# Single-atom verification of the noise-resilient and fast characteristics of
universal nonadiabatic noncyclic geometric quantum gates

J. W. Zhang1,3 L.-L. Yan2 J. C. Li1,3 G. Y. Ding1,3 J. T. Bu1,3 L. Chen1 S.-L.
Su2<EMAIL_ADDRESS>F. Zhou1<EMAIL_ADDRESS>M. Feng1,2,3,4
<EMAIL_ADDRESS>1 State Key Laboratory of Magnetic Resonance and Atomic
and Molecular Physics, Wuhan Institute of Physics and Mathematics, Innovation
Academy of Precision Measurement Science and Technology, Chinese Academy of
Sciences, Wuhan, 430071, China
2 School of Physics, Zhengzhou University, Zhengzhou 450001, China
3 School of Physics, University of the Chinese Academy of Sciences, Beijing
100049, China
4 Research Center for Quantum Precision Measurement, Institute of Industry
Technology, Guangzhou and Chinese Academy of Sciences, Guangzhou, 511458,
China

###### Abstract

Quantum gates induced by geometric phases are intrinsically robust against
noise due to their global properties of the evolution paths. Compared to
conventional nonadiabatic geometric quantum computation (NGQC), the recently
proposed nonadiabatic noncyclic geometric quantum computation (NNGQC) works in
a faster fashion, while still remaining the robust feature of the geometric
operations. Here, we experimentally implement the NNGQC in a single trapped
ultracold 40Ca+ ion for verifying the noise-resilient and fast feature. By
performing unitary operations under imperfect conditions, we witness the
advantages of the NNGQC with measured fidelities by quantum process tomography
in comparison with other two quantum gates by conventional NGQC and by
straightforwardly dynamical evolution. Our results provide the first evidence
confirming the possibility of accelerated quantum information processing with
limited systematic errors even in the imperfect situation.

Geometric quantum operations gqc1 , no matter adiabatic or nonadiabatic, are
intrinsically noise-resilient due to their global properties of the evolution
paths, which provide a promising paradigm for robust quantum information
processing gqc3 ; gqc4 ; gqc5 ; gqc6 ; gqc7 ; gqc8 ; gqc9 ; gqc10 ; gqc11 ;
gqc12 ; gqc13 ; gqc14 ; gqc15 ; gqc16 ; gqc17 . In particular, the
nonadiabatic geometric quantum computation (NGQC) ngqc1 ; ngqc2 ; ngqc3 ;
ngqc4 ; ngqc5 ; ngqc6 ; ngqc7 ; ngqc8 ; ngqc9 ; ngqc10 ; ngqc11 ; ngqc12 ;
ngqc13 ; ngqc14 ; ngqc15 ; ngqc16 ; ngqc17 ; ngqc18 , with less gating time
than the adiabatic counterpart, could effectively protect against environment-
induced decoherence. Various systems have already demonstrated such advantages
with currently available techniques Engqc1 ; Engqc2 ; Engqc3 ; Engqc4 ; Engqc5
; Engqc6 ; Engqc7 ; Engqc8 ; Engqc9 ; Engqc10 ; Engqc11 ; Engqc12 ; Engqc13 .

Figure 1: (Color online) Single-qubit NNGQC. (a) Level scheme of the trapped
40Ca+ ion, where the qubit is encoded in the ground state
$|4^{2}S_{1/2},m_{J}=+1/2\rangle$ and the metastable state
$|3^{2}D_{5/2},m_{J}=+1/2\rangle$. $|4^{2}P_{1/2}\rangle$ is the excited state
employed for qubit readout. (b) Conceptual sketch of the evolution
trajectories on the Bloch sphere for realizing single-qubit NNGQC gates, where
the noncyclic geometric phase is relevant to the solid angle enclosed by the
blue solid trajectory and the purple dashed geodesic between A (A’) and B
(B’), connecting the initial and final points. The realistic trajectories are
determined by the gate-relevant parameters SM  and the geodesic connection is
subject to the geodesic rule proposition new2 . The evolution trajectories of
$|\Psi_{1}\rangle$ and $|\Psi_{2}\rangle$ are orthogonal and simultaneously
reverse. (c1) Pulse sequences to realize the $U_{1}$ gate defined as
$U_{1}\equiv[1+i\ -1-i;1-i\
1-i]/2=(I+i\sigma_{z})/2-i(\sigma_{x}+\sigma_{y})/2$. (c2-c3) Experimental
results of $U_{1}$ gate reaching the fidelity $0.9796(18)$ in the evolution
started from the initial state $|\psi\rangle=(|g\rangle-i|e\rangle)/\sqrt{2}$,
where $P_{e}$ is the population in state $|e\rangle$ and $P_{ge}$ is the off-
diagonal term in the evolution matrix. (d) Observed fidelity of $U_{1}$ gate
for different input states
$|\psi(\theta)\rangle=\cos(\theta/2)|g\rangle-i\sin(\theta/2)|e\rangle$ with
an average fidelity $0.9784(23)$. The error bars indicating the statistical
standard deviation of the experimental data are obtained by 20000 measurements
for each data point.

The NGQC can be further accelerated if the time-optimal technology is involved
to minimize gating time under the framework of cyclic evolution OPgqc2 ;
OPgqc3 ; OPgqc4 . However, no matter with or without the time-optimal
technology, the NGQC should satisfy the cyclic condition, which are more
sensitive to decay and dephasing errors exc1 ; exc2 ; exc3 ; exc4  compared to
the conventional dynamical quantum computation (DQC). In addition, the NGQC
takes exactly the same amount of gating time regardless of the large or small
geometric rotation angle involved, which is subject to the cyclic condition,
but impossibly changed by optimizing other parameters. Actually, the cyclic
requirement is not always necessary in accomplishment of the NGQC, as
indicated in a recent proposal PRR-2-043130 , in which the proposed
nonadiabatic noncyclic geometric quantum computation (NNGQC) could reduce the
gating time, while retaining the robust property of the geometric gates. In
particular, compared to previously published noncyclic geometric models, e.g.,
new0 ; noncyclic1 ; noncyclic2 , the proposal using simpler pulse sequences
could be executed with a faster pace and more relevance to current laboratory
technique. Despite no experimental justification so far, the proposal presents
us with hopes that practical quantum computation would be available in a way
of high tolerance to errors and meanwhile high speed of gating.

Here we demonstrate a single-spin verification of this NNGQC implementation
via experimental manipulation of a trapped-ion system. Due to high-precision
control, our execution on a qubit (i.e., a single spin) encoded in a single
ultracold 40Ca+ ion provides an elaborate comparison of the NNGQC with others
conventionally employed, such as the NGQC and DQC. As sketched in Fig. 1(a),
our qubit is encoded in the ground state $|4^{2}S_{1/2},m_{J}=+1/2\rangle$
(labeled as $|g\rangle$) and the metastable state
$|3^{2}D_{5/2},m_{J}=+1/2\rangle$ (labeled as $|e\rangle$), where mJ is the
magnetic quantum number. Although our investigation below only focuses on this
qubit, to avoid thermal phonons yielding offsets of Rabi oscillation, we need
to cool the ion to be within the Lamb-Dicke regime ion-1 ; ion-2 ; ion-3 .

Before specifying the experimental details, we first elucidate briefly the
theory of constructing the single-qubit geometric gate using the basis states
$\\{|g\rangle,|e\rangle\\}$. For our purpose, the light-matter interaction is
given by the Hamiltonian in units of $\hbar=1$ PRR-2-043130 ,

$H=\frac{1}{2}\Omega(t)e^{i\varphi(t)}|e\rangle\langle g|+\rm H.C.,$ (1)

which can be achieved in a single trapped ion by laser irradiation under
carrier transitions. $\Omega(t)$ and $\varphi(t)$ are time dependent,
representing the driving amplitude (i.e., Rabi frequency) and the phase of the
laser, respectively. To understand the physics more clearly, we may map the
system’s states to two auxiliary states $|\Psi_{1,2}\rangle$. In terms of the
proposal PRR-2-043130 , realizing a NNGQC requires the amplitude and phase of
the laser in time evolution to satisfy a couple of differential equations as
below,

$\Omega(t)=-\frac{\dot{\xi}}{\sin(\varphi+\eta)},\quad\varphi(t)=-\eta+\arctan\frac{\dot{\xi}}{\dot{\eta}\tan\xi},$
(2)

where $\xi$ and $\eta$ are time-dependent variables defined by
$|\Psi_{1,2}\rangle$ spanning the space for constructing the evolution
operator regarding the NNGQC SM . To make the gates purely geometric, however,
we have to erase the diagonal dynamical phase produced in the state evolution,
implying $\varphi+\eta=-\pi/2$. In this case, the diagonal geometric phase is
relevant to the solid angle enclosed by the evolution trajectory of
$|\Psi_{1}\rangle$ or $|\Psi_{2}\rangle$ and the dashed geodesic connecting
the initial and final points, see Fig. 1(b). Compared to the NGQC that
requires the evolution trajectories going through the two polars of the Bloch
sphere, requirements for the NNGQC are much relaxed noncyclic1 ; new2 . As a
result, the NNGQC could take less time than that of the NGQC for the gate
implementation. For example, in U1 gate implementation, the NNGQC takes half
of time compared to the NGQC, and less than half of time than DQC SM .

For a universal set of single-qubit NNGQC gates, we may consider the following
parameter set

$\xi=\Omega_{0}t-\xi_{0},\quad\eta=\phi_{1}\varepsilon(t)+\phi_{0},$ (3)

where $\Omega_{0},\phi_{0},\phi_{1}$ and $\xi_{0}$ are constants dependent on
the designed phase gate, and $\varepsilon(t)$ is the step function dividing
the evolution with duration $\tau$ into two parts: $\varepsilon(t)=0$ for
$t\in[0,\xi_{0}/\Omega_{0}]$ and $\varepsilon(t)=1$ in the case of
$t\in[\xi_{0}/\Omega_{0},\tau]$. These constants constitute four parameters
$\eta_{\pm}$ and $\xi_{\pm}$, with $\eta_{+}=\phi_{1}+2\phi_{0}$,
$\eta_{-}=\phi_{1}$, $\xi_{+}=\Omega_{0}\tau$, and
$\xi_{-}=\Omega_{0}\tau-2\xi_{0}$, which are relevant to the rotating angles
$\theta$, $\alpha$ and $\beta$ regarding the NNGQC, as indicated below. The
total evolution operator is given by

$U(\theta,\alpha,\beta)=Z_{\beta}X_{\theta}Z_{\alpha},$ (4)

where the rotation operators regarding $z$ and $x$ axes of the Bloch sphere
are defined as $Z_{\beta}\equiv e^{-i\beta\sigma_{z}/2}$ and $X_{\theta}\equiv
e^{-i\theta\sigma_{x}/2}$. The angles are given by
$\theta=2\sin^{-1}\sqrt{\sin^{2}\gamma\sin^{2}(\xi_{+}/2)+\cos^{2}\gamma\sin^{2}(\xi_{-}/2)}$,
$\alpha=-\tan^{-1}\frac{\tan\gamma\cos(\xi_{+}/2)}{\cos(\xi_{-}/2)}-\tan^{-1}\frac{\tan\gamma\sin(\xi_{+}/2)}{\sin(\xi_{-}/2)}+\frac{\eta_{-}-\eta_{+}-\pi}{2}$,
and
$\beta=-\tan^{-1}\frac{\tan\gamma\cos(\xi_{+}/2)}{\cos(\xi_{-}/2)}+\tan^{-1}\frac{\tan\gamma\sin(\xi_{+}/2)}{\sin(\xi_{-}/2)}+\frac{\eta_{-}+\eta_{+}+\pi}{2}$
SM , where
$\gamma=\int_{0}^{\tau}i\langle\Psi_{1}(t)|(d/dt)|\Psi_{1}(t)\rangle=\phi_{1}/2$
is relevant to the NNGQC geometric phase. To execute $U_{1}$ (Hadamard) gate,
we choose $\phi_{0}=-\pi/2~{}(0)$, $\phi_{1}=\pi/2~{}(\pi/2)$,
$\xi_{0}=\pi/2~{}(\pi/2)$ and $\tau=\pi/\Omega_{0}~{}(3\pi/2\Omega_{0})$, as
proposed in PRR-2-043130 . These are exactly the gates we will execute
experimentally below.

Figure 2: (Color online) Experimental implementation of a single-qubit NNGQC
for (a1) $U_{1}$ gate as defined in Fig. 1 and (a2) Hadamard gate $H$, where
the input states are eigenvectors of the operator set
$\\{I,\sigma_{x},-i\sigma_{y},\sigma_{z}\\}$ regarding the basis states
$\\{|g\rangle,|e\rangle\\}$. (b-d) The experimental (black) and numerical
(red) bar charts for the real and imaginary parts of the process matrix
$\chi_{\rm exp,num}$, respectively, where the grey parts above the
experimental (black) bars denote the errors of the experimental data acquired
by 20000 measurements. The numbers labeled in the axes represent four input
states, i.e., $|g\rangle$, $|e\rangle$,
$|+\rangle=(|g\rangle+|e\rangle)/\sqrt{2}$, and
$|-\rangle=(|g\rangle-i|e\rangle)/\sqrt{2}$. (b,c) $U_{1}$ gate with the
process fidelity of $0.9658(34)$; (d,e) Hadamard gate with the process
fidelity of $0.9645(27)$.

Figure 3: (Color online) Comparison of NNGQC with NGQC and DQC. (a-c)
Schematic for processing a $U_{1}$ gate using the three gate schemes with
$\Omega_{m}$ the constant Rabi frequency as explained in SM . (d,e) Histogram
charts for the real and imaginary parts of $\chi$ under different quantum gate
processes, where focusing on the data around $0$ and $0.25$ is to display the
difference among different gate schemes and distinguish small errors of the
experimental data. To distinguish the slight differences, we increase the
measurement times to 50000 for each data. For a full comparison, we also plot
the ideal $U_{1}$ gate here, labelled as IQC.

In our experiment, the 40Ca+ ion is confined in a linear Paul trap with axial
frequency $\omega_{z}/2\pi=1.1$ MHz and radial frequency $\omega_{r}/2\pi=1.6$
MHz. We define a quantization axis along the axial direction by a magnetic
field of approximately 3.4 Gauss at the center of the trap, and manipulate the
qubit by a ultra-stable narrow linewidth 729-nm laser at an angle of 60
degrees to the quantization axis with the Lamb-Dicke parameter $\sim$0.1152.
The 729-nm laser is controlled by a double pass acousto-optic modulator. We
have the field programable gate array to control a direct digital synthesizer
for the frequency sources of the acousto-optic modulator, which provides the
phase and frequency control of the 729-nm laser during the experimental
operations. Before starting our experiment, we have cooled the ion, from the
three dimensions, down to near the ground state of the vibrational modes,
yielding the final average phonon number $\bar{n}<$1\. In addition to other
noises caused by the magnetic and electric field fluctuation, the qubit
suffers from the dephasing of $\Gamma=0.81(11)$ kHz, as measured by Ramsey
sequences. However, since the average Rabi frequency regarding the 729-nm
laser irradiation is $\Omega_{0}/2\pi=67.9(2)$ kHz and the decay rate ($<$ 1
Hz) is negligible, we estimate that the dephasing-induced infidelity is about
0.1$\%$ SM , which should be carefully treated in order to distinguish the
measured fidelities of the three different gate schemes as elucidated below.

To deliberate on the noise-resilient characteristic of the NNGQC, we also
introduce in our operations some other imperfection, respectively, regarding
the initial state preparation, the Rabi frequency and the resonance frequency,
which are the primary error sources in most trapped-ion experiments RMP-75-281
. We first carry out the $U_{1}$ gate by the NNGQC scheme in the presence of
imperfect initial state preparation, where the fidelity is defined by
$\mathcal{F}=\text{Tr}[\rho_{\rm exp}\rho_{\rm id}]$ with ’exp’ and ’id’
denoting the experimental and ideal states. The time sequences of the Rabi
frequency and the laser phase for achieving the NNGQC are designed in Fig.
1(c1), whose discontinuous features are also reflected in Fig. 1(c2,c3) as the
cusps found in the observed fidelities and the off-diagonal terms $P_{ge}$
regrading the output matrix of the state. Despite the small infidelity as
observed in Fig. 1(d), the experimental results coincide well with the
numerical results, whose average fidelity is $\mathcal{F}=0.9789(4)$.

To further characterize the performance of the NNGQC in this case, we employ
quantum process tomography (QPT) jmo-44-2455 ; Nie-Chu  for a full measurement
of the experimental process matrices $\chi_{\rm exp}$. With the specific input
state set $|\psi_{k}\rangle\in\\{|g\rangle,\ |e\rangle$,
$|+\rangle=(|g\rangle+|e\rangle)/\sqrt{2},\
|-\rangle=(|g\rangle-i|e\rangle)/\sqrt{2}\\}$, we have carried out the $U_{1}$
and Hadamard (i.e., $H$) gates under the pulse sequences as plotted in Fig.
2(a1, a2). We evaluate their process fidelities by
$\mathcal{F}=\text{Tr}[\chi_{\rm exp}\chi_{\rm id}]$, which are $0.9658(34)$
for $U_{1}$ gate as in Fig. 2(b,c) and $0.9645(27)$ for $H$ gate, see Fig.
2(d,e). This observation is consistent with the corresponding values of our
numerically predicted fidelities, i.e., $0.9681$ and $0.9667$.

As predicted in PRR-2-043130 , the NNGQC works faster and also owns a higher
fidelity than the NGQC and DQC in accomplishing the same quantum gating tasks.
The higher gating speed is easily understandable from Fig. 3(a,b,c) that the
NNGQC only takes half of the gating time compared to the NGQC, and takes much
less gating time than the DQC SM . However, distinguishing the fidelity
difference among the three gate schemes, which is theoretically predicted to
be less than $0.3\%$, is experimentally challenging SM . To display the
fidelity superiority of the NNGQC, we have increased the measurement times to
50000 for each data point, yielding better fidelities of the gate performance,
i.e., $0.9700(20)$, $0.9660(18)$ and $0.9642(17)$ for NNGQC, NGQC and DQC,
respectively. Correspondingly, we have the numerically calculated fidelities
to be $0.9681$, $0.9653$ and $0.9639$. These values show the similar trend
that both the geometric gates work with evidently higher fidelities than the
DQC, where the NNGQC also has a better performance than the NGQC even in the
presence of imperfection. After calibrating the data over the imperfection in
the initial state preparation, we have updated the process matrix elements as
shown in Fig. 3(d,e). The corresponding fidelities of $0.9978(16)$,
$0.9936(22)$ and $0.9918(16)$ are still consistent with the trend of the
calibrated numerical results $0.9959$, $0.9929$ and $0.9915$. This implies
that the NNGQC works better in both cases with and without the perfect
preparation of the initial state. It also indicates that our calibration is
beneficial to improve the fidelities of the gates, but without essentially
changing the main features of the gates in the comparison.

Figure 4: (Color online) Comparison of the robustness against systematic
errors based on the experimental fidelities of the output states after
undertaking the $U_{1}$ gates carried out, respectively, by the NNGQC, NGQC
and DQC schemes for the input state $|g\rangle$. (a) For the Rabi frequency
error $\delta\Omega$ of the laser and (b) for the resonance frequency error
$\Delta$ of the qubit. The experimental results are consistent with the
numerical simulation results. The error bars indicating the statistical
standard deviation of experimental data are obtained by 20000 measurements for
each data point.

In Fig. 4, we justify the advantages of the NNGQC experiencing two other
errors. To deliberate on this robustness, we have explored a wide range of the
deviation from the correct values. We can find from the measurements that the
exact resonance frequency is more important than the exact Rabi frequency
since a small deviation from the required resonance frequency affects the
fidelity more seriously than the counterpart of the Rabi frequency.
Nevertheless, the NNGQC owns a significant superiority over the other two
gating schemes in resisting both of the errors. For example, for the frequency
deviation of $|\Delta|/2\pi\leq 10$ kHz, which is actually a relatively large
error compared to the Rabi frequency $\Omega_{0}/2\pi=67.9$ kHz, the NNGQC
could still reach the fidelity larger than 97$\%$, much more robust than the
NGQC and the DQC. In fact, even for a large error, e.g., $\pm 20\%$ deviation
of $\Omega$ or $\pm$20 kHz of detuning, the NNGQC could still perform better
than the other two gates in the same situation.

For a more quantitative comparison, we assume the fluctuation errors to obey
the Gaussian distribution, i.e.,
$p(\epsilon)=\frac{1}{\sqrt{2\pi}\sigma}\exp(-\frac{\epsilon^{2}}{2\sigma^{2}})$
with $\sigma$ representing the standard deviation of the exemplified errors.
We evaluate these Gaussian standard deviations in Table 1, which fit well the
trend of the numerically calculated values and also indicate the prominent
advantage of the NNGQC in the error resistance regarding the imperfect Rabi
frequency and the inaccurate resonance frequency.

Table 1: Gaussian standard deviation (GSD) comparison of the NNGQC with the NGQC and DQC regarding the systematic errors from Rabi frequency and resonance frequency, where the GSD is estimated from the Gaussian distribution. Parameters | GSD | NNGQC | NGQC | DQC
---|---|---|---|---
Exp | Num | Exp | Num | Exp | Num
$\sigma_{\Omega}/\Omega_{0}$ | 0.1 | 0.66% | 0.65% | 0.75% | 0.77% | 1.22% | 1.15%
0.2 | 1.59% | 1.36% | 1.74% | 1.55% | 2.27% | 2.04%
$\sigma_{\Delta}$ (kHz) | 5 | 0.68% | 0.67% | 1.22% | 1.28% | 0.94% | 1.02%
10 | 2.30% | 2.38% | 4.13% | 4.43% | 3.13% | 3.50%

In summary, our experiment has provided the first single-spin evidence
confirming the fast and noise-resilient feature of the NNGQC in qubit
manipulation. In particular, our high-precision control at the fundamental
level of a single spin has displayed clearly the higher fidelity of the gate
performance accomplished by the NNGQC than by others under some imperfect
conditions. In contrast to the theoretical prediction that the NNGQC is
superior to NGQC under the same environmental decoherence due to shorter
gating time, our experimental observation has witnessed the advantages of the
NNGQC regarding the factors more than decoherence influence, which would
stimulate further study both theoretically and experimentally. Despite the
very small differences presented here in the comparison, the higher fidelity,
the higher tolerance to imperfection and the higher speed of the NNGQC gates
would be of a great impact in multi-sequences of gate operations in a
practical quantum information processing.

Considering the two-qubit controlled gates, which are the essential element in
constituting universal quantum computation, we have found that the two-qubit
controlled phase gates can be executed from an effective Hamiltonian very
similar to Eq. (1) SM . Thus, two-qubit gates of the NNGQC can be in principle
carried out with the similar operations to the single-qubit gate as
demonstrated above. This implies that our experimental verification here has
actually witnessed the advantages of both the single-qubit and two-qubit
operations required by the NNGQC explain . In particular, for the two-qubit
controlled gates, no matter whether they are achieved by direct coupling of
the two levels or by a Raman process, the characteristics of the NNGQC would
help for gating operations with higher-speed, higher-fidelity and lower error
fashion, under the same conditions or parameters, in comparison with the same
gating performance by other ways.

This work was supported by Key Research $\&$ Development Project of Guangdong
Province under Grant No. 2020B0303300001, by National Key Research $\&$
Development Program of China under grant No. 2017YFA0304503, by National
Natural Science Foundation of China under Grant Nos. 12074346, 12074390,
11835011, 11804375, 11804308, 91421111, 11734018, by Natural Science
Foundation of Henan Province under Grant No 202300410481 and by K. C. Wong
Education Foundation (GJTD-2019-15).

J. W. Z, L. L. Y. and J. C. Li contributed equally to this work.

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  * (68) The two-qubit NNQGC gate designed in SM  is built on the qubit encoded in the Zeeman sublevels of the S1/2, while our single-qubit NNQGC gate discussed above is executed between S1/2 and $D_{5/2}$. However, the single-qubit NNQGC gate can also be performed between the Zeeman sublevels of the S1/2, whose operations are mathematically equivalent to what were described in the present work. Therefore, we consider that our experimental witness, as a proof-of-principle work, is a full verification of the advantages of both the single-qubit and two-qubit NNQGC gates.

Supplementary material for ”Single-atom verification of the noise-resilient
and fast characteristics of universal nonadiabatic noncyclic geometric quantum
gates”

##  I Effect of dephasing on the fidelity of the gate

Time evolution of the system’s state under the dephasing effect can be
described by Master equation as below,

$\dot{\rho}=-i[H,\rho]+\frac{\Gamma}{2}(\sigma_{z}\rho\sigma_{z}-\rho),$ (S1)

where $\Gamma$ denotes the dephasing rate of the system. We define the
infidelity caused by the dephasing as

$\varepsilon=1-\mathcal{F},$ (S2)

where the fidelity of the quantum gate is defined as
$\mathcal{F}=\text{Tr}[\rho\rho_{\rm ideal}]$ with $\rho$ and $\rho_{\rm
ideal}$ corresponding to the real and ideal final states after accomplishing
the quantum gate. Considering the weak dephasing in our experiment, we only
plot the cases of $\Gamma\ll\Omega_{0}$ in Fig. S1, where to obtain an
infidelity smaller than $10^{-2}$, we require $\Gamma/\Omega_{0}<8.6\times
10^{-3}$. In our case, we have experimentally measured the Rabi frequency of
$\Omega/2\pi=67.9(2)$ kHz and the dephasing rate of $\Gamma=0.81(11)$ kHz,
implying that the infidelity is only 0.1$\%$.

Figure S1: Infidelity as a function of dephasing, where the parameters are
chosen as $\xi_{0}=\pi/2,\phi_{0}=-\pi/2,\phi_{1}=\pi/2$, and
$\tau=\pi/\Omega_{0}$ with the Rabi frequency $\Omega_{0}/2\pi=0.1$ MHz to
construct a $U_{1}$ geometric gate. The fitting curve is plotted by the
function of $\varepsilon=a(e^{-b\Gamma/\Omega_{0}}-1)$ with the fitting
parameters of $a=-0.4813$ and $b=2.447$, respectively.

##  II Errors caused by quantum projection measurement

Assuming the success probability of the measurement to be $p$, we have the
error caused by quantum projection measurement given by
$\epsilon=\sqrt{p(1-p)/N}$ with $N$ the times of the projection measurement.
Since the phase gate of a single qubit needs three different measurements of
the elements of the density matrix $\rho$, the gate error $\epsilon_{g}$ is
$\sqrt{3}$ times larger than a single projection measurement error. However,
the projection measurement errors of the gate are intrinsic and random, which
are impossibly calibrated. Nevertheless, due to the fact of
$\epsilon_{g}\propto 1/\sqrt{N}$, increasing the measurements is an effective
way to suppress this error.

In order to distinguish the small difference among the measured fidelities of
the three different gates as mentioned in the main text, we need to suppress
the projection measurement errors. Due to the difference being smaller than
0.5$\%$, we have made 50000 measurements for each data point in Fig. 3 of the
main text.

##  III Errors caused by imperfect preparation of the initial state

In our experiment, the qubit is initialized by the 397 nm laser, which
prepares the initial state in the ground state $S_{1/2}$ with the population
$p=0.9628(1)$ with imperfection $dp=1-p=0.0372(1)$. Thus, when we exclude this
imperfection, we calibrate the experimental data by the renormalized factor
$p$.

##  IV Some details for the gates by nonadiabatic noncyclic geometric quantum
computation

The rotation angles regarding the NNGQC, as mentioned in the main text, are of
the general forms as follows,

$\displaystyle\theta$ $\displaystyle=$ $\displaystyle
2\sin^{-1}(\sqrt{\Lambda_{2\gamma,\xi_{+}}^{2}+\nu_{\xi_{-},2\gamma}^{2}}),$
$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle-\tan^{-1}\frac{\nu_{2\gamma,\xi_{+}}}{\mu_{2\gamma,\xi_{-}}}-\tan^{-1}\frac{\Lambda_{2\gamma,\xi_{+}}}{\nu_{\xi_{-},2\gamma}}+\frac{\eta_{-}-\eta_{+}-\pi}{2},$
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle-\tan^{-1}\frac{\nu_{2\gamma,\xi_{+}}}{\mu_{2\gamma,\xi_{-}}}+\tan^{-1}\frac{\Lambda_{2\gamma,\xi_{+}}}{\nu_{\xi_{-},2\gamma}}+\frac{\eta_{-}+\eta_{+}+\pi}{2},$

where $\gamma$ and $\xi_{\pm}$ are angles as specified below, and other
corresponding parameters are defined as
$\mu_{k,j}\equiv\cos\frac{k}{2}\cos\frac{j}{2}$,
$\nu_{k,j}\equiv\sin\frac{k}{2}\cos\frac{j}{2}$,
$\Lambda_{k,j}\equiv\sin\frac{k}{2}\sin\frac{j}{2}$,
$\xi_{\pm}=\xi(\tau)\pm\xi(0)$, and $\eta_{\pm}=\eta(\tau)\pm\eta(0)$. To see
the realistic trajectory and geodesic connection, we have to introduce two
auxiliary basis states
$|\Psi_{1}\rangle=(\cos\frac{\xi}{2}e^{-i\frac{\eta}{2}},\sin\frac{\xi}{2}e^{i\frac{\eta}{2}})^{\rm
T}$ and
$|\Psi_{2}\rangle=(\sin\frac{\xi}{2}e^{-i\frac{\eta}{2}},-\cos\frac{\xi}{2}e^{i\frac{\eta}{2}})^{\rm
T}$ following the NNGQC theory, where $(\xi,~{}\eta)$ determine the vector
point on the Bloch sphere. The geometric phase is relevant to the solid angle
enclosed by the trajectory and the geodesic. As an example, the gate
$U_{1}=U_{c}(\pi/2,-\pi/2,0)$ is obtained by setting
$\phi_{0}=-\pi/2,\phi_{1}=\pi/2$, $\xi_{0}=\pi/2$ and $\tau=\pi/\Omega_{0}$,
with other parameters $\eta_{+}=-\pi/2$, $\eta_{-}=\pi/2$, $\xi_{+}=0$,
$\xi_{-}=\pi$ and the calculated
$\gamma=\int_{0}^{\tau}i\langle\Psi_{1}(t)|(d/dt)|\Psi_{1}(t)\rangle=\pi/4$.
In contrast, the Hadamard gate is achieved by setting $\phi_{0}=0$,
$\phi_{1}=\pi/2$, $\xi_{0}=\pi/2$ and $\tau=3\pi/2\Omega_{0}$, with other
parameters $\eta_{+}=\pi/2$, $\eta_{-}=\pi/2$, $\xi_{+}=\pi/2$,
$\xi_{-}=3\pi/2$ and the calculated $\gamma=\pi/4$. With the given parameters,
the trajectory and the geodesic connection for $U_{1}$ gate and Hadamard gate
are plotted in Fig. S2.

Figure S2: Illustrations of the trajectory and geodesic connection in the
Bloch sphere for the NNGQC $U_{1}$ and Hadamard gates. The evolution operator
for the NNGQC is
$U(\tau,0)=e^{i\gamma}|\Psi_{1}(\tau)\rangle\langle\Psi_{1}(0)|+e^{-i\gamma}|\Psi_{2}(\tau)\rangle\langle\Psi_{2}(0)|$.
With the parameters from the same group, $|\Psi_{1}\rangle$ and
$|\Psi_{2}\rangle$ evolve simultaneously along the trajectories plotted here.
In the case of the dynamical phase being zero, the geometric phase is
generally given by $\gamma+\arg\langle\Psi_{1}(0)|\Psi_{1}(\tau)\rangle$ new ;
ae2003 , where for the cyclic case (i.e.,
$|\Psi_{1}(\tau)\rangle=|\Psi_{1}(0)\rangle$) the geometric phase is simply
$\gamma$. The beginning and ending points of the trajectory for
$|\Psi_{1}\rangle$($|\Psi_{2}\rangle$) are $A(A^{\prime})$ and
$B(B^{\prime})$, respectively. It should be noted that since the dynamical
phase is zero throughout the evolution in our scheme, the realistic
trajectories also evolve along the geodesic lines, which is different from in
Refs. ae2003 ; sciadvgeodesic  involving non-zero dynamical phase during the
evolution process.

##  V Some details for the gates by nonadiabatic geometric quantum computation

To achieve a conventional NGQC, the evolution time $\tau$ is divided into
three intervals,

$\displaystyle\int_{0}^{\tau_{1}}\Omega(t)dt$ $\displaystyle=$
$\displaystyle\theta,\quad\ \ \quad\varphi=-\phi+\frac{\pi}{2},\qquad
t\in[0,\tau_{1}]$ $\displaystyle\int_{\tau_{1}}^{\tau_{2}}\Omega(t)dt$
$\displaystyle=$ $\displaystyle\pi,\quad\ \
\quad\varphi=-\phi-\gamma-\frac{\pi}{2},t\in[\tau_{1},\tau_{2}]$
$\displaystyle\int_{\tau_{2}}^{\tau}\Omega(t)dt$ $\displaystyle=$
$\displaystyle\pi-\theta,\quad\varphi=-\phi+\frac{\pi}{2},\qquad
t\in[\tau_{2},\tau]$

with $\theta\in[0,\pi]$. Therefore, the obtained single-qubit gate after the
total geometric evolution process is given by

$\displaystyle U_{c}(\theta,\gamma,\phi)$ $\displaystyle=$
$\displaystyle\exp(i\gamma\vec{n}\cdot\vec{\sigma})$ (S3) $\displaystyle=$
$\displaystyle\cos\gamma+i\sin\gamma\cos\theta\sigma_{z}+i\sin\gamma\sin\theta\cos\phi\sigma_{x}$
$\displaystyle+i\sin\gamma\sin\theta\sin\phi\sigma_{y}$

which corresponds to a rotation around the axis
$\vec{n}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ by an angle
$-2\gamma$.

In our case, the $U_{1}$ gate is written as
$U_{1}=[1+i,-1-i;1-i,1-i]/2=(I+i\sigma_{z})/2-i(\sigma_{x}+\sigma_{y})/2$. To
construct the $U_{1}$ gate, we require the parameters to satisfy
$U_{c}(\theta,\gamma,\phi)=U_{1}$ which means

$\cos\gamma=\frac{1}{2},\cos\theta=\frac{1}{\sqrt{3}},\sin\phi=-\frac{1}{\sqrt{2}},\cos\phi=-\frac{1}{\sqrt{2}}.$

Then the parameters are determined as

$\gamma=\frac{\pi}{3},\theta=\arccos\frac{\sqrt{3}}{3},\phi=\frac{5\pi}{4}.$
(S4)

Choosing the constant Rabi frequency, i.e., $\Omega(t)=\Omega_{m}$ with
$\Omega_{m}$ denoting the constant Rabi frequency, we obtain the total time of
a quantum gate to be $\tau=2\pi/\Omega_{m}$.

##  VI Some details for the gates by dynamical Quantum Computation

In our experiment, the DQC is carried out based on the evolution of the qubit
state under carrier transitions, as described by the Hamiltonian,

$H=\frac{1}{2}\Omega_{0}e^{i\varphi}|e\rangle\langle g|+\rm H.C.,$ (S5)

with $\Omega_{0}$ and $\varphi$ denoting the Rabi frequency and the phase of
the laser. The corresponding unitary evolution operator is

$U_{d}(\theta,\varphi)=\cos\frac{\theta}{2}-i\sin\frac{\theta}{2}(\sigma_{+}e^{i\varphi}+\sigma_{-}e^{-i\varphi}),$
(S6)

with $\theta=\Omega_{0}t$. Thus we generate the $U_{1}$ gate by following
pulse sequences

$U_{1}=U_{d}(\frac{\pi}{2},0)U_{d}(\pi,-\frac{\pi}{4})U_{d}(\pi,\frac{\pi}{2}),$
(S7)

which means that the process needs a time duration of $\tau=5\pi/2\Omega_{0}$.

##  VII Tomography of a single qubit state

Using the Stokes parameters, the arbitrary state of a single qubit can be
uniquely represented as

$\rho=\frac{1}{2}\sum_{i=0}^{3}S_{i}\sigma_{i}$ (S8)

which is constituted by Pauli operators, i.e., $\sigma_{0}=I$,
$\sigma_{1}=\sigma_{x}$, $\sigma_{2}=\sigma_{y}$, and $\sigma_{3}=\sigma_{z}$.
We have $S_{i}=\rm Tr[\sigma_{i}\rho]$ with $S_{0}=1$, and three parameters as
below need to be measured.

$S_{1}=P_{e}^{x}-P_{g}^{x},\quad S_{2}=P_{e}^{y}-P_{g}^{y},\quad
S_{3}=P_{e}-P_{g},$ (S9)

where Px is a projection measurement on
$|g\rangle^{x}=\frac{1}{\sqrt{2}}(|e\rangle-|g\rangle)$ and
$|e\rangle^{x}=\frac{1}{\sqrt{2}}(|e\rangle+|g\rangle)$. Py is a projection
measurement on $|g\rangle^{y}=\frac{1}{\sqrt{2}}(|e\rangle-i|g\rangle)$ and
$|e\rangle^{y}=\frac{1}{\sqrt{2}}(|e\rangle+i|g\rangle)$. Then the tomography
of a qubit state under the carrier transition is carried out by following
steps,

I. Measurement of $S_{3}$: Measure $\rho$ in the state $|e\rangle$ to obtain
$P_{e}$; Then swap the states $|g\rangle$ and $|e\rangle$ by applying a
resonant $\pi$ pulse and then measure $\rho$ in the state $|g\rangle$ to
obtain $P_{g}$. Finally, calculate $S_{3}$ using the relation
$S_{3}=P_{e}-P_{g}$.

II. Measurement of $S_{1}$: Firstly, rotate $\rho$ by applying a resonant
pulse with $\theta=\phi=\pi/2$ (i.e., $U_{x}$). Secondly, make the measurement
in $|e\rangle$ to obtain $P_{e}^{x}$; Then swap the states $|g\rangle$ and
$|e\rangle$ by applying a resonant $\pi$ pulse, and then make the measurement
in $|e\rangle$ to acquire $P_{g}^{x}$. Finally, calculate $S_{1}$ using the
relation $S_{1}=P_{e}^{x}-P_{g}^{x}$.

III. Measurement of $S_{2}$: First rotate $\rho$ by applying a resonant pulse
with $\theta=\pi/2,\phi=0$ (i.e., $U_{y}$). Then make the measurement in
$|e\rangle$ and $|g\rangle$ to obtain $P_{e}^{y}$ and $P_{g}^{y}$,
respectively. Then, calculate $S_{1}$ using the relation
$S_{2}=P_{e}^{y}-P_{g}^{y}$.

Assuming that the ideal and experimental states are defined by the vector
$\vec{S}^{k}\equiv(S_{0}^{k},S_{1}^{k},S_{2}^{k},S_{3}^{k})^{T}$ with
$k=exp,id$, respectively, regarding
$\rho_{k}=\frac{1}{2}\sum_{i=0}^{3}S^{k}_{i}\sigma_{i}$, we have the fidelity
of the quantum gate calculated as

$\mathcal{F}=\text{Tr}[\rho_{\rm exp}\rho_{\rm id}]=\frac{1}{2}\vec{S}^{\rm
exp}\cdot\vec{S}^{\rm id}.$ (S10)

Correspondingly, the infidelity is given by

$\epsilon_{\mathcal{F}}=\sqrt{\frac{1}{4}\sum_{k=0}^{3}(d\vec{S}^{\rm
exp}_{k}\cdot\vec{S}^{\rm id}_{k})^{2}},$ (S11)

where the vector $d\vec{S}^{\rm exp}$ denotes the experimental error vector of
$\vec{S}^{\rm exp}$.

##  VIII Tomography for the quantum gate process

A quantum operation process can be described as jmo-44-2455

$\varepsilon(\rho)=\sum_{mn}\chi_{mn}A_{m}\rho A_{n}^{\dagger},$ (S12)

where $\chi_{mn}$ is a process matrix and the operators $A_{m}$ consist of a
fixed operator set to describe the process operator $\varepsilon$. In the one-
qubit case, the operator set is defined as

$A_{0}=I,\ A_{1}=\sigma_{x},\ A_{2}=-i\sigma_{y},\ A_{3}=\sigma_{z}.$ (S13)

Then, the process matrix $\chi$ can be measured by making a tomography of the
density matrix for the corresponding output states by choosing four different
input states,

$|g\rangle,\ |e\rangle,\ |+\rangle=(|g\rangle+|e\rangle)/\sqrt{2},\
|-\rangle=(|g\rangle-i|e\rangle)/\sqrt{2}.$ (S14)

The four density matrices are determined by the following forms

$\displaystyle\rho_{1}$ $\displaystyle=$
$\displaystyle\varepsilon(|g\rangle\langle g|),$ $\displaystyle\rho_{2}$
$\displaystyle=$ $\displaystyle\varepsilon(|e\rangle\langle e|),$
$\displaystyle\rho_{3}$ $\displaystyle=$
$\displaystyle\varepsilon(|+\rangle\langle+|)-i\varepsilon(|-\rangle\langle-|)-\frac{(1-i)(\rho_{1}+\rho_{2})}{2},$
$\displaystyle\rho_{4}$ $\displaystyle=$
$\displaystyle\varepsilon(|+\rangle\langle+|)+i\varepsilon(|-\rangle\langle-|)-\frac{(1+i)(\rho_{1}+\rho_{2})}{2}.$

Actually, the states $\rho_{3,4}$ correspond to the output matrices of the
input states $|g\rangle\langle e|$ and $|e\rangle\langle g|$.

Therefore, the process matrix $\chi$ can be expressed as

$\chi=\Lambda\left[\begin{matrix}\rho_{1}&\rho_{4}\\\
\rho_{3}&\rho_{2}\end{matrix}\right]\Lambda,\rm{with}\
\Lambda=\frac{1}{2}\left[\begin{matrix}I&\sigma_{x}\\\
\sigma_{x}&-I\end{matrix}\right].$ (S15)

Then, the fidelity of the gate process for the ideal and experimental results
is written as

$\mathcal{F}_{gp}=\text{Tr}[\chi_{\rm exp}\chi_{\rm id}].$ (S16)

##  IX Supplemental figures regarding the main text

We present in Figs. 3-6 the experimental data of the Stokes parameters for
different output states in the gate processes by different schemes.

Figure S3: Stokes parameters $S_{k}$ ($k=x,y,z$) regrading the experimental
data of Fig. 1 in the main text. (a) For the evolution process of the gate
under different time durations (main text Fig. 1(c)). (b) For the output
states with different input states (main text Fig. 1(d)). Figure S4: Stokes
parameters $S_{k}$ ($k=x,y,z$) regrading the experimental data of Fig. 2 in
the main text. (a) For the $U_{1}$ gate process and (b) For the Hadamard gate
process under different input states.  Figure S5: Stokes parameters $S_{k}$
($k=x,y,z$) for the experimental data of $U_{1}$ gate process in Fig. 3 of the
main text. (a-c) For NNGQC, NGQC and DQC, respectively.  Figure S6: Stokes
parameters $S_{k}$ ($k=x,y,z$) for the experimental data of $U_{1}$ gate
process in Fig. 4 of the main text. (a-c) Regarding different Rabi frequency
errors for NNGQC, NGQC and DQC, respectively, and (d-f) Regarding different
resonance frequency errors corresponding to NNGQC, NGQC and DQC, respectively.

##  X Scheme for the two-qubit NNGQC

We elucidate below that the two-qubit NNGQC could be essentially carried out
by the similar way to the single-qubit counterpart.

Consider two ions simultaneously driven by two 729-nm lasers and coupled by
the vibrational mode of motion. As plotted in Fig. S7, the two lasers couple
the levels $|g\rangle$ and $|e\rangle$ to the auxiliary level $|s\rangle$ with
the corresponding detunings $\Delta_{1,2}$ and Rabi frequencies
$\Omega_{1,2}$, which can be described, in the interaction picture, by the
Hamiltonian as

$H_{I}=\sum_{j=1}^{2}\Omega_{j}\sigma_{+}^{j}e^{-i\Delta_{j}t+i\phi_{j}}\exp[i\tilde{\eta}_{j}(a^{\dagger}e^{i\omega
t}+ae^{-i\omega t})]+H.C.,$ (S17)

where $\omega$ is the vibrational frequency of center-of-mass mode, $\phi_{j}$
denotes the laser phase, $\sigma_{+}^{1}=|s\rangle\langle g|$,
$\sigma_{+}^{2}=|s\rangle\langle e|$,
$\Delta_{1}=\omega_{1}+\omega_{g}-\omega_{s}$,
$\Delta_{2}=\omega_{2}+\omega_{e}-\omega_{s}$ and the Lamb-Dicke parameters
are defined as $\tilde{\eta}_{j}=k\cos\theta_{j}\sqrt{\hbar/2m\omega}$ with
$\theta_{j}$ denoting the intersection angle between the $j$-th laser
irradiation and the vibrational direction. In our case, we choose
$\theta_{1}=0$ and $\theta_{2}=\pi/2$. By setting $\Delta_{1}=-\omega-\Delta$
and $\Delta_{2}=-\Delta$ with $\Delta\ll\omega$, we rewrite the above
Hamiltonian, under the rotating wave approximation ($\Omega_{1,2}\ll\omega$),
as

$H_{I}=(\Delta|s\rangle+\Omega
e^{-i\bar{\phi}}|e\rangle+\frac{g}{2}a^{\dagger}|g\rangle)\langle s|+H.C.,$
(S18)

where $\Omega=\Omega_{2}$, $g=2\eta_{1}\Omega_{1}$ and we have set
$\phi_{1}=-\pi/2,\phi_{2}=\bar{\phi}$. Meanwhile, the scheme requires that the
phase of the second laser on different ions can be independently adjusted.
Thus, the total Hamiltonian of the system can be written as

$H_{t}=\sum_{k=1}^{2}(\Delta|s\rangle_{k}+\bar{\Omega}_{k}|e\rangle_{k}+\frac{g}{2}a^{\dagger}|g\rangle_{k})\langle
s|+H.C.$ (S19)

where $\bar{\Omega}_{k}=\Omega e^{-i\bar{\phi}_{k}}$ and $k$ denotes the $k$th
ion.

Figure S7: (a) Schematic of two trapped ions driven by two 729-nm lasers,
where one laser (red beam) irradiates along the ions’ vibrational direction,
i.e., $\theta_{1}=0$ and the other irradiation (black beams) perpendicular to
the vibrational direction (i.e., $\theta_{2}=\pi/2$) with the requirement of
individually addressing. (b) Level scheme of the ion, where the Zeeman
sublevels of S1/2 are labeled as $|g\rangle$ and $|e\rangle$, and one Zeeman
level $m_{J}=3/2$ of D5/2 is employed as the auxiliary state $|s\rangle$.

Given the system in the ground state of vibration, the initial state of the
system will be restricted in the space spanned by
$|\psi_{1}\rangle=|ee0\rangle,|\psi_{2}\rangle=|eg0\rangle,|\psi_{3}\rangle=|ge0\rangle,|\psi_{4}\rangle=|gg0\rangle$,
where the first and the second terms of the ket denotation denote the state of
two ions and the third term represents the phonon states of the system. When
the system is initially in $|\psi_{1}\rangle$, the evolution will occur in the
subspace
$C_{ee0}=\\{|\psi_{1}\rangle,|\psi_{5}\rangle,|\psi_{6}\rangle,...,|\psi_{12}\rangle\\}$
with

$\displaystyle|\psi_{5}\rangle$ $\displaystyle=$ $\displaystyle|se0\rangle,\
|\psi_{6}\rangle=|es0\rangle,\ |\psi_{7}\rangle=|ge1\rangle,\
|\psi_{8}\rangle=|eg1\rangle$ $\displaystyle|\psi_{9}\rangle$ $\displaystyle=$
$\displaystyle|ss0\rangle,\ |\psi_{10}\rangle=|gs1\rangle,\
|\psi_{11}\rangle=|sg1\rangle,\ |\psi_{12}\rangle=|gg2\rangle.$

The Hamiltonian of $C_{ee0}$ can be written as
$\hat{\mathcal{H}}_{ee0}=\hat{\mathcal{H}}_{g}+\hat{\mathcal{H}}_{\bar{\Omega}}+\hat{\mathcal{H}}_{\Delta}$
with

$\displaystyle\hat{\mathcal{H}_{g}}$ $\displaystyle=$ $\displaystyle
g(\left|\psi_{5}\right\rangle\left\langle\psi_{7}\right|+\left|\psi_{6}\right\rangle\left\langle\psi_{8}\right|+\left|\psi_{9}\right\rangle\left\langle\psi_{10}\right|+\left|\psi_{9}\right\rangle\left\langle\psi_{11}\right|$
$\displaystyle+\left|\psi_{10}\right\rangle\left\langle\psi_{12}\right|+\left|\psi_{11}\right\rangle\left\langle\psi_{12}\right|)+\rm
H.c.$ $\displaystyle\hat{\mathcal{H}}_{\bar{\Omega}}$ $\displaystyle=$
$\displaystyle\bar{\Omega}_{1}(\left|\psi_{1}\right\rangle\left\langle\psi_{5}\right|+\left|\psi_{1}\right\rangle\left\langle\psi_{9}\right|+\left|\psi_{8}\right\rangle\left\langle\psi_{11}\right|)$
$\displaystyle+\bar{\Omega}_{2}(\left|\psi_{1}\right\rangle\left\langle\psi_{6}\right|+\left|\psi_{5}\right\rangle\left\langle\psi_{9}\right|+\left|\psi_{7}\right\rangle\left\langle\psi_{10}\right|)+{\rm
H.c.}$ $\displaystyle\hat{\mathcal{H}}_{\Delta}$ $\displaystyle=$
$\displaystyle\Delta(\left|\psi_{5}\right\rangle\left\langle\psi_{5}\right|+\left|\psi_{6}\right\rangle\left\langle\psi_{6}\right|+\sum_{j=9}^{11}\left|\psi_{j}\right\rangle\left\langle\psi_{j}\right|).$
(S20)

For a quantum Zeno dynamical process under the condition
$\hat{\mathcal{H}}_{g}\gg\ \hat{\mathcal{H}}_{\bar{\Omega}}$, i.e.,
$g\gg\bar{\Omega}_{1,2}$, the system will evolve within the subspace of the
initial eigenstates, and the bosonic modes have three zero-eigenvalue
eigenstates {
$\left|\Phi_{d1}\right\rangle=(\left|\psi_{9}\right\rangle+\left|\psi_{12}\right\rangle)/\sqrt{2}$,
$\left|\Phi_{d2}\right\rangle=(\left|\psi_{10}\right\rangle+\left|\psi_{11}\right\rangle)/\sqrt{2}$,
$\left|\psi_{1}\right\rangle$}. Therefore, there is no effective coupling
among $\left|\Phi_{d1}\right\rangle$, $\left|\Phi_{d2}\right\rangle$ and
$\left|\psi_{1}\right\rangle$ and the system will be trapped in the initial
state $|\psi_{1}\rangle=|ee0\rangle$ all the time.

If the initial state is $|\psi_{2}\rangle$ or $|\psi_{3}\rangle$, the
evolution will be remaining in the subspace
$C_{eg0}=\\{|\psi_{2}\rangle,|\psi_{3}\rangle,|\psi_{13}\rangle,|\psi_{14}\rangle,|\psi_{15}\rangle\\}$
with

$|\psi_{13}\rangle=|eg0\rangle,\ |\psi_{14}\rangle=|sg0\rangle,\
|\psi_{15}\rangle=|gg1\rangle,$

the Hamiltonian in the subspace $C_{eg0}$ can be expressed as
$\hat{H}_{eg0}=\hat{H}_{g}+\hat{H}_{\bar{\Omega}}+\hat{H}_{\Delta}$ with

$\displaystyle\hat{H}_{g}$ $\displaystyle=$ $\displaystyle
g(\left|\psi_{13}\right\rangle\left\langle\psi_{14}\right|+\left|\psi_{14}\right\rangle\left\langle\psi_{15}\right|+\rm
H.c.,$ $\displaystyle\hat{H}_{\bar{\Omega}}$ $\displaystyle=$
$\displaystyle\bar{\Omega}_{1}\left|\psi_{2}\right\rangle\left\langle\psi_{13}\right|+\bar{\Omega}^{*}_{2}\left|\psi_{15}\right\rangle\left\langle\psi_{3}\right|+{\rm
H.c.},$ (S21) $\displaystyle\hat{H}_{\Delta}$ $\displaystyle=$
$\displaystyle\Delta(\left|\psi_{13}\right\rangle\left\langle\psi_{13}\right|+\left|\psi_{15}\right\rangle\left\langle\psi_{15}\right|).$

Given the initial state $\left|\psi_{4}\right\rangle=\left|gg0\right\rangle$,
the system keeps unchanged based on the Hamiltonian in Eq. (S19).

Thus, in the picture of the eigenstates of $\hat{H}_{g}$:
$\\{\left|\Psi_{\pm}\right\rangle=(\left|\psi_{13}\right\rangle\pm\sqrt{2}\left|\psi_{14}\right\rangle+\left|\psi_{15}\right\rangle)/2,\
\left|\Psi_{d}\right\rangle=(\left|\psi_{13}\right\rangle-\left|\psi_{15}\right\rangle)/\sqrt{2}\\}$
corresponding to the eigenvalues $\pm\sqrt{2}g$ and 0, the Hamiltonian can be
represented as

$\displaystyle\hat{H}_{\bar{\Omega}}$ $\displaystyle=$
$\displaystyle\bar{\Omega}_{1}\left|\psi_{2}\right\rangle\left\langle\Phi_{+}\right|+\bar{\Omega}_{2}\left|\psi_{3}\right\rangle\left\langle\Phi_{-}\right|+\rm
H.c.,$ $\displaystyle\hat{H}_{\Delta}$ $\displaystyle=$ $\displaystyle
2\Delta\left|\Psi_{d}\right\rangle\left\langle\Psi_{d}\right|+\Delta\left|\Phi_{+}\right\rangle\left\langle\Phi_{-}\right|+\rm
H.c.,$ (S22)

where the shorthand states
$\left|\Phi_{\pm}\right\rangle\equiv(e^{i\sqrt{2}gt}\left|\Psi_{+}\right\rangle+e^{-i\sqrt{2}gt}\left|\Psi_{-}\right\rangle\pm\sqrt{2}\left|\Psi_{d}\right\rangle)/2$.
Due to the condition of $g\gg\bar{\Omega}_{1,2}$, the states
$\left|\Psi_{\pm}\right\rangle$ will be decoupled from
{$\left|\Psi_{d}\right\rangle,\left|\psi_{2}\right\rangle,\left|\psi_{3}\right\rangle$},
and then the effective Hamiltonian is obtained as

$\hat{H}_{\rm
eff}=\Delta\left|\Psi_{d}\right\rangle\left\langle\Psi_{d}\right|+\frac{1}{\sqrt{2}}(\bar{\Omega}_{1}\left|\psi_{2}\right\rangle-\bar{\Omega}_{2}\left|\psi_{3}\right\rangle)\left\langle\Psi_{d}\right|+\rm
H.c.$

Under the large detuning condition $\Delta\gg\bar{\Omega}_{1,2}$ and using the
James-Jerke method JJ , the above equation can be rewritten as

$\displaystyle\hat{H}_{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{|\bar{\Omega}_{1}|^{2}}{2\Delta}(\left|\psi_{2}\right\rangle\left\langle\psi_{2}\right|-\left|\Psi_{d}\right\rangle\left\langle\Psi_{d}\right|)+\frac{|\bar{\Omega}_{2}|^{2}}{2\Delta}(\left|\psi_{3}\right\rangle\left\langle\psi_{3}\right|$
(S23)
$\displaystyle-\left|\Psi_{d}\right\rangle\left\langle\Psi_{d}\right|)-\frac{\bar{\Omega}_{1}^{*}\bar{\Omega}_{2}}{2\Delta}\left|\psi_{3}\right\rangle\left\langle\psi_{2}\right|+\rm
H.c.,$

where the first two terms are caused by the Stark shifts and
$\left|\Psi_{d}\right\rangle$ is decoupled from $\left|\psi_{2}\right\rangle$
and $\left|\psi_{3}\right\rangle$. Choosing
$\widetilde{\Omega}\equiv\lvert\bar{\Omega}_{1}\rvert=\lvert\bar{\Omega}_{2}\rvert$
and ignoring the global phase factor, the effective Hamiltonian in the
interaction picture can be further reduced to

$\hat{H}_{\rm
eff}=\frac{\widetilde{\Omega}^{2}e^{i\varphi}}{2\Delta}\left|\psi_{3}\right\rangle\left\langle\psi_{2}\right|+\rm
H.c.$ (S24)

with the time-dependent phase difference
$\varphi=\pi+\bar{\phi}_{1}-\bar{\phi}_{2}$. This Hamiltonian is actually of
the similar form to Eq. (1) in the main text, which in principle can be
carried out by the steps of NNGQC, and enjoy the advantages as we verified.
Considering $\hat{U}^{\prime}(\theta,~{}\alpha,~{}\beta)$ as the evolution
operator relevant to Eq. (24), we may have the two-qubit evolution operator,
which is unitarily equivalent to
$\hat{U}^{\prime}(\theta,~{}\alpha,~{}\beta)$, constituting a universal two-
qubit controlled gate within the subspace spanned by $|\psi_{1}\rangle$,
$|\psi_{2}\rangle$, $|\psi_{3}\rangle$ and $|\psi_{4}\rangle$. Note that
$\hat{U}^{\prime}(\theta,~{}\alpha,~{}\beta)$ owns the same matrix form as
$\hat{U}(\theta,~{}\alpha,~{}\beta)$ in the main text, but only works on the
states $|\psi_{2}\rangle$ and $|\psi_{3}\rangle$. $|\psi_{1}\rangle$ and
$|\psi_{4}\rangle$ remain unchanged in this two-qubit controlled gate. The
NNGQC operational details as well as the advantages regarding higher-speed,
higher-fidelity and lower error in comparison with the same gating operations
by other ways under the same conditions/paramters will be published elsewhere.

## References

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