by Eqs. (S3a)–(S3c). The matrix form of the
Hamiltonian in the adiabatic basis
$\\{|E_{0}(t)\rangle,|E_{+}(t)\rangle,|E_{-}(t)\rangle\\}$ reads

$\displaystyle
H_{a}(t)=\left[\begin{array}[]{ccc}0&-i\dot{\phi}(t)\sin{\varphi(t)}&-i\dot{\phi}(t)\cos{\varphi(t)}\\\
i\dot{\phi}(t)\sin{\varphi(t)}&\Omega(t)\cot{\varphi(t)}&i\dot{\varphi}(t)\\\
i\dot{\phi}(t)\cos{\varphi(t)}&-i\dot{\varphi}(t)&-\Omega(t)\tan{\varphi(t)}\end{array}\right],$
(S7)

where the overdot represents a time derivative. Obviously, to make the dark
state $|d(t)\rangle$ decouple to other adiabatic states, we have to set
$\dot{\phi}(t)=0$, and the solution is

$\displaystyle{\phi}(t)=\mathrm{constant}.$ (S8)

This means the SP pulses must be synchronized. In fact, this synchronization
between the SP pulses over time is an essential condition to make one state
decouple to other states. In the following, we give a brief proof for it.

Consider a general unitary transform $R$ on this system, and the matrix form
of $R$ in the basis $\\{|g\rangle,|f\rangle,|e\rangle\\}$ is

$\displaystyle R(t)$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}\cos{\phi(t)}e^{-i\vartheta}&\sin{\phi(t)}&0\\\\[4.30554pt]
-\sin{\phi(t)}&\cos{\phi(t)}e^{i\vartheta}&0\\\\[4.30554pt]
0&0&1\cr\end{array}\right].$ (S12)

We now study the system dynamics in a rotation frame with respect to $R$,
where the basis are $\\{|r_{1}(t)\rangle,|r_{2}(t)\rangle,|e\rangle\\}$ with
$|r_{1}(t)\rangle=\cos{\phi(t)}e^{-i\vartheta}|g\rangle-\sin{\phi(t)}|f\rangle$
and
$|r_{2}(t)\rangle=\sin{\phi(t)}|g\rangle+\cos{\phi(t)}e^{i\vartheta}|f\rangle$.
Here, the undetermined parameters $\phi(t)$ and $\vartheta$ are used to ensure
that one basis is decoupled from other two bases in this rotation frame. We
first calculate the expression of the Hamiltonian in the basis
$\\{|r_{1}(t)\rangle,|r_{2}(t)\rangle,|e\rangle\\}$, i.e.,

$\displaystyle H_{r}(t)$ $\displaystyle=$ $\displaystyle
R^{\dagger}(t)H(t)R(t)-iR^{\dagger}(t)\dot{R}(t),$ (S16) $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}0&h_{12}&h_{13}\\\\[4.30554pt]
h_{21}&0&h_{23}\\\\[4.30554pt] h_{31}&h_{32}&h_{33}\cr\end{array}\right]$

where

$\displaystyle h_{12}$ $\displaystyle=$ $\displaystyle
h_{21}^{*}=-i\dot{\phi}(t)e^{-i\vartheta},$ (S17a) $\displaystyle h_{13}$
$\displaystyle=$ $\displaystyle
h_{31}^{*}=\Omega_{p}(t)\cos\phi(t)e^{i(\theta_{p}+\vartheta)}-\Omega_{s}(t)\sin\phi(t)e^{i\theta_{s}},$
(S17b) $\displaystyle h_{23}$ $\displaystyle=$ $\displaystyle
h_{32}^{*}=\Omega_{p}(t)\sin\phi(t)e^{i\theta_{p}}+\Omega_{s}(t)\cos\phi(t)e^{i(\theta_{s}-\vartheta)},$
(S17c) $\displaystyle h_{33}$ $\displaystyle=$ $\displaystyle\Delta(t).$
(S17d)

It can be found that it is required $h_{12}=h_{13}=0$ to decouple
$|r_{1}(t)\rangle$ from $|r_{2}(t)\rangle$ and $|e\rangle$, or
$h_{12}=h_{32}=0$ to decouple $|r_{2}(t)\rangle$ from $|r_{1}(t)\rangle$ and
$|e\rangle$. For $h_{12}=0$, we have $\dot{\phi}(t)=0$, and the solution is

$\displaystyle{\phi}(t)=\mathrm{constant}.$ (S18)

On the other hand, the equation $h_{13}=0$ can be simplified as

$\displaystyle\tan\phi(t)=\frac{\Omega_{p}(t)}{\Omega_{s}(t)}e^{i(\theta_{p}-\theta_{s}+\vartheta)},$
(S19)

while the equation $h_{23}=0$ is simplified as

$\displaystyle\cot\phi(t)=-\frac{\Omega_{p}(t)}{\Omega_{s}(t)}e^{i(\theta_{p}-\theta_{s}+\vartheta)}.$
(S20)

As a result, we acquire that $\vartheta=\theta_{s}-\theta_{p}=\theta_{sp}$ and
the Stokes and pump pulses must be synchronized, since $\phi(t)$ is real
constant.

##  S2 SII. Perfect transitionless quantum driving by two SP pulse pairs

In the section, we demonstrate that perfect transitionless quantum driving is
obtained by merely concatenating two SP pulse pairs with a well-designed phase
difference.

Due to the decoupling between the dark state $|b(t)\rangle$ and the states
$|b(t)\rangle$ and $|e\rangle$, the general expression of the propagator for
the single SP pulse pair in the basis
$\\{|d(t)\rangle,|b(t)\rangle,|e\rangle\\}$ is written as

$\displaystyle U_{1}(\bm{\theta})$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}1&0&0\\\\[2.15277pt]
0&se^{i\alpha}&re^{-i\theta_{s}}\\\\[2.15277pt]
0&-r^{*}e^{i\theta_{s}}&se^{-i\alpha}\end{array}\right],$ (S24)

where $r$ and $\alpha$ are determined by the parameters of the SP pulse pair
and $s=\sqrt{1-|r|^{2}}$. Thus, the propagator for two SP pulse pairs is given
by

$\displaystyle
U_{2}(\bm{\theta}_{1}^{\prime})=U_{1}(\bm{\theta}_{2})U_{1}(\bm{\theta}_{1})=\left[\begin{array}[]{ccc}1&0&0\\\\[4.30554pt]
0&{U}_{2,bb}&{U}_{2,be}\\\\[4.30554pt]
0&{U}_{2,eb}&{U}_{2,ee}\end{array}\right],\\!$ (S28)

where

$\displaystyle{U}_{2,bb}$ $\displaystyle=$
$\displaystyle{U}^{*}_{2,ee}=-|r|^{2}e^{i(\theta_{s,1}-\theta_{s,2})}+s^{2}e^{i2\alpha},$
(S29a) $\displaystyle{U}_{2,eb}$ $\displaystyle=$
$\displaystyle-{U}^{*}_{2,be}=-r^{*}s\left[e^{-i(\alpha-\theta_{s,1})}+e^{i(\alpha+\theta_{s,2})}\right],$
(S29b)

and $\theta_{s,1}$ ($\theta_{s,2}$) are the phase of the first (second) Stokes
pulse. To perfectly implement transitionless quantum driving, we need to
nullify the matrix element ${U}_{2,eb}$, and thus the equation becomes

$\displaystyle e^{-i(\alpha-\theta_{s,1})}+e^{i(\alpha+\theta_{s,2})}=0.$
(S30)

It is easily to obtain the solution of Eq. (S30), i.e.,

$\displaystyle\theta_{s,2}-\theta_{s,1}=\pi-2\alpha.$ (S31)

This is actually Eq. (5) in the main text. As a result, the propagator given
by Eq. (S28) becomes diagonalized, i.e.,

$\displaystyle
U_{2}(\bm{\theta}_{1}^{\prime})=\left[\begin{array}[]{ccc}1&0&0\\\\[2.15277pt]
0&e^{i2\alpha}&0\\\\[2.15277pt]
0&0&e^{-i2\alpha}\end{array}\right].\\!~{}~{}~{}$ (S35)

Therefore, we can perform perfect transitionless quantum driving by only two
SP pulse pairs regardless of whether it exists the nonadiabatic transition in
the single SP pulse pair.

##  S3 SIII. Robustness against nonadiabatic transition induced by various
systematic errors

In this section, we demonstrate how to concatenate $N_{2}$ propagators
$U_{2}(\bm{\theta}_{n_{2}}^{\prime})$ with different constant phases to
achieve robust transitionless quantum driving in the presence of systematic
errors, i.e., the second concatenation in the green box of Fig. 1 in the main
text.

When the quantum system exhibits various errors (e.g., pulse amplitude error,
pulse duration error, and detuning error etc.), it leads to the deviation of
two quantities $r$ and $\alpha$ in the propagator
$U_{2}(\bm{\theta}_{n_{2}}^{\prime})$. In the following, let us study this
issue separately.

(1) _The deviation of the quantities_ $r$ ($s$): $r\rightarrow(1+\delta_{r})r$
[$s\rightarrow(1+\delta_{s})s$]. The propagator $U_{1}(\bm{\theta}_{1})$ for
the single SP pulse pair becomes

$\displaystyle U_{1}(\bm{\theta}_{1})$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}1&0&0\\\\[2.15277pt]
0&(1+\delta_{s})se^{i\alpha}&(1+\delta_{r})re^{-i\theta_{s,1}}\\\\[2.15277pt]
0&-(1+\delta_{r})r^{*}e^{i\theta_{s,1}}&(1+\delta_{s})se^{-i\alpha}\end{array}\right],$
(S39)

Then, the propagator for two SP pulse pairs is given by

$\displaystyle
U_{2}(\bm{\theta}_{1}^{\prime})=U_{1}(\bm{\theta}_{2})U_{1}(\bm{\theta}_{1})=\left[\begin{array}[]{ccc}1&0&0\\\\[4.30554pt]
0&{U}_{2,bb}&{U}_{2,be}\\\\[4.30554pt]
0&{U}_{2,eb}&{U}_{2,ee}\end{array}\right],$ (S43)

where

$\displaystyle{U}_{2,bb}$ $\displaystyle=$
$\displaystyle{U}^{*}_{2,ee}=-(1+\delta_{r})^{2}|r|^{2}e^{i(\theta_{s,1}-\theta_{s,2})}+(1+\delta_{s})^{2}s^{2}e^{i2\alpha},$
(S44a) $\displaystyle{U}_{2,eb}$ $\displaystyle=$
$\displaystyle-{U}^{*}_{2,be}=-(1+\delta_{r})(1+\delta_{s})r^{*}s\left[e^{-i(\alpha-\theta_{s,1})}+e^{i(\alpha+\theta_{s,2})}\right].$
(S44b)

Obviously, the solution of Eq. (S44b) is

$\displaystyle\theta_{s,2}-\theta_{s,1}=\pi-2\alpha,$ (S45)

which is independent of the deviation in the quantity $r$ ($s$). Therefore,
once the phase difference satisfy Eq. (S45) in the two SP pulse pairs, one can
completely eliminate the effect of the deviation in the quantity $r$ ($s$) no
matter how large it is.

(2) _The deviation of the quantity $\alpha$_:
$\alpha\rightarrow(1+\delta_{\alpha})\alpha$. In this case, the propagator
$U_{1}(\bm{\theta}_{1})$ becomes

$\displaystyle
U_{1}(\bm{\theta}_{1})=\left[\begin{array}[]{ccc}1&0&0\\\\[4.30554pt]
0&se^{i(1+\delta_{\alpha})\alpha}&re^{-i\theta_{s,1}}\\\\[4.30554pt]
0&-r^{*}e^{i\theta_{s,1}}&se^{-i(1+\delta_{\alpha})\alpha}\end{array}\right].$
(S49)

As a result, the propagator $U_{2}(\bm{\theta}_{1}^{\prime})$ is

$\displaystyle
U_{2}(\bm{\theta}_{1}^{\prime})=U_{1}(\bm{\theta}_{2})U_{1}(\bm{\theta}_{1})=\left[\begin{array}[]{ccc}1&0&0\\\\[4.30554pt]
0&e^{2i\alpha}(|r|^{2}+e^{2i\alpha\delta_{\alpha}}s^{2})&2irs\sin(\alpha\delta_{\alpha})e^{i(\alpha-\theta_{s,1})}\\\\[4.30554pt]
0&2ir^{*}s\sin(\alpha\delta_{\alpha})e^{i(\theta_{s,1}-\alpha)}&e^{-2i\alpha}(|r|^{2}+e^{-2i\alpha\delta_{\alpha}}s^{2})\end{array}\right].$
(S53)

Obviously, Eq. (S53) demonstrates that the nonadiabatic transition is induced
by the presence of the deviation $\delta_{\alpha}$.

It is worth mentioning that the unfavorable effect caused by the deviation
$\delta_{\alpha}$ can only be sharply reduced rather than completely canceled
in this situation. Next, we elaborate the implementation of robust
transitionless quantum driving by designing the phase differences in a train
of $N_{2}$ propagators $U_{2}(\bm{\theta}_{n_{2}}^{\prime})$, where the
recombination propagator $U_{3}({\theta}_{1}^{\prime\prime})$ reads

$\displaystyle
U_{3}({\theta}_{1}^{\prime\prime})=U_{2}(\bm{\theta}_{N_{2}}^{\prime})\cdots
U_{2}(\bm{\theta}^{\prime}_{1})=\left[\begin{array}[]{ccc}1&0&0\\\\[4.30554pt]
0&{U}_{3,bb}&{U}_{3,be}\\\\[4.30554pt]
0&{U}_{3,eb}&{U}_{3,ee}\end{array}\right],$ (S57)

with $\bm{\theta}_{n_{2}}^{\prime}=(\theta_{s,{n_{2}}},\theta_{p,{n_{2}}})$,
$n_{2}=1,\dots,N_{2}$. Through the Tayor-Maclaurin expansion for the
propagator (S57) with respect to $\delta_{\alpha}$, we can obtain the series
of the element ${U}_{3,eb}$. Then, to achieve robustness against the deviation
$\delta_{\alpha}$, we require to design the phase differences to nullify the
derivatives in a increasing order as many as possible, i.e.,

$\displaystyle\partial^{k}_{\delta_{\alpha}}\\!U_{3,eb}=0,~{}~{}~{}k=0,1,2,3,\dots$
(S58)

where $\partial^{k}_{\delta_{\alpha}}\\!U_{3,eb}$ represents the
$k^{\mathrm{th}}$ derivative of $U_{3,eb}$ with respect to $\delta_{\alpha}$,
and $k=0$ corresponds to the zero-error value. When the number of the
propagators $U_{2}(\bm{\theta}_{n_{2}}^{\prime})$ is small, we can
analytically obtain the solution of Eq. (S58). For a longer sequence, the
solution can be given by concatenating the obtained solutions of short
sequences. Alternatively, we can directly adopt numerical calculations to
present the numerical solution. In the following, we address this issue.

We first study the case of a concatenation of two propagators:

$\displaystyle
U_{3}({\theta}_{1}^{\prime\prime})=U_{2}(\bm{\theta}_{2}^{\prime})U_{2}(\bm{\theta}^{\prime}_{1})=U_{2}^{[1]}({\theta}_{1}^{[1]}),$
(S59)

where $U_{2}^{[1]}({\theta}_{1}^{[1]})$ is used in this subsection for the
sake of distinction from the third concatenation in Sec. SIV. The equation to
be solved reads

$\displaystyle\partial^{1}_{\delta_{\alpha}}\\!U_{3,eb}=-2ie^{-i3\alpha}r^{*}s\alpha\left[e^{i\left(4\alpha+\theta^{\prime}_{s,2}\right)}+e^{i\theta^{\prime}_{s,1}}\right]=0,$
(S60)

Obviously, the solution of the phase difference is

$\displaystyle\theta^{\prime}_{2,1}=\theta^{\prime}_{s,2}-\theta^{\prime}_{s,1}=\pi-4\alpha.$
(S61)

As a result, the sequence is accurate to the second order of the deviation
[i.e., $\mathcal{O}(\delta^{2}_{\alpha}$)].

By concatenating three propagators, the form of the total propagator is

$\displaystyle
U_{3}({\theta}_{1}^{\prime\prime})=U_{2}(\bm{\theta}_{3}^{\prime})U_{2}(\bm{\theta}_{2}^{\prime})U_{2}(\bm{\theta}^{\prime}_{1}),$
(S62)

where

$\displaystyle\partial^{1}_{\delta_{\alpha}}\\!U_{3,eb}=-2ie^{-i5\alpha}r^{*}s\alpha\left[e^{i\left(8\alpha+\theta^{\prime}_{s,3}\right)}+e^{i\left(4\alpha+\theta^{\prime}_{s,2}\right)}+e^{i\theta^{\prime}_{s,1}}\right]=0,$
(S63)

One solution of Eq. (S63) can be expressed as

$\displaystyle\theta^{\prime}_{21}=2\arctan{\left[\frac{\sqrt{3}-2\sin
4\alpha}{2\cos
4\alpha-1}\right]},~{}~{}~{}~{}\theta^{\prime}_{32}=2\arctan{\left[\frac{\sqrt{3}+2\sin
4\alpha}{1-2\cos 4\alpha}\right]}.$ (S64)

Therefore, this sequence is still accurate to the second order of the
deviation.

When $N_{3}>4$, it is difficult to obtain the analytical expression of phase
differences by directly solving Eq. (S58). However, the idea of the
concatenation method demonstrated in the main text can be also applied here to
obtain the analytical expressions. For example, there are two ways to obtain
the phase differences of the six-propagator sequence. One is directly
numerically solving Eq. (S58), and the other is concatenating Eqs. (S61) and
(S64). Next, we take $2^{M}$ propagators $U_{2}(\bm{\theta}_{n_{2}}^{\prime})$
as the example to elaborate the concatenated process.

First of all, we directly regard the propagator
$U_{2}^{[1]}({\theta}_{1}^{[1]})$ as the new propagator, and then perform two
propagators recombination again. In this case, the total propagator becomes

$\displaystyle
U_{3}({\theta}_{1}^{\prime\prime})=U_{2}^{[2]}({\theta}_{1}^{[2]})$
$\displaystyle=$ $\displaystyle
U_{2}^{[1]}({\theta}_{2}^{[1]})U_{2}^{[1]}({\theta}_{1}^{[1]})$ (S65)
$\displaystyle=$ $\displaystyle
U_{2}(\bm{\theta}_{4}^{\prime})U_{2}(\bm{\theta}^{\prime}_{3})U_{2}(\bm{\theta}_{2}^{\prime})U_{2}(\bm{\theta}^{\prime}_{1}),$
(S66)

where the phase difference must satisfy

$\displaystyle{\theta}_{2}^{[1]}-{\theta}_{1}^{[1]}=\pi-8\alpha.$ (S67)

At this time, the sequence is accurate to the third order of the deviation.

In a similar concatenation way, when the number of pulses is $2^{M}$, the
total propagator reads

$\displaystyle U_{3}({\theta}_{1}^{\prime\prime})$ $\displaystyle=$
$\displaystyle U_{2}^{[M]}({\theta}_{1}^{[M]})$ (S68) $\displaystyle=$
$\displaystyle
U_{2}^{[M-1]}({\theta}_{2}^{[M-1]})U_{2}^{[M-1]}({\theta}_{1}^{[M-1]})$ (S69)
$\displaystyle=$
$\displaystyle\underbrace{U_{2}^{[M-2]}({\theta}_{4}^{[M-2]})U_{2}^{[M-2]}({\theta}_{3}^{[M-2]})}\underbrace{U_{2}^{[M-2]}({\theta}_{2}^{[M-2]})U_{2}^{[M-2]}({\theta}_{1}^{[M-2]})}$
$\displaystyle=$
$\displaystyle\underbrace{\underbrace{U_{2}^{[M\\!-\\!3]}(\\!{\theta}_{8}^{[M\\!-\\!3]}\\!)U_{2}^{[M\\!-\\!3]}(\\!{\theta}_{7}^{[M\\!-\\!3]}\\!)}\underbrace{U_{2}^{[M\\!-\\!3]}(\\!{\theta}_{6}^{[M\\!-\\!3]}\\!)U_{2}^{[M\\!-\\!3]}(\\!{\theta}_{5}^{[M\\!-\\!3]}\\!)}}\underbrace{\underbrace{U_{2}^{[M\\!-\\!3]}(\\!{\theta}_{4}^{[M\\!-\\!3]}\\!)U_{2}^{[M\\!-\\!3]}(\\!{\theta}_{3}^{[M\\!-\\!3]}\\!)}\underbrace{U_{2}^{[M\\!-\\!3]}(\\!{\theta}_{2}^{[M\\!-\\!3]}\\!)U_{2}^{[M\\!-\\!3]}(\\!{\theta}_{1}^{[M\\!-\\!3]}\\!)}}$
$\displaystyle=$ $\displaystyle\cdots$ $\displaystyle=$
$\displaystyle\underbrace{\underbrace{U_{2}(\bm{\theta}_{2^{M}}^{\prime})U_{2}(\bm{\theta}^{\prime}_{2^{M-1}})}\underbrace{U_{2}(\bm{\theta}_{2^{M-2}}^{\prime})U_{2}(\bm{\theta}^{\prime}_{2^{M-3}})}}_{\dots}\dots\underbrace{\underbrace{U_{2}(\bm{\theta}_{4}^{\prime})U_{2}(\bm{\theta}^{\prime}_{3})}\underbrace{U_{2}(\bm{\theta}_{2}^{\prime})U_{2}(\bm{\theta}^{\prime}_{1})}}_{\dots}.~{}~{}~{}~{}~{}~{}~{}$
(S72)

The phase differences must satisfy the following relations:

$\displaystyle\mathrm{The~{}first~{}concatenation}:~{}$
$\displaystyle\theta^{\prime}_{2n_{2},2n_{2}-1}=\theta_{s,2n_{2}}-\theta_{s,2n_{2}-1}=\pi-4\alpha,$
(S73a) $\displaystyle\mathrm{The~{}second~{}concatenation}:~{}$
$\displaystyle\theta^{\prime}_{4n_{2}-1,4n_{2}-3}=\theta_{s,4n_{2}-1}-\theta_{s,4n_{2}-3}=\pi-8\alpha,$
(S73b) $\displaystyle\mathrm{The~{}third~{}concatenation}:~{}$
$\displaystyle\theta^{\prime}_{8n_{2}-3,8n_{2}-7}=\theta_{s,8n_{2}-3}-\theta_{s,8n_{2}-7}=\pi-16\alpha,$
(S73c)
$\displaystyle\dots~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\mathrm{The~{}\emph{M}^{th}~{}concatenation}:~{}$
$\displaystyle\theta^{\prime}_{2^{M}n_{2}-2^{M}+1,2^{M}n_{2}-2^{M-1}+1}=\theta_{s,2^{M}n_{2}-2^{M}+1}-\theta_{s,2^{M}n_{2}-2^{M-1}+1}=\pi-2^{M+1}\alpha,~{}~{}~{}~{}~{}~{}$
(S73d)

where the sequence has a deviation compensation up to the order
$O(\delta_{\alpha}^{M+1})$. Notice that the first phase $\theta_{1}$ of the
first SP pulse pair is arbitrary, which can be modulated for other
hierarchical concatenations (e.g., the third concatenation in the main text).

Figure S1 presents the amplitude of the element $|U_{N_{2},eb}|$ against the
deviation $\delta_{\alpha}$ for different $N_{2}$, where the case from
$N_{2}=4$ to $N_{2}=8$ is derived by numerical calculations. As shown by the
curves in Fig. S1(a), the robust performance of transitionless quantum driving
becomes superior overall with the pulse number increasing. On the other hand,
we can also find that the $N_{2}=2$ sequence has a better robustness than the
$N_{2}=3$ sequence. This is because both of their Taylor series are only
ensured to be accurate to the second order of $\delta_{\alpha}$, while the
amplitude of $\delta_{\alpha}^{2}$ in the former is smaller than the latter.
Therefore, the paired pulses perform more excellent in the second
concatenation, and a similar result is also obtained by a comparison between
$N_{2}=4$ and $N_{2}=5,6$. In Fig. S1(b), we compare the robustness
performance achieved by the numerical method with the concatenated method.
Generally speaking, both methods can obtain a similar robust performance of
transitionless quantum driving. While it is more convenient and efficient to
acquire all phases by merely matching the phase differences from Eq. (S73)
instead of finding the solution via lots of tedious numerical calculations.

Figure S1: The amplitude of the element $|U_{N_{2},eb}|$ vs the deviation $\delta_{\alpha}$ by different $N_{2}$ sequences. We set $r=0.1$ and $\alpha=1.029$, and all phase differences is presented in Table SI. Table SI: Phase differences for the second concatenation of SP pulse pairs by the numerical method. $N_{2}$ |  $\theta^{\prime}_{21}$ |  $\theta^{\prime}_{32}$ |  $\theta^{\prime}_{43}$ |  $\theta^{\prime}_{54}$ |  $\theta^{\prime}_{65}$ |  $\theta^{\prime}_{76}$ |  $\theta^{\prime}_{87}$
---|---|---|---|---|---|---|---
4 |  $-$0.9750 |  2.1678 |  $-$0.9744 |  – |  – |  – |  –
5 |  4.8034 |  $-$2.2925 |  0.3437 |  $-$0.4690 |  – |  – |  –
6 |  5.0645 |  $-$4.4539 |  $-$1.3636 |  1.8294 |  $-$1.2188 |  – |  –
7 |  $-$0.7739 |  $-$3.8410 |  0.2731 |  0.2287 |  2.6760 |  $-$0.8487 |  –
8 |  $-$1.1366 |  1.8387 |  $-$2.0145 |  $-$3.3636 |  4.2688 |  $-$4.4445 |  5.1465

##  S4 SIV. High precision quantum operations

After the second concatenation of SP pulse pairs, the nonadiabatic transition
induced by various errors is dramatically suppressed, and thus the propagator
$U_{3}({\theta}^{\prime\prime}_{1})$ can be approximately written in a
diagonalized form,

$\displaystyle U_{3}({\theta}^{\prime\prime}_{1})\approx|d(t)\rangle\langle
d(t)|+\exp[{i2\beta(1+\delta_{\beta})}]|b(t)\rangle\langle
b(t)|+\exp[{-i2\beta^{\prime}(1+\delta_{\beta^{\prime}})}]|e\rangle\langle
e|.$ (S74)

Actually, the deviation $\delta_{\beta}$ in Eq. (S74) would decline the
precision of quantum operations. To show it more clearly, we transform the
propagator $U_{3}({\theta}^{\prime\prime}_{1})$ back to the original
representation spanned by $\\{|g\rangle,|f\rangle,|e\rangle\\}$. In the basis
$\\{|g\rangle,|f\rangle\\}$, the form of the propagator
$U_{3}({\theta}^{\prime\prime}_{1})$ is given by

$\displaystyle
U_{3}({\theta}^{\prime\prime}_{1})=\exp\left[{-\frac{i}{2}\beta(1+\delta_{\beta})\bm{n}\cdot\bm{\sigma}}\right],$
(S75)

where
$\bm{n}=(-\sin\phi(t)\cos\theta_{sp},\sin\phi(t)\sin\theta_{sp},\cos\phi(t))$.
It is easily observed from Eq. (S75) that the rotation angle would produce
errors in the Bloch sphere due to the presence of the deviation
$\delta_{\beta}$.

To further restrain the influence of the deviation $\delta_{\beta}$ on the
precision of quantum operations, we need to a third concatenation of SP pulse
pairs. That is, we concatenate ${N_{3}}$ propagators
$U_{3}({\theta}_{n_{3}}^{\prime\prime})$ to form a new one,

$\displaystyle
U_{4}({\theta}_{1}^{\prime\prime\prime})=U_{3}({\theta}_{N_{3}}^{\prime\prime})\cdots
U_{3}({\theta}^{\prime\prime}_{1})=\left[\begin{array}[]{ccc}1&0&0\\\\[4.30554pt]
0&{U}_{4,bb}&{U}_{4,be}\\\\[4.30554pt]
0&{U}_{4,eb}&{U}_{4,ee}\end{array}\right],$ (S79)

where ${\theta}^{\prime\prime}_{n_{3}}$ is the adjustable phases in this
situation, $n_{3}=1,\dots,N_{3}$. Taking similar steps in Sec. SIII, we
modulate the phases ${\theta}_{n_{3}}^{\prime\prime}$ to nullify the
coefficients $\partial^{k}_{\delta_{\beta}}\\!|U_{4,eb}|^{2}$ in a increasing
order as many as possible, where
$\partial^{k}_{\delta_{\beta}}\\!|U_{4,eb}|^{2}$ represents the
$k^{\mathrm{th}}$ derivative of $|U_{4,eb}|^{2}$ with respect to
$\delta_{\beta}$, $k=1,2,\dots$ It is worth mentioning that the zero-order
coefficient $\partial^{0}_{\delta_{\beta}}\\!|U_{4,eb}|^{2}$ is used to
control the population of the state $|f\rangle$.

In the following, we take the case $N_{3}=3$ as the example to demonstrate the
design of the phases ${\theta}^{\prime\prime}_{n_{3}}$. The form of the total
propagator reads

$\displaystyle
U_{4}({\theta}_{1}^{\prime\prime\prime})=U_{3}(\theta_{3}^{\prime\prime})U_{3}(\theta_{2}^{\prime\prime})U_{3}(\theta_{1}^{\prime\prime}),$
(S80)

and the equations to be solved are

$\displaystyle\partial^{0}_{\delta_{\beta}}\\!|U_{4,eb}|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[2+\cos(\theta_{21}^{\prime\prime}+\theta_{32}^{\prime\prime})-\cos(\theta_{21}^{\prime\prime}-\theta_{32}^{\prime\prime})\right]=P_{f},$
(S81a) $\displaystyle\partial^{1}_{\delta_{\beta}}\\!|U_{4,eb}|^{2}$
$\displaystyle=$
$\displaystyle-\frac{\pi}{4}\left[2\cos\theta_{32}^{\prime\prime}\cos^{2}\frac{\theta_{21}^{\prime\prime}}{2}+\cos\theta_{21}^{\prime\prime}\right]=0,$
(S81b)

where $P_{f}$ represents the population of the state $|f\rangle$. It readily
obtains the solution of Eq. (S81), which are

$\displaystyle\theta_{21}^{\prime\prime}$ $\displaystyle=$ $\displaystyle
2\arctan\pm\left(\sqrt{1-P_{f}^{2}}\pm\sqrt{2P_{f}-P_{f}^{2}}\right),$ (S82a)
$\displaystyle\theta_{32}^{\prime\prime}$ $\displaystyle=$ $\displaystyle
2\arctan\pm\left(\sqrt{1-P_{f}^{2}}\mp\sqrt{2P_{f}-P_{f}^{2}}\right).$ (S82b)

When ${N_{3}}$ is large, it is convenient to adopts numerical methods, and a
group of numerical solutions for different $N_{3}$ are presented in Table SII,
while the corresponding excitation profiles of the state $|f\rangle$ are
plotted in Fig. S2. Obviously, the excitation profile for the single
propagator $U_{3}({\theta}_{1}^{\prime\prime})$ cannot keep the target value
in the presence of the deviation $\delta_{\beta}$, while the excitation
profile for the case $N_{3}=3$ shows a primary flat platform around
$\delta_{\beta}=0$. To achieve better robustness, we can continue to
concatenate more propagators $U_{3}({\theta}_{n_{3}}^{\prime\prime})$ to
generate a longer sequence. Of course, the long sequence can also be
concatenated by short sequences. In this way, more free phases
${\theta}^{\prime\prime}_{n_{3}}$ are introduced to nullify more high-order
coefficients $\partial^{k}_{\delta_{\beta}}\\!|U_{4,eb}|^{2}$. As expected,
the flat central platform of the excitation profile around $\delta_{\beta}=0$
becomes wider with the increase of $N_{3}$; see the dotted yellow and dotted-
dashed violet curves in Fig. S2.

Figure S2: Population $P_{f}$ vs deviation $\delta{\beta}$. Here, the target population of the state $|f\rangle$ is $P_{f}$=0.5, $\phi(t)=\pi/4$, and the other parameters are given in Table SII. Table SII: Phase differences for the third concatenation of SP pulse pairs. $N_{3}$ | $\theta^{\prime\prime}_{21}$ | $\theta^{\prime\prime}_{32}$ | $\theta^{\prime\prime}_{43}$ | $\theta^{\prime\prime}_{54}$
---|---|---|---|---
3 | 2.0944 | 0 | – | –
4 | 2.3562 | 0 | 0.7854 | –
5 | 2.5133 | 0 | 1.2566 | 0

##  S5 SV. Implementation of robust transitionless quantum driving by
different pulse shapes

As we have demonstrated in the main text, the current approach is suitable for
arbitrary pulse shapes. In the main text, we employ the Gaussian shape to
implement a specific rotation operation, where the relevant parameters can be
found in Table SIII. Here, we illustrate other typical and regular pulse
shapes, such as the sinusoidal pulse, the sawtooth pulse, the triangle pulse,
and the trapezoidal pulse. In Table SIV, we give out the function expressions
for the Stokes pulse, where the parameter $\tau$ denotes the time reaching the
peak of the pulse. In the trapezoidal pulse, $\tau_{1}$ and $\tau_{2}$ are the
start and final moment of the peak value. For simplicity, we set the detuning
as constant. Certainly, it is feasible to simultaneously set the coupling
strength and detuning as time-dependent functions as well.

Table SIII: The relevant parameters for the $\mathcal{R}_{\frac{\pi}{4}}(N_{1}\times N_{2},N_{3})$ sequence with the Gaussian shape (in units of $1/\Omega_{0}$), where $\theta_{p,1}=\theta_{s,1}$. $\mathcal{R}_{\frac{\pi}{4}}(N_{1}\times N_{2},N_{3})$ | $A$ | $\Delta$ | $\alpha$ | $\theta_{s,1}$
---|---|---|---|---
$\mathcal{R}_{\frac{\pi}{4}}(2,1)$ | 1.099 | 0.6574 | 1.1868 | 1.5708
$\mathcal{R}_{\frac{\pi}{4}}(4,3)$ | 2.381 | 0.2802 | 0.4479 | 0.5237
$\mathcal{R}_{\frac{\pi}{4}}(4,5)$ | 2.381 | 0.2802 | 0.4479 | $-$0.9425
$\mathcal{R}_{\frac{\pi}{4}}(8,1)$ | 1.1299 | 0.1640 | 0.2957 | 1.5708
$\mathcal{R}_{\frac{\pi}{4}}(8,5)$ | 1.1299 | 0.1640 | 0.2957 | $-$0.9425
Table SIV: The expressions of some common waveforms. Waveforms |  Expressions
---|---
Gaussian |  $\Omega_{s}(t)=A\Omega_{0}e^{-(t-\tau)^{2}/T^{2}}$
Sinusoidal |  $\Omega_{s}(t)=A\Omega_{0}|\sin(\pi t/T+\tau)|$
Sawtooth |  $\Omega_{s}(t)\\!=\\!\displaystyle\frac{A\Omega_{0}}{T}t,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}0\leq t\leq T$
Triangle |  $\Omega_{s}(t)\\!=\\!\left\\{\begin{aligned} &\displaystyle\frac{A\Omega_{0}}{\tau}t,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}0\leq t\leq\tau\\\\[4.30554pt] &\displaystyle\frac{A\Omega_{0}}{\tau-T}(t-T),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\tau<t\leq T\\\\[4.30554pt] \end{aligned}\right.$
Trapezoidal |  $\Omega_{s}(t)\\!=\\!\left\\{\begin{aligned} &\displaystyle\frac{A\Omega_{0}}{\tau_{1}}t,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}0\leq t\leq\tau_{1}\\\\[4.30554pt] &A\Omega_{0},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\tau_{1}<t\leq\tau_{2}\\\\[4.30554pt] &\displaystyle\frac{A\Omega_{0}}{\tau_{1}+\tau_{2}-T}(t-T),~{}~{}\tau_{2}<t\leq T\\\\[4.30554pt] \end{aligned}\right.$

We first simulate perfect transitionless quantum driving by different pulse
shapes without systematic errors. Specifically, we plot in Fig. S3 the
population evolution of three states, and the corresponding waveforms of the
Stokes pulse, where the target state is
$|\Psi_{T}\rangle=1/\sqrt{2}(|g\rangle+|f\rangle)$. At the evolution time
$t=2T$, regardless of the waveform of the SP pulse pair, the maximum
superposition of the state $|g\rangle$ and $|f\rangle$ can be successfully
achieved. This result gives the evidence that the leakage to the state
$|e\rangle$ has been completely forbidden by matching the phase difference
$\theta_{21}=\pi-2\alpha$.

Figure S3:  Population evolution (top panels) of three states and corresponding waveform (bottom panels) of the Stokes pulse by two SP pulse pairs. The target state is $|\Psi_{T}\rangle=1/\sqrt{2}(|g\rangle+|f\rangle)$, and the relevant parameters of different shapes are given in Table SV. Definitely, all shapes can be used for perfect transitionless quantum driving.  Table SV: The parameter for different pulse shapes (in units of $1/\Omega_{0}$). Waveforms |  $A$ |  $\Delta$ |  $T$ |  $\tau$ |  $\tau_{2}$
---|---|---|---|---|---
Sinusoidal |  3.044 |  1.98 |  1 |  0 |  –
Sawtooth |  3.935 |  1.865 |  1 |  – |  –
Triangle |  3.884 |  2.18 |  1 |  0.5 |  –
Trapezoidal |  3.249 |  2.04 |  1 |  0.2 |  0.2

When the system exhibits the errors, we take the
$\mathcal{R}_{\frac{\pi}{4}}(8,5)$ sequence to demonstrate the feasibility of
the robustness with respect to the errors by different shapes in Fig. S4,
where $\mathcal{R}_{\frac{\pi}{4}}(8,5)$ represents the rotation around the
axis $\bm{n}$ with the angle ${{\pi}/{4}}$. For different pulse shapes, it is
available to achieve the high-fidelity quantum operation with favorable
robustness against systematic errors, as shown by the red regions on the top
panel in Fig. S4.

Figure S4:  Fidelity $F$ of the target state $|\Psi_{T}\rangle=1/\sqrt{2}(|g\rangle+|f\rangle)$ (top panels) and population $P_{e}$ of the state $|e\rangle$ (bottom panels) vs pulse amplitude error $\delta_{\Omega_{s}}$ and detuning error $\delta_{\Delta}$ by the $\mathcal{R}_{\frac{\pi}{4}}(8,5)$ sequence with (a–b) the sinusoidal, (c–d) the sawtooth, (e–f) the triangle, and (g–h) the trapezoidal pulses. The parameters of different shapes are given in Table SVI. Table SVI: The parameters for different pulse shapes (in units of $1/\Omega_{0}$). Waveforms |  $A$ |  $\Delta$ |  $\tau$ |  $\tau_{2}$ |  $\alpha$ |  $\theta_{p,1}$
---|---|---|---|---|---|---
Sinusoidal |  3.1445 |  0.496 |  3 |  – |  0.0518 |  $-$0.9425
Sawtooth |  4.0070 |  0.46 |  3 |  – |  0.0337 |  $-$0.9416
Triangle |  4.0040 |  0.546 |  0.5 |  – |  0.0766 |  $-$0.9416
Trapezoidal |  4.0535 |  0.6615 |  0.2 |  0.2 |  0.1343 |  $-$0.9416