# Verification of Bell Nonlocality by Violating Quantum Monogamy Relations

Yan-Han Yang1, Xin-Zhu Liu1,Xing-Zhou Zheng1,Shao-Ming Fei2,3, Ming-Xing
Luo1,4 1 School of Information Science and Technology, Southwest Jiaotong
University, Chengdu 610031, China
2 School of Mathematical Sciences, Capital Normal University, Beijing 100048,
China
3 Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
4 CAS Center for Excellence in Quantum Information and Quantum Physics, Hefei,
230026, China

###### Abstract

Quantum nonlocality as a witness of entanglement plays a crucial role in
various fields. Existing quantum monogamy relations rule out the possibility
of simultaneous violations of any Bell inequalities with partial statistics
generated from one Bell experiment on any multipartite entanglement or post-
quantum sources. In this paper, we report an efficient method to construct
multipartite Bell test based on any Bell inequalities. We demonstrate that
violating these monogamy relations can dynamically witness simultaneous Bell
nonlocalities of partial systems. We conduct a tripartite experiment to verify
quantum nonlocalities by violating a tripartite monogamy relation using a
maximally entangled two-photon state.

Keywords: Quantum nonlocality; multipartite entanglement; quantum monogamy
relations; dynamical witness; entangled two-photon

## Introduction

Quantum entanglement is a fascinating phenomenon that has attracted
significant attention in the field of quantum mechanics. Local measurements on
bipartite entangled systems can give rise to joint correlations that exhibit
nonlocal features, which cannot be explained by classical physics under the
assumptions of locality and causality EPR ; Bell . These bipartite nonlocal
correlations can be witnessed using Bell inequalities CHSH ; Li ; Son ; BCPS ,
which provide device-independent methods for verifying quantum nonlocality
CHSH ; BCPS . In multipartite scenarios, there are richer nonlocal
correlations. one example is a simple extension of bipartite nonlocal
correlations to characterize the bipartition of all the systems. The other is
the genuine multipartite correlations Sy  that can witness the most strongest
global nonlocal correlations generated from the multipartite entanglement to
rule the classical mixture of any biseparable correlations. This has been
extended for GHZ states Aolita2012  and graph states Guhne2005 , operational
frameworks Gallego2012 , network entanglement Luo2021a ; Armin2022  or
robusted entanglement Luo2021b . Quantum entanglement has potential
applications in quantum computing Raussendorf ; Linden ; Zurel ; McArdle ;
Hangleiter , quantum cryptography Curty ; Lo ; Ma ; Vazirani ; Yin ; Zhang2 ;
Gisin ; Xu ; Portmann , and communication ZZH ; HHH ; Kim ; Bennett ;
Cavalcanti ; Lee ; Hu .

Witnessing the quantum nonlocality of entanglement can be equivalently
realized by testing the winning probability of specific nonlocal games BCPS ;
SMA . A nonlocal game is a hypothetical game in which several players, each
receive a question from a referee, and then respond with an answer. The
referee randomly selects the questions according to a known distribution, and,
upon receiving answers from all players, decides whether they win or lose. The
violation of a Bell inequality indicates that quantum players using quantum
resources have a higher probability of winning in an equivalent nonlocal game
than those using classical resources. One example is the Clauser-Horne-
Shimony-Holt (CHSH) game CHSH  for evaluating the Boolean equation: $a\oplus
b=x\wedge y$. This has been extended to witness nonlocality of general
entanglement RV ; EWL ; Luo2019 .

All the previous Bell tests have assumed that the observers share a classical
variable or quantum state. This setting in principle makes use of static
implementations of Bell experiments, i.e., the source generates the fixed
states and then distributes to observers. This is reasonable for standard
statistic assumption of independent and identical distributions. But for
multipartite quantum entanglement, it exhibits local monogamy features Coffman
; Terhal ; Osborne2006 , which limit the sharing of local entanglement between
different subsets of particles. For example, although two particles can be
entangled with each other, but they cannot simultaneously share the same level
of entanglement with another particle in a multipartite entangled system. This
implies that the bipartite nonlocality verified in local Bell test will rule
out the possibilities of all the other bipartite nonlocal correlations even if
they may jointly exhibit the genuine multipartite nonlocality. This can be
formally illustrated using a tailed CHSH inequality for each pair in a
multipartite Bell test, where any quantum or post-quantum resources show no
advantage beyond classical resources Toner2009 ; RH ; Regula2014 ; PB ; Fei ;
LiF . This raises a question of how the monogamous nature of global
multipartite entanglement affects Bell tests of bipartite nonlocalities Kurzy
; Tran .

Figure 1: (Color online) Schematically tripartite Bell test.

A. All observers share a variable $\lambda$ or any quantum state $\rho$ on
Hilbert space $\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C}$.
B. Each pair of two observers receive a bipartite state $\rho$ conditional on
inputs $x,y,z\in\\{0,1\\}$. Each player receives a state for the input
$x,y,z=2$. There are two outputs $a,b,c\in\\{-1,1\\}$. All the measurement
devices can be regarded as black boxes.

In this paper, we propose a unified method to construct multipartite monogamy
relations from any linear Bell inequalities that are useful to witness general
entanglement. We compare these Bell experiments in static and dynamical
manners, as shown in Figure 1. We show that when all observers share any
global quantum entanglement or post-quantum resources statically, their
measurement statistics have no quantum nonlocality beyond classical scenarios
Fei ; PB ; RH . We further demonstrate that by dynamically distributing a
small-size entangled state, their measurement statistics show quantum
nonlocality by violating the present multipartite monogamy relations. We
implement a tripartite experiment using two-photon entangled states. This
allows implementing a measurement-device independent witness of Bell
nonlocality beyond previous Bell nonlocality experiments Freedman ; Weihs ;
Kwiat ; Zhang ; Zhong ; Vertesi ; Rowe ; Big ; Storz2023 ; Ansmann ; Mao ; Xue
; Wu .

## Results

## Dynamical verification of Bell nonlocalities

Consider a standard Bell test with two parties Alice (A) and Bob (B). Let $x$
and $y$ be the measurement settings of Alice and Bob respectively, and $a$ and
$b$ for their outcomes. The joint probability of all the outputs $a$ and $b$
conditional on the inputs $x$ and $y$ allows the following decomposition Bell
:

$\displaystyle P(a,b|x,y)=\int p(a|x,\lambda)p(b|y,\lambda)d\mu(\lambda),$ (1)

where the conditional probabilities $p(a|x,\lambda)$ and $p(b|y,\lambda)$
depend on their input and the shared variable $\lambda$ with the probability
distribution $\mu(\lambda)$. A general Bell inequality with correlations of
local outcomes observed is defined as

$\displaystyle\mathcal{L}(\textsf{A},\textsf{B})\equiv\sum_{x,y}\sum_{a,b}\alpha_{x,y}^{a,b}P(a,b|x,y)\leq
c,$ (2)

where parameters $\alpha_{x,y}^{a,b}$ can be chosen positive and $c$ is the
classical bound in hidden variable model Bell .

In quantum scenarios, two parties share an entangled state $\rho$ on Hilbert
space $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. Both parties perform positive-
operator valued measurements (POVMs). Alice uses the measurement operators
$\\{M_{a|x}\\}$ satisfying $M_{a|x}\geq 0$ and $\sum_{a}M_{a|x}=\mathbbm{1}$
with the identity operator $\mathbbm{1}$, and Bob uses $\\{M_{b|y}\\}$
satisfying $M_{b|y}\geq 0$ and $\sum_{b}M_{b|y}=\mathbbm{1}$. From the Born’s
rule, the joint probability of outputs is defined as

$\displaystyle P_{Q}(a,b|x,y)={\rm{}Tr}(M_{a|x}\otimes M_{b|y}\rho).$ (3)

Violating the inequality (2) by the quantum statistics witnesses the
nonlocality of the shared entanglement. Now, consider an $n$-partite Bell
experiment with separated parties, $\textsf{A}_{1},\cdots,\textsf{A}_{n}$. For
any two parties $\textsf{A}_{i}$ and $\textsf{A}_{j}$ they can verify the
bipartite nonlocality from their shared bipartite entanglement violating the
Bell’s inequality (2) with their measurement statistics. But remarkably, these
kinds of violations cannot be simultaneously realized even if all the parties
$\textsf{A}_{i}$’s share an $n$-partite entanglement. Especially, we have the
following monogamy relation

$\displaystyle\sum_{1\leq i<j\leq n}{\cal
L}(\textsf{A}_{i},\textsf{A}_{j})\leq\frac{1}{2}n(n-1).$ (4)

where all the correlations of ${\cal L}(\textsf{A}_{i},\textsf{A}_{j})$ are
included in one multipartite Bell experiment. This monogamy relation can be
proved by using the inequality PB : $\sum_{i=2}^{n}{\cal
L}(\textsf{A}_{1},\textsf{A}_{i})\leq(n-1)c$. This inequality holds for all
the parties share post-quantum sources in terms of any every non-signaling
theory. This means that in any standard multipartite Bell test any local
bipartite correlations shows a total similar behaviours regardless of their
shared resources. Instead, the violation of the inequality (4) will rule out
the non-signaling principle.

The present monogamy of Bell inequalities can be further extended to
characterize more than two parties. Let
$P(a_{1},\cdots,a_{m}|x_{1},\cdots,x_{m})$ be the joint probability that all
parties $\textsf{A}_{1},\cdots,\textsf{A}_{m}$ observe the outcomes
$a_{1},\cdots,a_{m}$ conditional on the inputs $x_{1},\cdots,x_{m}$,
respectively. Consider any linear Bell inequality

$\displaystyle\mathcal{L}(\textsf{A}_{1},\cdots,\textsf{A}_{m})\equiv\sum_{x_{1},\cdots,x_{m}}\sum_{a_{1},\cdots,a_{m}}\alpha_{x_{1},\cdots,x_{m}}^{a_{1},\cdots,a_{m}}P(a_{1},\cdots,a_{m}|x_{1},\cdots,x_{m})\leq
c^{\prime},$ (5)

which holds in the local hidden variable model Bell  or biseparable model Sy ,
and $c^{\prime}$ is a constant. Inspired by recent result PB  we have the
following monogamy relation as

$\displaystyle\sum_{1\leq i_{1}\leq\cdots\leq i_{m}\leq n}{\cal
L}(\textsf{A}_{i_{1}},\cdots,\textsf{A}_{i_{m}})\leq cC^{n}_{m},$ (6)

where the constant $C_{m}^{n}$ denotes the combination number, i.e.,
$C_{m}^{n}=n!/(n-m)!m!$. Although violating the inequality (5) witnesses the
Bell nonlocality of specific multipartite entanglement, simultaneous
violations are not allowed for any genuine multipartite entanglement. This
demonstrates the trade-off among strengths of violations of any linear Bell
inequalities under the non-signaling principle.

Now, we present a unified way to violate the monogamy inequality (6) by using
dynamical implementations while still adhering to the non-signaling principle.
To show main idea, we consider a tripartite Bell test $\mathcal{G}_{3}$ shown
in Figure 1. There are three parties involved in this test: Alice (A), Bob
(B), and Charlie (C). They will utilize a shared source to complete the test.
Each party possesses three measurement settings denoted as $x(y,z)\in{0,1,2}$.
Additionally, they have two outputs, denoted as $a(b,c)\in{0,1}$, for each
measurement setting, respectively. Suppose three parties share a classical
measurable variable. Inspired by the CHSH inequality CHSH , we obtain the
following Bell inequality as

$\displaystyle\mathcal{B}(\textsf{A},\textsf{B},\textsf{C})$ $\displaystyle:=$
$\displaystyle A_{0}B_{0}C_{2}+A_{0}B_{1}C_{2}+A_{1}B_{0}C_{2}$ (7)
$\displaystyle-A_{1}B_{1}C_{2}+A_{0}B_{2}C_{0}+A_{0}B_{2}C_{1}$
$\displaystyle+A_{1}B_{2}C_{0}-A_{1}B_{2}C_{1}+A_{2}B_{0}C_{0}$
$\displaystyle+A_{2}B_{0}C_{1}+A_{2}B_{1}C_{0}-A_{2}B_{1}C_{1}\leq 6$

where $A_{i},B_{j},C_{k}$ are measurement observables with two outcomes
$\\{-1,1\\}$. This can be proved by the fact that for each set of inputs
$x,y,z$, the measurement statistic will violate the CHSH inequality if any two
parties who have the inputs less than 2 win the CHSH game CHSH , see the proof
in SI SI . This means the inequality (7) holds when all the parties share
classical variables, quantum states, or post-quantum resources because of the
monogamous features of nonlocalities RH ; PB ; Fei . This means in the present
setting, all the generated correlations consist of the same set regardless of
the their shared resources. Moreover, this inequality holds for all the
involved parties. This provides a form of global monogamy relationships going
beyond all the previous results with local monogamy relationships RH ; PB ;
Fei ; Toner2009 ; RH ; Regula2014 .

In quantum scenarios, each party in the tripartite test $\mathcal{G}_{3}$
receives one particle from the source. Since the observers do not have prior
knowledge about the state of the received particle, we impose a special
restriction on each party: their local quantum measurement devices are black-
boxes. This means that each party can only choose a local measurement box
based on the input state, but they cannot modify the internal workings of
their measurement devices. Under this assumption, we consider the present
scheme as a measurement device-independent implementation of Bell test. There
are three cases as follows:

Firstly, assuming that the source distributes a fixed three-particle state
$\rho_{ABC}$ on Hilbert space
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C}$ to all parties,
where Alice receives particle A, Bob receives particle B, and Charlie receives
particle C. Similar to the standard Bell test this case depends on the
independent identical assumption of shared quantum states. The players can
exploit any quantum correlations present in the shared state. However, all the
quantum correlations satisfy the inequality (7), i.e., there is no quantum
advantage beyond classical strategies.

Secondly, the source sends the particles passing through a swapping device.
This can be regarded as a tripartite nonlocal game where the referee sends the
swapped states to all observers conditional on their questions. The swapping
operation $S=\sum_{i,j}|ij\rangle\langle ji|$ exchanges the states of two
particles in the given state. For the input $z=2$, the source generates the
state $\rho_{ABC}$ and sends it to all observers. For the input $y=2$, the
source performs a swapping operation on particles B and C to get the state
$\rho^{{}_{(y)}}_{ABC}\equiv S_{B,C}\rho S_{B,C}^{\dagger}$ and then sends it
to all observers. For the input $x=2$, the source performs swapping operation
on particles A and C to get the state $\rho^{{}_{(x)}}_{ABC}\equiv S_{A,C}\rho
S^{\dagger}_{A,C}$ and sends it out. Suppose the source dynamically
distributes a separable state
$\rho_{ABC}=|\Phi\rangle\langle\Phi|_{AB}\otimes\rho_{C}$, where the
generalized EPR state $|\Phi\rangle$ is defined by
$|\Phi\rangle=\cos\theta|00\rangle+\sin\theta|11\rangle$ with
$\theta\in(0,\frac{\pi}{4})$, and $\rho_{C}$ is a single state of particle C.
This allows all observers simultaneously witness the Bell nonlocality of
$|\Phi\rangle$ by violating the monogamy inequality (7) (see the proof in SI
SI ) as

$\displaystyle\mathcal{B}_{epr}>\mathcal{B}_{q}=\mathcal{B}_{c}=6$ (8)

for any $\theta$ satisfying $\sin
2\theta>\frac{2}{3}\sqrt{\frac{2}{5}}-\frac{1}{3}\approx 0.0866$. Here, the
optimal winning probability is consistent with the maximal violation of the
inequality (7) with the maximally entangled EPR states. This shows that the
dynamical distribution of the EPR state allows all observers to achieve a
quantum advantage beyond the static experiments with a fixed classical
resource or any tripartite entangled states Sy . The noise robustness of
Werner state Werner  is easily followed for the linear inequality (7). We
extend the analysis to multipartite Bell tests based the CHSH inequality or
general Bell inequalities in SI SI . This provides the first Bell experiment
by combining quantum resources with dynamical resource distributing strategies
beyond any others with single utilization of classical resources, quantum
resources, or post-quantum resources.

Figure 2: Sketch of the experimental setup.

The SPDC source distributes maximally entangled two-photon pairs to build a
probabilistic triangle quantum network. At one time, each player receives one
photon and performs single-qubit measurements in Eq.(9). The SPDC source is
based on a PPKTP crystal and is pumped by an ultraviolet pump pulse (405-nm,
80-MHz, 140-fs). Here, key elements include polarization Sagnac interferometer
(PSI), half-wave plate (HWP), quarter-wave plate (QWP), splitting photon plate
(SPP), interference filter (IF), polarizing beam splitter (PBS), quantum
switch (QS) based on quantum random number generator (QRNG), quantum detector
(QD), and single photon detector (SD).

## Experimental results

To verify the Bell nonlocality, we first prepare a maximally entangled two-
photon entangled state $|\psi\rangle=(|HH\rangle+|VV\rangle)/\sqrt{2}$, as
shown in Figure 2. We exploit the spontaneous parametric down-conversion
(SPDC) source to generate polarimetric entangled photons Pan2012 . We encode
horizontal (H) and vertical (V) polarization states as qubits $|0\rangle$ and
$|1\rangle$, respectively. The reconstructed density matrix is illustrated in
Figure 3. The fidelity of the state
$F=\langle\phi^{+}|\rho_{exp}|\phi^{+}\rangle$ is calculated to be $0.9905\pm
0.0004$. The standard deviation is estimated assuming Poisson statistics. The
$P$ value is $10^{-12}$ in our experiment.

Figure 3: Tomographic results of the EPR source.

The left subplot represents the real part, while the right subplot represents
the imaginary part of the reconstructed density matrices.

Figure 4 presents a graphical representation of the experimental coincidence
rate behavior of our EPR source. The overlaid dotted curves are the best
sinusoidal fits to the corresponding data. We achieve this by fixing the
polarizer angle $\theta_{1}$ at 0 and $\pi/4$, while varying the polarizer
angle $\theta_{2}$ to align with the theoretical prediction. We measure the
experimental visibilities of $V_{HV}=0.9956\pm 0.0009$ and $V_{DA}=0.9818\pm
0.0019$. The standard deviation is estimated according to Poisson statistics.
These experimental visibilities exhibit a substantial and consistent deviation
from the predictions of classical physics which predicts that $V\leq 0.71$.

Figure 4: Experimental visibilities of $\theta_{1}$ against the polarizer
angle of $\theta_{2}$.

The gray region corresponds to $\theta_{1}$ being fixed at an angle of 0,
while the blue region corresponds to $\theta_{1}$ being fixed at an angle of
$\pi/4$. Combining a HWP and a PBS in front of the detectors acts as a
polarizer box, where the polarizer angle is given by
$\theta^{\rm{pol}}=2\theta^{\rm{HWP}}$.

During the experiment, we let the QRNG dynamically distribute the maximally
entangled EPR state $|\Phi\rangle=(|HH\rangle+|VV\rangle)/\sqrt{2}$ to two out
of the three parties, Alice, Bob, and Charlie. QRNG generates a set of
sequences (consisting of 0, 1, and 2) according to uniform random
distribution, where 0 encodes the case that the entangled source will be sent
to Alice and Bob with the experimental probability, 1 encodes the case for Bob
and Charlie and 2 encodes the case for Alice and Charlie. The experimental
coincidence probabilities are shown as $p_{0}=0.33549$, $p_{1}=0.33241$, and
$p_{2}=0.33210$.

In our experiment, we record the basis choices $(x,y,z)$ and the corresponding
readout result $(a,b,c)$ for each of the $n$ trials. Utilizing these recorded
values, we calculate the correlator $\langle A_{x}B_{y}C_{z}\rangle$ for all
possible combinations of measurement bases. In each of the twelve consecutive
experiments, we took 400s to complete the evaluation. We perform a post-
selection process, retaining 16 sets of coincidences, and present the
corresponding experimental values in Table 1. The coincidence rates are shown
in Tables S1-S3 in SI SI . Notably, the experimental measurement (7) yields a
value of 7.5348, which violates the maximal bound stipulated in the inequality
(7). In our experiment, we acknowledge the presence of two common loopholes
that are frequently encountered in nonlocality experiments: the locality
loophole and the postselection loophole AKAZ .

Table 1: Experimental correlators $\langle A_{x}B_{y}C_{z}\rangle$ for twelve combinations of basis choices $(x,y,z)$. Correlator | Experimental values
---|---
$\langle A_{0}B_{0}C_{2}\rangle$ | $0.6856\pm 0.0062$
$\langle A_{0}B_{1}C_{2}\rangle$ | $0.6972\pm 0.0060$
$\langle A_{1}B_{0}C_{2}\rangle$ | $0.6794\pm 0.0063$
$\langle A_{1}B_{1}C_{2}\rangle$ | $-0.6698\pm 0.0064$
$\langle A_{0}B_{2}C_{0}\rangle$ | $0.9710\pm 0.0012$
$\langle A_{0}B_{2}C_{1}\rangle$ | $0.0215\pm 0.0129$
$\langle A_{1}B_{2}C_{0}\rangle$ | $0.0190\pm 0.0130$
$\langle A_{1}B_{2}C_{1}\rangle$ | $0.9674\pm 0.0015$
$\langle A_{2}B_{0}C_{0}\rangle$ | $0.7012\pm 0.0060$
$\langle A_{2}B_{0}C_{1}\rangle$ | $0.6823\pm 0.0061$
$\langle A_{2}B_{1}C_{0}\rangle$ | $0.6737\pm 0.0062$
$\langle A_{2}B_{1}C_{1}\rangle$ | $-0.6716\pm 0.0062$

## Discussion

We have presented tripartite monogamy relationships of violating CHSH
inequality CHSH , which can be regarded as a wired Bell inequalities. This
kind of Bell inequalities provide the trade-off among strengths of violations
of partial Bell inequalities under only the non-signaling condition Fei ; PB ;
RH . We have further shown that there are quantum violations when a source
generates a flying EPR state EPR . Although the shared state has changed
during in each round of experiment, it is compatible with the independent and
identical distribution assumptions under specific input distribution. The
swapping operation being performed by the source can be regarded as a post-
selection strategy in a standard Bell experiments with three generated states
Bell ; BCPS . This provides a witness of Bell nonlocality beyond previous
experiments Freedman ; Weihs ; Kwiat ; Zhang ; Zhong ; Vertesi ; Rowe ; Big ;
Storz2023 . If we regard the present Bell tests as computation tasks that are
distributed across a network of interconnected nodes dynamically adapting to
changing conditions, the present quantum advantage means that combining
quantum resources with dynamical resource strategies are going beyond any
other single utilization of resources. This harnesses the collective
computational power of distributed systems to dynamical changes and
effectively tackle large-scale problems, and provides a new insight in
designing quantum distributed tasks or quantum cryptography on dynamical
quantum Internet Kim . This also intrigues to explore the inherent
nonclassicality of general multipartite Bell tests. We present an extension
with any linear Bell inequality in SI.

In conclusion, we have presented a unified way to construct monogamy
relationships using any Bell inequality. Violating these monogamous
relationships is distinguished for all parties sharing general sources. The
present model is useful for witnessing any entangled pure state by lifting it
into a larger space. This can be regarded as a feature of entangled states
combined with implementation strategies beyond previous static quantum
entanglement experiments. Our results are interesting in Bell theory,
theoretical computer science, and distributive computations.

## Experimental procedures

## Resource Availability

### Lead Contact

Further information and requests for resources should be directed to the lead
contact Ming-Xing Luo (mxluo@swjtu.edu.cn).

### Materials available

This paper did not generate new materials.

### Data and code availability

The experimental data are available from the corresponding author upon
reasonable request. This paper did not report original code.

## Experimental setup

We build the dynamical implementation of tripartite nonlocal game
$\mathcal{G}_{3}$ with a biseparable state by a spontaneous parametric down-
conversion (SPDC) source, which produces polarisation entangled photon pairs
to supply the entangled photonic link. The experimental setup is illustrated
in Figure 2. To create a probabilistic triangle network, a fixed EPR source
distributes pairs of two photons to Alice, Bob, and Charlie. The allocation of
these photon pairs to the parties is determined by using the QS based on QRNG
with uniform distribution.

The source of EPR states is generated by a cavity-stabilized Ti:sapphire pump
laser operating at a central wavelength of 405 nm. This laser generates signal
and idler photons with wavelengths symmetrically distributed around 810 nm
through the process of SPDC. The SPDC source comprises a 15*2*1 mm
periodically poled KTiOPO4 (PPKTP) crystal, which is cut in beam-like type-II
phase matching configurations, providing advantages in terms of high entangled
photon brightness (0.34 MHz) and efficient light collection ($60\%$).

To generate polarization-entangled photon pairs, we showcase a variation of
the bidirectionally pumped down-conversion source that utilizes a PSI. The
pump pulse is divided into two beams (horizontally and vertically polarized
pump) with equal power using QWPs, HWPs, a PBS, and a dual-wavelength
polarizing beam splitter. Each of these beams, with a pump power of 18 mW, is
used to stimulate the SPDC source to generate EPR states. A pump photon has an
equal chance of undergoing downconversion in the PPKTP crystal, resulting in
photon pairs with the polarization state $|H\rangle_{1}|V\rangle_{2}$. If the
pair is generated by the horizontally polarized pump component, it goes
through a dual-wavelength half-wave plate, which causes a rotation to the
state $|V\rangle_{1}|H\rangle_{2}$. After passing through the PSI, the second
particle of the output state undergoes a polarization rotation of $\pi/2$ at
the HWP. After compensating for spatial and temporal factors, the final state
of the photon pair is given by
$(|H\rangle_{1}|H\rangle_{2}+|V\rangle_{1}|V\rangle_{2})/\sqrt{2}$.

The qubits are encoded in the polarisation degree of freedom of the photons.
Following the rules of nonlocal game in SI, different inputs are associated
with different CHSH operators. The measurement device at each party is a
polarisation analysis system which consists of a QWP, an HWP, a PBS, an IF,
and a QD.

## Experimental method

Each party’s detector has three settings $x,y,z\in\\{0,1,2\\}$ with three
outcomes for each setting. For two measurement settings $\\{0,1\\}$, the
measurement bases corresponding to the detector settings of each party are
defined as

$\displaystyle
A_{0}=C_{0}=\sigma_{z},A_{1}=C_{1}=\sigma_{x},B_{y}=\frac{1}{\sqrt{2}}(\sigma_{z}+(-1)^{y}\sigma_{x}).$
(9)

For the measurement setting $x_{i}=3$, each party will output a bit based on a
classical random number generator. Here, the quantum measurement device for
each party is regarded as a black box which is controlled by a quantum
switcher. Especially, for a given input $x_{i}$, the quantum switcher will
transfer the state into the corresponding input channel of the measurement
device.

For the setting $z=2$, we evaluate the tripartite correlation $\langle
A_{x}B_{y}C_{z}\rangle=\langle A_{x}B_{y}\rangle\langle C_{z}\rangle$. This
allows us to evaluate the experimental correlators using the photonic
coincidence probabilities of measurements made locally by each pair of two
quantum observers on their respective subsystems. In our case, one is the
measurement outcomes on the signal and idler photons from the SPDC source, and
the other is from a classical random signal. Especially, we compute the
tripartite correlation as

$\displaystyle\langle A_{x}B_{y}C_{z}\rangle=\frac{\sum_{a,b=\pm
1}(-1)^{s(a,b)}N^{a,b}_{x,y}}{\sum_{a,b=\pm
1}N^{a,b}_{x,y}}\times\frac{\hat{N}^{+}_{2}-\hat{N}^{-}_{2}}{\hat{N}^{+}_{2}+\hat{N}^{-}_{2}}$
(10)

where $N_{x,y}^{a,b}$ denote photon coincidences, $s(a,b)$ denotes a function
equal to $1$ if there are odd number of $-1$, and 2 otherwise, and
$\hat{N}^{\pm}_{2}$ denotes the coincidence of classical signal which is
evaluated by using random number generator. Similar evaluations are for $x=2$
or $y=2$. With these correlations, we then calculate the measurement operator
(7) and winning probability.

Early Bell test experiments utilized the standard deviation as a metric to
gauge the statistical significance of a Bell inequality violation Aspect1981 .
This approach is constrained by two assumptions: the Gaussian distribution
assumption and the independence of each trial result. To address these
limitations, we adopt a statistical analysis that relies on the evaluation of
the $P$ value HTP .

## Acknowledgements

This work was supported by the National Natural Science Foundation of China
(Nos.62172341, 61772437, 12075159), Sichuan Natural Science Foundation
(No.2023JQ00447), Academy for Multidisciplinary Studies, Capital Normal
University, and the Academician Innovation Platform of Hainan Province.

## Author Contributions

Y.H. and M.X. conceived the study. Y.H. and M.X. designed the experiment. Y.H.
and X.Z. conducted the experiments. All authors wrote and reviewed the paper.

## Declaration of Interests

The authors declare no competing interests.

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# Atom-Orbital Qubit under Non-Adiabatic Holonomic Quantum Control

Hongmian Shui${}^{\color{blue}\ddagger}$ State Key Laboratory of Advanced
Optical Communication System and Network, Department of Electronics, Peking
University, Beijing 100871, China  Shengjie Jin${}^{\color{blue}\ddagger}$
State Key Laboratory of Advanced Optical Communication System and Network,
Department of Electronics, Peking University, Beijing 100871, China  Zhihan
Li${}^{\color{blue}\ddagger}$ State Key Laboratory of Advanced Optical
Communication System and Network, Department of Electronics, Peking
University, Beijing 100871, China  Fansu Wei  State Key Laboratory of Advanced
Optical Communication System and Network, Department of Electronics, Peking
University, Beijing 100871, China  Xuzong Chen  State Key Laboratory of
Advanced Optical Communication System and Network, Department of Electronics,
Peking University, Beijing 100871, China  Xiaopeng Li
<EMAIL_ADDRESS>State Key Laboratory of Surface Physics, Institute of
Nanoelectronics and Quantum Computing, and Department of Physics, Fudan
University, Shanghai 200438, China  Shanghai Qi Zhi Institute, Shanghai
200030, China  Xiaoji Zhou<EMAIL_ADDRESS>State Key Laboratory of Advanced
Optical Communication System and Network, Department of Electronics, Peking
University, Beijing 100871, China  Collaborative Innovation Center of Extreme
Optics, Shanxi University, Taiyuan, Shanxi 030006, China

###### Abstract

Quantum computing has been attracting tremendous efforts in recent years. One
prominent application is to perform programmable quantum simulations of
electron correlations in large molecules and solid-state materials, where
orbital degrees of freedom are crucial to quantitatively model electronic
properties. Electron orbitals unlike quantum spins obey crystal symmetries,
making the atomic orbital in optical lattices a natural candidate to emulate
electron orbitals. Here, we construct an atom-orbital qubit by manipulating
$s$\- and $d$-orbitals of atomic Bose-Einstein condensation in an optical
lattice. Noise-resilient single-qubit gates are achieved by performing
holonomic quantum control, which admits geometrical protection. We find it is
critical to eliminate the orbital leakage error in the system. Our work opens
up wide opportunities for atom-orbital based quantum information processing,
of vital importance to programmable quantum simulations of multi-orbital
physics in molecules and quantum materials.

Orbital degrees of freedom are essential to the quantitative description of
electrons in large molecules Bartlett and Musiał (2007) and solid state
materials Hwang  _et al._ (2012). The complex interplay of spin, charge, and
orbital, is key to the emergence of novel electron phenomena such as
multiferroics Hwang  _et al._ (2012), unconventional superconductivity Stewart
(2011), and exotic molecular spin filtering Li  _et al._ (2020); Liu  _et al._
(2021). Incorporating orbitals lies at the heart of accurate quantum chemistry
calculations Bartlett and Musiał (2007), and adds substantial computation
complexity in simulating many-body electron correlation. With quantum
computing, the overall computation complexity of quantum algorithms for
quantum chemistry calculation has been reduced to polynomial Aspuru-Guzik  _et
al._ (2005); McArdle  _et al._ (2020), which has triggered much recent
research effort on experimental demonstration of such quantum algorithms with
superconducting qubits Kandala  _et al._ (2017); Quantum  _et al._ (2020) and
trapped ions Hempel  _et al._ (2018); Nam  _et al._ (2020). However, using
these qubits to emulate electron orbitals meets experimental challenge of
expensive qubit encoding Kandala  _et al._ (2017) and demanding Hamiltonian
engineering to impose precise orbital symmetry McClean  _et al._ (2016), and
consequently the captured multi-orbital effect in the experiments is rather
limited Kandala  _et al._ (2017); Quantum  _et al._ (2020).

With ultracold atoms confined in optical lattices, the atom-orbital
wavefunction obeys the same crystalline symmetry as the electron orbitals Li
and Liu (2016), making atom-orbital an ideal qubit candidate to perform
programmable quantum simulations of electron orbitals in molecules and solid
materials. There has been fascinating progress accomplished in controlling
atom orbitals in optical lattices for quantum simulations of exotic
superfluids Isacsson and Girvin (2005); Liu and Wu (2006); Kuklov (2006);
Wirth  _et al._ (2011); Soltan-Panahi  _et al._ (2012); Jin  _et al._ (2021)
and topological quantum states Sun  _et al._ (2012); Li  _et al._ (2013); Liu
_et al._ (2014); Khamehchi  _et al._ (2016). However, their gate-based
universal quantum control is still lacking, leaving orbital qubit quantum
computing so-far unachieved.

Figure 1: Atom orbital qubit in a one-dimensional optical lattice. (a) The
Bloch wave function amplitude of the $s$, $p$, and $d$ bands at zero quasi-
momentum. The $s$\- and $d$\- states constitute our orbital qubit. The dashed
line represents the optical lattice potential. (b) The band gap structures in
our experimental optical lattice. The energy bands from bottom to top are $s$,
$p$, $d$, $f$ and $g$ bands respectively. (c) Measurement of relaxation time,
$T_{\rm relax}$, in the decay dynamics of $d$-band population. The red dots
correspond to the time evolution of the atom number $N_{d}$ in the $d$ band by
time-of-flight measurement. The blue line represents fitting to a function
$N_{d}(t)=N_{0}\exp(-t/T_{\rm relax})$. The relaxation time is determined as
$T_{\rm relax}=4.5$ ms. (d) Measurement of dephasing time, $T_{2}$, through
Ramsey interference. The red dots show the measured time evolution of the
proportion of atoms $P_{d}$ in the $d$-band by band mapping method. The blue
line represents a fitting function $P_{d}=A\exp(-t/T_{2})\sin(\omega
t+\phi)+B$, with the black line the envelope of the Ramsey oscillations. We
have a dephasing time $T_{2}=2.1$ ms by fitting.

In this Letter, we construct an atom-orbital qubit using $s$\- and
$d$-orbitals of a one-dimensional optical lattice. We have measured the
orbital relaxation time ($T_{\rm relax}$) and the dephasing time ($T_{2}$),
finding $T_{\rm relax}=4.5\pm 0.1$ ms (milliseconds), and $T_{2}=2.1\pm 0.1$
ms in our experiment. By programming lattice modulation, we reach universal
nonadiabatic holonomic quantum gate control Lewis and Riesenfeld (1969);
Güngördü  _et al._ (2014) over the atom-orbital qubit, which exhibits noise-
resilience against laser fluctuations due to geometrical protection. We
demonstrate the holonomic quantum control of Hadamard and $\pi/8$ gates, which
form a universal gate-set for single-qubit rotation. The lattice modulation
pulses are programmed to minimize orbital leakage error, which is the key to
reach high fidelity holonomic quantum control of atom-orbital qubit. We
implement quantum process tomography (QPT) on the orbital qubit to measure the
full density matrix, from which the obtained average gate fidelity is
$98.36(10)\%$.

Atom-orbital qubit.— Our experiment is based on a 87Rb Bose-Einstein
condensate (BEC) with $2\times 10^{5}$ atoms confined in a one-dimensional
(1D) optical lattice. The lattice potential takes a form of
$V_{p}(x)=V_{0}\cos^{2}(2\pi x/l)$, with $V_{0}$ the lattice depth, and $l$
the wavelength of the laser forming the lattice, which is $1064$ nm in this
experiment. The lattice depth $V_{0}$ is five times of the laser recoil energy
($E_{r}=h^{2}/2ml^{2}$), with $m$ the atomic mass. With lattice confinement,
atoms acquire orbital degrees of freedom, as each lattice site contains
localized Wannier orbitals—$s$, $p$, $d$, … Li and Liu (2016), whose quantum
superposition gives multiple Bloch bands. Having atoms condense at lattice
quasi-momentum $k=0$, the orbital quantum state is described by a density
matrix

$\textstyle\rho=\sum_{\nu\nu^{\prime}}\rho_{\nu\nu^{\prime}}|\nu\rangle\langle\nu^{\prime}|,$
(1)

where $|\nu=s,p,d,\ldots\rangle$ represents the Bloch mode at $k=0$ of the
$\nu$-orbital band. In order to control the orbital state, we apply lattice
modulation with a particular frequency $\omega$ to resonantly couple $s$ and
$d$ orbitals (see Fig. 1), which introduces an additional potential,

$\Delta V(x,t)=A\sin(\omega t+\varphi)V_{p}(x),$ (2)

with both of amplitude $A$, and phase $\varphi$ programmable in our experiment
(see Supplementary Material). With leakage to other orbitals neglected, the
system corresponds to a two-level system, defining our atom-orbital qubit,
with $|d\rangle$ and $|s\rangle$ orbital states identified as the qubit basis
states, $|0\rangle$ and $|1\rangle$. In our experimental lattice setup, the
energy gap between $s$\- and $d$-bands has a large detuning from the energy
gap between other orbitals except the gap from the $s$\- to $p$-band. Despite
the absence of the energy suppression of the $s$ to $p$ transition, this
transition is forbidden due to inversion symmetry present during the lattice
modulation. The transitions from the $s$-orbital to the undesired orbitals are
thus avoided, either by energy suppression or by symmetry.

For qubit initialization, we selectively load the atomic BEC into the lattice
using our previously developed shortcut preparation method Zhou  _et al._
(2018), with which we are able to initialize the qubit to an arbitrary state,
$\left\lvert\psi\right\rangle=\cos\theta\left\lvert 0\right\rangle+\sin\theta
e^{i\phi}\left\lvert 1\right\rangle$, within $250$ $\mu$s. For qubit readout,
we implement a time-of-flight quantum state tomography (TOFQST), which
extracts the full information of the density matrix $\rho_{\nu\nu^{\prime}}$
from the atomic momentum distribution (see Supplemental Material). In the
experiment, we prepare a set of six complementary initial states,
$\mathbb{S}\equiv\\{\left\lvert 0\right\rangle,\left\lvert
1\right\rangle,\left\lvert+\right\rangle=\frac{1}{\sqrt{2}}(\left\lvert
0\right\rangle+\left\lvert
1\right\rangle),\left\lvert-\right\rangle=\frac{1}{\sqrt{2}}(\left\lvert
0\right\rangle-\left\lvert
1\right\rangle),\left\lvert+i\right\rangle=\frac{1}{\sqrt{2}}(\left\lvert
0\right\rangle+i\left\lvert
1\right\rangle),\left\lvert-i\right\rangle=\frac{1}{\sqrt{2}}(\left\lvert
0\right\rangle-i\left\lvert 1\right\rangle)\\}$, whose measured state
fidelities are $99.96(2)\%$, $99.88(5)\%$, $99.30(11)$, $99.37(10)\%$,
$99.83(5)\%$, and $99.98(1)\%$. The averaged fidelity is $99.72(7)\%$.

By initializing to the excited $d$-orbital state, namely $|0\rangle$, we
measure the time ($t$) evolution of the atom number $N_{d}$ in the $d$ band,
and obtain $N_{d}(t)$. Fitting this function to $e^{-t/T_{\rm relax}}$, we
extract the relaxation $T_{\rm relax}$ time, which is $4.5\pm 0.1$ ms (Fig.
1(c)). To measure the dephasing time $T_{2}$, we prepare an orbital
superposition state $\left[|0\rangle+|1\rangle\right]/\sqrt{2}$. For the
energy splitting between $s$, and $d$ orbital bands in the lattice, this
superposition state develops Ramsey interference fringes in dynamics. The time
dependence of the off-diagonal term has a form of (Fig. 1(d))

${\rm Re}(\rho_{sd}(t))\sim e^{-t/T_{2}}\cos(\Omega t+\phi),$ (3)

with $\Omega$, $\phi$, $T_{2}$ determined by fitting the experimental data
gotten by band mapping method. We obtain a $T_{2}$ time of $2.1\pm 0.1$ ms.

Figure 2: Simulated time evolution of the $|0\rangle$ state on the Bloch
sphere under the holonomic $X$ gate. North and south poles of Bloch sphere
denote the $|d\rangle$ and $|s\rangle$ orbital states. The red curve shows the
ideal state trajectory under the holonomic control according to the idealized
qubit model (Eq. (4)). The blue (green) curve shows the state trajectory by
the multi-orbital numerical simulation, with (without) the orbital leakage
error eliminated.

Nonadiabatic holonomic orbital gate construction.— With the optical potential
in Eq. (2), we have a qubit control Hamiltonian $H(t)$ of the form,

$\displaystyle\frac{1}{2}\Delta\sigma_{z}+\frac{1}{2}\lambda\left(-\cos(\omega
t+\varphi)\sigma_{y}+\sin(\omega t+\varphi)\sigma_{x}\right),$ (4)

where $\Delta$ is the gap between the $s$ and $d$-bands at quasi-momentum
$k=0$ ($\Delta=5.23E_{r}$ in our experiment), and the induced coupling by the
lattice modulation is

$\textstyle\lambda=A\int dxV_{p}(x)\phi_{d}(x)\phi_{s}(x),$ (5)

with $\phi_{\nu=s,d}(x)$ the Bloch function of $\nu$-orbital band at $k=0$.
With a resonant coupling $\omega\approx\Delta$, the overall quantum gate
operation time is determined by $1/\lambda$. To avoid non-resonant transitions
to other bands, predominately the $g$-band, which has an energy gap from the
$s$-band, $\Delta_{sg}$ ($=16.73E_{r}\gg\Delta$), it is required that
$\lambda\ll|\Delta_{sg}-\omega|$, which sets an upper limit for the amplitude
of lattice modulation. For this requirement, it becomes more desirable to
construct nonadiabatic quantum gates, as the adiabatic control would be too
slow—the total adiabatic evolution time is required to be much longer than
$1/\lambda$ to maintain quantum adiabaticity.

We implement nonadiabatic holonomic orbital control based on a dynamical
invariant of the Hamiltonian in Eq. (4),
$I=(\Delta-\omega)\sigma_{z}+\lambda\left(-\cos(\omega
t+\varphi)\sigma_{y}+\sin(\omega t+\varphi)\sigma_{x}\right)$ Lewis and
Riesenfeld (1969); Güngördü  _et al._ (2014), with its instantaneous
eigenstates $|\psi_{\pm}(t)\rangle$. Through one period ($T=2\pi/\omega$) of
quantum evolution, an initial quantum state $\sum_{n}c_{n}|\psi_{n}\rangle$ is
transformed to
$\sum_{n}c_{n}e^{i(\gamma_{n}^{g}+\gamma_{n}^{d})}|\psi_{n}\rangle$, with the
accumulated geometrical phase
$\gamma_{n}^{g}=\int_{0}^{T}dt\left\langle\psi_{n}(t)|i\partial_{t}|\psi_{n}(t)\right\rangle$,
and the dynamical phase
$\gamma_{n}^{d}=-\int_{0}^{T}dt\left\langle\psi_{n}(t)|H(t)|\psi_{n}(t)\right\rangle$.
The wavefunction evolution in the dynamical invariant eigenbasis during the
holonomic quantum control resembles quantum adiabatic dynamics Lewis and
Riesenfeld (1969); Güngördü  _et al._ (2014). The geometrical phase only
depends on the solid angle enclosed by the evolution path of
$|\psi_{n}(t)\rangle$ on the Bloch sphere Berry (1984). The holonomic gate is
thus a generalization of adiabatic geometrical gate Zhu and Wang (2002). This
gate has intrinsic resilience against experimental control errors for
geometrical protection Johansson  _et al._ (2012); Berger  _et al._ (2013),
which has been demonstrated with liquid NMR Feng  _et al._ (2013); Li  _et
al._ (2017), solid-state Zu  _et al._ (2014); Arroyo-Camejo  _et al._ (2014);
Yale  _et al._ (2016); Sekiguchi  _et al._ (2017); Kleißler  _et al._ (2018);
Huang  _et al._ (2019), neutral atoms Zhao  _et al._ (2017); Liao  _et al._
(2019); Liu  _et al._ (2020), ions Duan  _et al._ (2001); Toyoda  _et al._
(2013a); Leibfried  _et al._ (2003); Ai  _et al._ (2020); Zhang  _et al._
(2021) and superconducting qubits Abdumalikov Jr  _et al._ (2013); Yan  _et
al._ (2019, 2019); Xu  _et al._ (2018, 2020); Zhang  _et al._ (2019); Zhao
_et al._ (2021). To exploit the geometrical protection, the dynamical phase
has to be cancelled, which corresponds to

$\textstyle\lambda^{2}+\Delta(\Delta-\omega)=0.$ (6)

This gives the holonomic gate condition Güngördü  _et al._ (2014), which is
explicitly satisfied in our lattice modulation design. The SU(2) rotation
through one period of lattice modulation is then
$U_{\beta\varphi}(T)=\sum_{\pm}e^{i\gamma_{\pm}^{g}}|\psi_{\pm}(0)\rangle\langle\psi_{\pm}(0)|$.
This single-qubit rotation can be rewritten as

$\textstyle
U_{\beta\varphi}=-e^{i\pi\sin\beta\left[-\sin\varphi\cos\beta\sigma_{x}+\cos\varphi\cos\beta\sigma_{y}+\sin\beta\sigma_{z}\right]}$
(7)

with $\beta$ determined by the lattice modulation
frequency—$\cos^{2}\beta=\Delta/\omega$ ($\beta\in[0,\pi/2]$), and the angle
$\varphi$ controllable by programming the potential (Eq. (2)). An arbitrary
target of holonomic SU(2) rotation, $U_{\rm target}$, is approached by
combining multiple periods ($M$ times) of lattice modulation, which gives a
concatenated unitary
$U=U_{\beta_{M}\varphi_{M}}U_{\beta_{M-1}\varphi_{M-1}}...U_{\beta_{2}\varphi_{2}}U_{\beta_{1}\varphi_{1}}$.
The parameter sequence ($\beta_{j},\varphi_{j}$) is determined by optimizing
the gate fidelity $F=|\operatorname{tr}\left(U^{\dagger}U_{\text{ target
}}\right)|/2$. We then obtain a control sequence of lattice modulation
frequency, amplitude, and phase, denoted by
$\Theta_{j}\equiv(\omega_{j},A_{j},\varphi_{j})$. With this scheme, we expect
gate fidelities above $99.999\%$ using $M\leq 5$ for $X$, $Y$, $Z$, Hadamard,
and $\pi$/8 gates under the idealized model in Eq. (4)).

Figure 3: Experimental realization of the holonomic quantum gates. (a) Experimental procedures of performing holonomic $X$ gate on the initial state, $\left\lvert-i\right\rangle$. The numbers on top of the square pulses in (a) represent the time durations in units of microseconds. (b) Time-of-flight patterns at different evolution time. The condensed numbers of atoms at momentum $0\hbar K$, and $\pm 2\hbar K$ are extracted following standard analysis in cold atom experiments. (c) Oscillation dynamics of the atom number proportion at $0\hbar K$ and $\pm 2\hbar K$. We average over four experimental runs for each data point, with the error bar denoting the standard statistical error. Here we set the lattice confinement $V_{0}=5E_{r}$.  Initial state | $|0\rangle$ | $|1\rangle$ | $|+\rangle$ | $|-\rangle$ | $|+i\rangle$ | $|-i\rangle$ | Average
---|---|---|---|---|---|---|---
X-gate | 99.92(3) | 99.08(8) | 98.87(11) | 93.28(34) | 98.75(9) | 99.13(7) | 98.17(16)
Y-gate | 99.19(13) | 99.71(6) | 98.29(13) | 96.91(20) | 98.73(11) | 99.15(8) | 98.66(13)
Z-gate | 97.87(14) | 99.97(3) | 97.47(18) | 94.83(29) | 99.14(11) | 97.75(16) | 97.84(17)
Hadamard | 98.24(18) | 98.38(11) | 99.45(7) | 97.90(20) | 99.08(8) | 96.77(23) | 98.30(16)
$\pi$/8-gate | 99.21(21) | 99.76(7) | 97.78(30) | 95.94(41) | 99.69(8) | 99.28(24) | 98.61(25)
Table 1: Measured fidelities of the experimental holonomic quantum gates by
TOFQST.

However, in comparing the results of the ideal model with a more precise
multi-orbital numerical simulation that incorporates all continuous degrees of
freedom of the lattice (Supplementary Material), we find significant
discrepancy. Fig. 2 illustrates one example of state evolution under a
constructed holonomic $X$ gate. In the example shown in Fig. 2(a), the gate
fidelity obtained by the multi-orbital numerical simulation is below $85\%$.
The sizeable difference from the ideal model is attributed to the leakage to
other unwanted orbitals. Even a small fraction of $g$-band population below
$5\%$ is found to strongly disturb the qubit evolution on the Bloch sphere
(Fig. 2(a)). To resolve the problem of orbital leakage, we develop an orbital
leakage elimination protocol (Supplemental Material), where the leakage error
is minimized by optimizing the control sequence $\Theta_{j}$. With the optimal
control sequence designed for holonomic $X$, $Y$, $Z$, Hadamard, and $\pi/8$
gates (Supplemental Material), the gate fidelity is improved to above $98\%$
in the multi-orbital numerical simulation. The orbital leakage elimination
protocol is implemented in our experiment for high fidelity realization of
holonomic gates.

Figure 4: Representations of the experimentally accomplished quantum
processes. (a) The schematic illustration of the quantum processes. The red
lines, gray arrows, and red dots, represent the rotation axis, direction and
angle of the corresponding SU(2) rotation, respectively. (b) Process matrices
of the implemented holonomic $X$, $Y$, $Z$, Hadamard, and $\pi/8$ gates by QPT
measurements.

Holonomic quantum orbital gate fidelities.— In the experiment, we implement
holonomic orbital gates taking the control sequences as designed with our
orbital leakage elimination protocol. The band population dynamics during the
gate operation has a very good agreement with our multi-orbital numerical
simulation (Supplementary Figure S5). Fig. 3 shows the experimental
realization of the holonomic $X$ gate. The atomic BEC is initialized in the
quantum state $|-i\rangle$. We then perform the holonomic $X$ gate control.
The occupation of momentum $\tilde{p}=0$ and $\tilde{p}=2\hbar K$ (with
$K=2\pi/l$) states are measured through time-of-flight (TOF), by which the
full orbital qubit state is mapped out through TOFQST (Supplemental Material).
This experimental procedure is repeated for all the initial states in
$\mathbb{S}$. The fidelity ($F$) averaged over the final states is
$98.17(16)\%$ (Table 1), as calculated from the overlap of density matrix
measured in the experiment ($\rho_{\rm exp}$) with the theoretical
construction ($\rho_{\rm th}$), i.e., $F=\left\lvert{\rm Tr}(\rho_{\rm
exp}\rho_{th}^{\dagger})\right\rvert/\sqrt{{\rm Tr}(\rho_{\rm exp}\rho_{\rm
exp}^{\dagger}){\rm Tr}(\rho_{\rm th}\rho_{\rm th}^{\dagger})}$. We also
implement the holonomic $Y$, $Z$, Hadamard, and $\pi/8$ gates in the
experiment, whose averaged final state fidelities are $98.66(13)\%$,
$97.84(17)\%$, $98.30(16)\%$, and $98.61(25)\%$.

One way to construct universal holonomic gate for single qubit rotation is to
combine Hadamard and $\pi/8$ gates Boykin  _et al._ (1999). The other is to
search for the control sequence numerically, whose universality is confirmed
through our multi-orbital numerical simulation. We randomly sample a hundred
Haar random SU(2) rotations, and find all can be constructed by the
concatenated unitary sequence with $M\leq 5$, producing final state fidelities
all above $98\%$ (Supplementary Figure S2).

To characterize the holonomic quantum gates directly, we further perform
quantum process tomography (QPT). We initialize the atomic BEC in the six
orbital states in $\mathbb{S}$, and map out the final quantum states following
the holonomic quantum gates by TOFQST. The $\chi$ matrices, which represent
the quantum gate operations, are reconstructed by the QPT method Nielsen and
Chuang (2010); Howard  _et al._ (2006). The results are shown in Fig. 4. For
completeness we also provide quantum process fidelities (QPFs) as defined by
$\left\lvert{\rm Tr}(\chi_{\rm exp}\chi_{\rm
th}^{\dagger})\right\rvert/\sqrt{{\rm Tr}(\chi_{\rm
exp}\chi_{exp}^{\dagger}){\rm Tr}(\chi_{\rm th}\chi_{\rm th}^{\dagger})}$,
with $\chi_{\rm exp}$ and $\chi_{\rm th}$, the $\chi$ matrix reconstructed
from experimental measurements and the theoretical expectation, respectively.
The measured QPFs are $98.47(9)\%$, $98.35(11)\%$, $97.81(13)\%$,
$98.53(8)\%$, $98.63(15)\%$, for the X, Y, Z, Hadamard, and $\pi$/8 gates in
the experiment.

Discussion.— We have constructed an orbital qubit using atomic Bose-Einstein
condensation in a one-dimensional optical lattice. The atom orbital qubit is
equipped with nonadiabatic holonomic quantum control, that shows noise
resilience against laser intensity fluctuations due to geometrical protection
(Supplementary Figure S3). In order to further improve the gate fidelity in
the experiment, we expect it is helpful to incorporate the inhomogenity
produced by the trap, which can be treated by integrating spin echo like pulse
or optimal control protocols Toyoda  _et al._ (2013b); Liu  _et al._ (2019);
Song  _et al._ (2020).

To achieve universal quantum computing with the orbit qubit setup, we still
need to construct two-qubit gates, which can be implemented through an atom
collision scheme (Supplementary Material). We expect the orbital quantum
control techniques developed here can be generalized to full quantum control
of all orbitals. For orbital transitions that respect parity symmetry, one may
still take the lattice modulation protocol but using different resonant
frequencies. To control orbital transitions violating parity symmetry would
require lattice shaking techniques Parker  _et al._ (2013). The orbital
lattice based quantum computing would provide novel opportunities for
programmable quantum simulations. For example, the demonstrated arbitrary
coupling between $s$ and $d$ orbitals may be used to perform programmable
quantum simulations of spin-orbital interaction analogue and the consequent
topological physics Li  _et al._ (2013).

Acknowledgements.— We acknowledge helpful discussion with Yidun Wan. This work
is supported by National Natural Science Foundation of China (Grants No.
61727819, 11934002, 91736208, and 11774067), National Program on Key Basic
Research Project of China (Grant No. 2016YFA0301501, Grant No.
2017YFA0304204), Shanghai Municipal Science and Technology Major Project
(Grant No. 2019SHZDZX01).

${}^{\color{blue}\ddagger}$ These authors contributed equally to this work.

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Supplementary Material

##  S-1 Experimental lattice setup

Our experiment is based on a Bose-Einstein condensate (BEC) of 87Rb atoms
confined in a harmonic trap, with trapping frequencies
[$(\omega_{x},\omega_{y},\omega_{z})=2\pi\times(24{\rm Hz},48{\rm Hz},55{\rm
Hz})$] along the three spatial directions. The BEC is loaded into a one-
dimensional lattice formed by counter-propagating laser beams along the $x$
direction. To achieve the holonomic quantum gate manipulation on the atom-
orbital qubit, on the one hand, the lattice laser is required to be rapidly
turned on and off in order to form square wave pulses in the state
preparation, on the other hand, the variation of lattice depth must accurately
follow the sinusoidal form in the gate operation. Our experimental setup is
carefully designed with two AOMs cooperating (Fig. S1) to improve control
precision. The AOM1 works with the laser power feedback control loop,
consisting of PD (Photodetector), AWG1 (Arbitrary wave generator), PI
(Proportional-Integral) feedback circuit, and variable power RF (Radio
frequency) source1. A voltage signal in PD for programming the lattice depth
$V(t)=V_{0}+\Delta V(t)$ is achieved by the diffracted light of AOM1, where
the time-dependent part $\Delta V(t)$ corresponds to a sinusoidal wave pulse.
AWG1 generates the control signal $V_{AWG1}\propto V(t)$. Both signals are
connected into PI circuit, and its output controls the power of RF source1 so
that the diffraction efficiency of AOM1 is delicately adjusted in order to
keep the value of $V_{AWG1}$ stable. The AOM2 works with the fast control
module, consisting of RF switch, RF source2 and AWG2 that generates square
wave pulses. According to RF source2, the RF switch can control AOM2 to
realize the sharp variation of lattice depth corresponding to square wave
pulses. With this feedback control design using two AOMs, the rise and fall
time is maintained below 100ns, the modulation amplitude uncertainty within
2$\%$ and the laser power fluctuation below $0.2\%@1$s.

Figure S1: Experimental set-up. The optical lattice is formed by 1064 nm
laser, and its optical power (lattice depth) is controlled by two AOMs. The
red solid line with arrow indicates the optical path and the yellow dashed
line shows the direction of electrical signal transmission.

##  S-2 Qubit initialization

We use a shortcut method Zhou  _et al._ (2018) for atom-orbital qubit
initialization. After preparation of an atomic Bose-Einstein condensate in a
harmonic trap, we load the atoms directly into the desired orbital state of
the optical lattice. For a given targeting quantum superposition state of
$|s\rangle$ and $|d\rangle$ orbitals, a pulse sequence of the lattice-forming
laser is designed to maximize the state preparation fidelity (see an example
in Fig. 3(a)). The shortcut method typically involves a few cycles of turn-on
and off of the laser intensity, with a time duration $\tau_{{\rm on},i}$, and
$\tau_{{\rm off},i}$ ($i$ labeling different cycles), respectively. The time
duration $\tau_{{\rm on},i}$ and $\tau_{{\rm off}_{i}}$ defining the pulse
sequence are variationally determined. The pulse sequences for preparation in
the initial states ($\mathbb{S}\equiv\\{\left\lvert 0\right\rangle,\left\lvert
1\right\rangle,\left\lvert+\right\rangle,\left\lvert-\right\rangle,\left\lvert+i\right\rangle,\left\lvert-i\right\rangle\\}$)
are provided in Table S1. This shortcut method is demonstrated an efficient
protocol for atom-orbital qubit initialization.

Initial state | $\tau_{{\rm on},1}$ | $\tau_{{\rm off},1}$ | $\tau_{{\rm on},2}$ | $\tau_{{\rm off},2}$ | $\tau_{{\rm on},3}$ | $\tau_{{\rm off},3}$ | $(\mu s)$
---|---|---|---|---|---|---|---
$|0\rangle$ | 64.5 | 40.7 | 62.1 | 41.4 |  |  |
$|1\rangle$ | 28.1 | 35.1 | 40.6 | 27.0 |  |  |
$|+\rangle$ | 31.0 | 7.7 | 1.5 | 41.1 | 12.1 | 8.7 |
$|-\rangle$ | 38.4 | 61.7 | 38.6 | 30.7 |  |  |
$|+i\rangle$ | 52.0 | 80.0 | 3.0 | 4.0 |  |  |
$|-i\rangle$ | 23.3 | 43.7 | 56.5 | 41.8 |  |  |
Table S1: The laser pulse sequences for atom-orbital qubit initialization.

##  S-3 Pulse sequence design

In constructing the atom-orbital qubit and the holonomic quantum control, we
have neglected atomic interactions, which is valid in a relatively shallower
lattice. The requirement of having unharmonic band structures to avoid
unwanted inter-band transitions also favors shallow lattice. At the same time,
the lattice cannot be too shallow for the corresponding coupling $\lambda$
would be too small to maintain the validity of the rotating wave approximation
as used in constructing the holonomic dynamical invariant (Eq. (4)). Taking
these constraints into account, we choose a moderate lattice depth of
$V_{0}=5E_{r}$. We have also tried other lattice depths in the experiment, and
the choice of $5E_{r}$ is found to perform the best.

Through numerical tests, we observe that a single holonomic unitary operation
given Eq. (6) is not adequate to produce universal SU(2) quantum gates. To
resolve this problem, we consider a concatenated unitary composed of a
sequence of holonomic unitary operations,

$U=U_{\beta_{M}\varphi_{M}}U_{\beta_{M-1}\varphi_{M-1}}...U_{\beta_{2}\varphi_{2}}U_{\beta_{1}\varphi_{1}}$
(S1)

in order to construct generic quantum gates. The control sequences
($\beta_{j},\varphi_{j}$) are obtained by optimizing the gate fidelity,

$F=|\operatorname{tr}\left(U^{\dagger}U_{\text{ target }}\right)|/2,$ (S2)

with $U_{\rm target}$ the targeting gate. Having ($\beta_{j},\varphi_{j}$),
the phase of the laser pulse in Eq.(2) is given, and the modulation amplitude
is given by

$\displaystyle A=\frac{|\Delta\tan\beta|}{\int
dxV_{p}(x)\phi_{d}(x)\phi_{s}(x)}.$ (S3)

Considering the physical setup, we have $A\in[0,1]$, this sets a constraint on
$\beta$, $\beta\in[0,0.2]$. In absence of the phase $\varphi$ controlability,
the lattice setup would be too restrictive to produce a proper holonomic
control unitary with a satisfactory fidelity (Eq. (S2)), which is the reason
for us to introduce the control over the phase $\varphi$ in the experiment.
Having both $\beta$ and $\varphi$ under control, we find it is adequate to
construct the holonomic $X$, $Y$, $Z$, Hadamard, and $\pi/8$ gates with $M\leq
5$. The corresponding control sequences are listed in Table. S2.

Table S2: Sequences of the holonomic quantum gates before the orbital leakage elimination. Each pulse is applied to the system for one entire period of time $(2\pi/\omega)$.  gate |  | pulse 1 | pulse 2 | pulse 3 | pulse 4 | pulse 5
---|---|---|---|---|---|---
X-gate |  $\omega$/(2$\pi\cdot$kHz) | 10.768 | 10.777 | 10.769 | 10.772 |
$A$ | 0.6332 | 0.6505 | 0.6351 | 0.6417 |
$\varphi$/rad | 1.5377 | 1.7139 | 1.5816 | 1.2258 |
Y-gate |  $\omega$/(2$\pi\cdot$kHz) | 10.634 | 10.736 | 10.892 | 10.998 |
$A$ | 0.2949 | 0.5708 | 0.8311 | 0.9697 |
$\varphi$/rad | 1.0246 | 0.0415 | 0.0118 | 0.0141 |
Z-gate |  $\omega$/(2$\pi\cdot$kHz) | 10.892 | 10.886 | 10.864 | 10.889 | 10.896
$A$ | 0.8318 | 0.8231 | 0.7913 | 0.8274 | 0.8375
$\varphi$/rad | 0.5454 | 0.7938 | 1.5684 | 2.3366 | 2.5971
Hadamard |  $\omega$/(2$\pi\cdot$kHz) | 10.815 | 10.783 | 10.770 | 10.796 | 10.836
$A$ | 0.7147 | 0.6611 | 0.6375 | 0.6828 | 0.7493
$\varphi$/rad | 2.3780 | 2.1068 | 1.5938 | 1.0489 | 0.6591
$\pi$/8-gate |  $\omega$/(2$\pi\cdot$kHz) | 11.023 | 11.023 | 11.023 | 11.023 | 11.023
$A$ | 0.9987 | 0.9987 | 0.9987 | 0.9987 | 0.9987
$\varphi$/rad | 1.4075 | 1.4860 | 1.5645 | 1.6431 | 1.7216

##  S-4 Time-of-flight quantum state tomography

In our experiment, the momentum distribution $n(\tilde{p})$ (with $\tilde{p}$
the momentum along the lattice direction) is obtained with a standard tight-
of-flight measurement. For the lattice BEC with orbital degrees of freedom,
the momentum distribution is related to the density matrix in Eq. (1) by,

$n(\tilde{p})\propto\sum_{\ell,\nu,\nu^{\prime}}\delta_{\tilde{p}l,4\ell\pi}\rho_{\nu\nu^{\prime}}\tilde{w}_{\nu}(\tilde{p})\tilde{w}^{*}_{\nu^{\prime}}(\tilde{p}),$
(S4)

with $\tilde{w}_{\nu}(\tilde{p})$ the Fourier transform of of the Wannier
function of the $\nu$-orbital band. The momentum distribution has coherent
peaks at $\tilde{p}=4\ell\pi/l$. In principle, the density matrix can be fully
reconstructed from the measured momentum distribution. But in practice, the
higher order momentum peaks with $|\ell|\geq 2$ have rather low signal-to-
noise ratio compared to $\ell=0,\pm 1$. Having the momentum distribution at
$\ell=0,\pm 1$ only is not adequate to map out the density matrix.

For quantum state tomography, we let the state to measured evolve in the
static lattice for a certain amount of time $t_{\rm evo}$. The time dependence
of the momentum distribution $n(\tilde{p};t_{\rm evo})$ is given by Eq. (S4)
with $\rho_{sd}$ ($\rho_{ds}$) replaced by $\rho_{sd}e^{-i\Delta\times t_{\rm
evo}/\hbar}$ ($\rho_{ds}e^{i\Delta\times t_{\rm evo}/\hbar}$). With
experimental data for $n(\tilde{p};t_{\rm evo})$, the full density matrix
$\rho_{\nu\nu^{\prime}}$ is mapped out by fitting. This quantum state
tomography scheme for orbital states is analogous to a quench protocol
developed for tomography of band insulator topology Hauke  _et al._ (2014).

##  S-5 Construction of random unitary gates

To justify the universality of holonomic gate construction for all single-
qubit gates, we randomly sample the SU(2) unitary gates, and obtain its
fidelity in atom-orbital gate realization through our multi-orbital simulation
method, which has been found to model the experiment quantitatively. For an
SU(2) unitary operation, parameterized as $e^{i\sigma({\theta,\phi})\beta/2}$,
with $\beta$ the rotation angle on the Bloch sphere, and $\theta$ and $\phi$
defining the rotation
axis—$\sigma(\theta,\phi)=\cos(\theta)\sigma_{z}+\sin(\theta)\left[\cos(\phi)\sigma_{x}+\sin(\phi)\sigma_{y}\right]$.
We randomly sample the angles according to a uniform distribution,
$\theta\in[0,\pi)$, $\phi\in[0,2\pi)$, and $\beta\in[0,2\pi)$. Through the
orbital leakage error elimination protocol, we confirm all these generated
random unitary gates can be performed with our atom-orbital qubit, with final
state fidelities all above $98\%$ (Figure S2).

Figure S2: Final state fidelities of $100$ random SU(2) rotations. We randomly
sample rotation axis (represented by the polar angle $\theta_{gate}$ and the
azimuth angle $\phi_{gate}$) and the rotation angle $\beta_{gate}$. For each
of these quantum gates, we theoretically design the holonomic quantum control
sequence and calculate the corresponding final state fidelity using our multi-
orbital numerical simulations. All the gates are achieved with the pulse step
$M\leq 5$. All fidelities are above $98\%$.  Figure S3: Comparison between
holonomic gates and non-holonomic dynamic gates. (a) Gate robustness against
laser fluctuation noise. The ratio ($\delta A/A$) represents the relative
fluctuation strength in the laser intensity. DC-noise is applied here to
account for long-term drift of laser intensities that may potentially happens
in experiments. The red points (blue squares) represent the experimental
fidelity of the holonomic gates (dynamic gates). The fidelity is obtained by
the QPT method. The red lines (blue lines) are the theoretical results
(simulated with the multi-orbital numerical method) for the fidelity of the
holonomic gates (dynamic gates). (a1) to (a5) correspond to the X-, Y-, Z-,
Hadamard-, and $\pi/8$\- gate, respectively. (b) Time sequence of the non-
holonomic dynamic gate. The dynamic gate is constructed by a series of square
wave pulses. The duration and interval for each pulse are designed and
optimized to realize different gates. The pulse sequence details are provided
in (c).

##  S-6 Robustness against laser fluctuations

Having robustness against noise in the quantum control protocol is critical to
reaching high quality quantum information processing. We characterize the
robustness of the holonomic quantum gates by deliberately introducing a slight
deviation on the optical lattice potential amplitude in Eq. (2) with an amount
of $\delta A$. The results of the robustness study are provided in Fig. S3. As
the lattice modulation amplitude varies up to $10\%$, we find the average
experimental fidelity of the holonomic gates remains above $90\%$. The
experimental results have a quantitative agreement with the multi-orbital
numerical simulation (Fig. S3(a)).

In order to further illustrate the advantage of the holonomic gates, we also
construct a non-holonomic dynamic gate, and show their comparison in Fig.
S3(a). As Fig. S3(b) shows, the non-holonomic dynamic gate is constructed by a
series of square wave pulses for the lattice depth, and its dynamic phase is
not cancelled. The time duration and interval in each pulse are optimized to
reach the targeting gate operation. The specific time sequences are shown in
Fig. S3(c), and the total time is close to the holonomic gate for a fair
comparison. Both gates can achieve high fidelity without the laser intensity
noise (when $\delta A/A=0$). However, by introducing laser intensity noise
with $\delta A/A\neq 0$, we find that the holonomic gates exhibit better
resistance against fluctuations. For example, the QPT fidelity of the
holonomic X-gate is above 92% when the fluctuation $\delta A/A$ reaches
$20\%$. In sharp contrast, the fidelity of the non-holonomic dynamic X-gate is
below $60\%$. The noise resilience of the holonomic quantum gates on orbital
qubits is thus confirmed in our experiment.

In principle, imperfect control of the pulse durations may also affect the
gate quality. But in optical lattice experiments, the control precision of
pulse duration is typically maintained below the level of $10^{-4}$, which
could only cause fidelity decreasing by about $0.01\%$ through our numerical
simulation.

Figure S4:  Leakage to the $g$-band. The red dashed and blue dashed lines
represent the modulation sequences of the Z gate with and without our orbital
leakage elimination scheme, respectively. The red solid and blue solid lines
illustrate the proportion of leakage to the g-band during the Z gate process
with and without the orbital leakage elimination. The red and blue arrows mark
the maximum proportion of the $g$-band for the two cases. Here we choose the
initial state to be $|1\rangle$.

##  S-7 Orbital leakage elimination

In the experiment, we observe that the orbital leakage error strongly
compromises the orbital qubit gate fidelity. Although we try to avoid the
transitions to the unwanted orbitals by energy suppression and symmetry, the
leakage error is still present and harmful. As an example, we show the process
of Z-gate operation on the initial state $|1\rangle$ in Fig. S4. Through our
multi-orbital numerical simulation, we find that $g$-band proportion reaches
above $5\%$, and the final state fidelity is about $71\%$.

In order to resolve the orbital leakage problem, we develop an orbital leakage
elimination technique, to be described in the next paragraph. With the orbital
leakage elimination technique, the proportion of the g-band drops down below
$1.3\%$ (Fig. S4), and the fidelity reaches above $99\%$ for the Z-gate
operation, in our numerical simulation. The orbital leakage elimination
technique is implemented in our experiment. This technique is the key for us
to reach the high gate fidelities as reported in the main text.

Here we describe the details of orbital leakage elimination technique. For a
target holonomic quantum gate, we first obtain the parameter sequence
($\theta_{j},\varphi_{j}$) by maximizing the idealized gate fidelity (the
index $j$ labels the lattice modulation periods). Then at each lattice
modulation period, the quantum state evolution $U^{\star}(t)$ ($t\in[0,T]$) is
fully specified by $\theta_{j}$ and $\varphi_{j}$. The orbital leakage
elimination scheme is to search for a most suitable choice of lattice
modulation pulse, i.e., ($\omega$, $A$, $\varphi$), which gives a quantum
evolution that best approximates $U^{\star}(t)$. In order to reach this
solution, we propose a loss function

$\displaystyle L(\Theta)$ $\displaystyle=$ $\displaystyle\int
dt\left\\{\Sigma_{nm\in\\{S,D\\}}|U^{\star}_{nm}(t)-U_{nm}(t;\Theta)|^{2}\right.+$
$\displaystyle\left.\Sigma_{n\in\\{S,D\\},m\notin\\{S,D\\}}|U_{nm}(t;\Theta)|^{2}+|U_{mn}(t;\Theta)|^{2}\right\\},$

where $\Theta$ represents $(\omega,A,\varphi)$, and $U(t;\Theta)$ the quantum
evolution matrix incorporating all orbitals in our calculation. Through
minimization, we obtain a most suitable $\Theta$, with which the leakage error
caused by the unwanted orbitals is minimized. The control sequences designed
with the orbital leakage error elimination protocol for $X$, $Y$, $Z$,
Hadamard, and $\pi/8$ gates are listed below as Table. S3.

gate |  | pulse 1 | pulse 2 | pulse 3 | pulse 4 | pulse 5
---|---|---|---|---|---|---
X-gate |  $\omega$/(2$\pi\cdot$kHz) | 10.768 | 10.777 | 10.769 | 10.772 |
$A$ | 0.4749 | 0.4555 | 0.5626 | 0.4765 |
$\varphi$/rad | 1.6142 | 1.6383 | 1.5791 | 1.2129 |
Y-gate |  $\omega$/(2$\pi\cdot$kHz) | 10.634 | 10.736 | 10.892 | 10.998 |
$A$ | 0.2027 | 0.3952 | 0.6141 | 0.7747 |
$\varphi$/rad | 0.6247 | 0.2082 | 0.1576 | 0.1429 |
Z-gate |  $\omega$/(2$\pi\cdot$kHz) | 10.892 | 10.886 | 10.864 | 10.889 | 10.896
$A$ | 0.6639 | 0.5293 | 0.5407 | 0.5680 | 0.5428
$\varphi$/rad | 0.7953 | 0.9442 | 1.5551 | 2.2610 | 2.4972
Hadamard |  $\omega$/(2$\pi\cdot$kHz) | 10.815 | 10.783 | 10.770 | 10.796 | 10.836
$A$ | 0.5153 | 0.5106 | 0.5445 | 0.5281 | 0.4986
$\varphi$/rad | 2.3395 | 2.0068 | 1.5531 | 1.1157 | 0.8583
$\pi$/8-gate |  $\omega$/(2$\pi\cdot$kHz) | 11.023 | 11.023 | 11.023 | 11.023 | 11.023
$A$ | 0.8012 | 0.8030 | 0.7970 | 0.8176 | 0.6218
$\varphi$/rad | 1.5058 | 1.4766 | 1.5636 | 1.6471 | 1.6892

Table S3: The control sequences of the holonomic quantum gates with the
orbital leakage elimination protocol. Each pulse is applied to the system for
one entire period of time $(2\pi/\omega)$.  Figure S5: Comparison between
experimental observation with numerical simulation for the quantum state
evolution during the holonomic quantum control. The dynamical process
corresponds to a holonomic $X$-gate on the initial state, $|0\rangle$
($d$-orbital state). (a) and (b) show the time evolution of the proportion of
the $d$-band population, $P_{d}$. (c) and (d) show the time evolution of the
phase difference between $|0\rangle$ ($d$-orbital state) and $|1\rangle$
($s$-orbital state). The red lines represent the ideal state evolution of the
holonomic control. The blue and green stars correspond to the experimental
results with and without the orbital leakage elimination protocol. The green
and blue lines are the corresponding simulation results.

##  S-8 Multi-orbital simulation method

In the experiment, we find the discrepancy in the simplified two-orbital model
in Eq. (4)—the simulated results have significant disagreement with the
experiment. We thus develop a multi-orbital simulation method, which treats
the non-interacting quantum dynamics exactly. Having atoms confined in a
dynamical potential $V(x,t)=V(t)\cos^{2}(Kx)$, the Bose-Einstein condensate
state evolves in a subspace spanned by plane wave states $|q\rangle$, with its
wave function $e^{2iqKx}$. In this subspace, the kinetic part of the
Hamiltonian is diagonal given by

$H_{K,qq^{\prime}}=\frac{\hbar^{2}}{2M}(2qK)^{2}\delta_{qq^{\prime}}.$ (S6)

The potential term is off-diagonal given by

$H_{P,qq^{\prime}}=\frac{1}{2}V(t)\times\left(\delta_{q,q^{\prime}+1}+\delta_{q,q^{\prime}-1}\right).$
(S7)

The quantum state evolution is described by a time-dependent column vector
$\psi(t)$, according to

$\psi(t+\delta t)=e^{-iH_{K}\delta t/2}e^{-iH_{P}(t+\delta t/2)\delta
t}e^{-iH_{K}\delta t/2}\psi(t)+{\cal O}(\delta t^{3}).$ (S8)

Since $H_{K}$ is diagonal, the multiplication of $e^{-iH_{K}\delta t}$ on a
vector can be efficiently implemented as multiplication of an element-wise
phase. The matrix $H_{P}$ is off-diagonal, but can be diagonalized by a
Fourier transformation. The matrix multiplication of $e^{-iH_{K}\delta t}$ can
be efficiently performed by combining with a Fast Fourier transformation
algorithm. Choosing a high-momentum cutoff $|q|<q_{\rm max}$, the computation
complexity of this multi-orbital simulation method is ${\cal O}(q_{\rm
max}\log q_{\rm max})$. The simulated results have very good agreement with
our experimental measurements (Fig. S5). With numerical simulation, we find
the orbital leakage error strongly damages the quantum gate operations. With
our orbital leakage elimination protocol, the time evolution taking the
corrected control pulses approaches to the ideal holonomic quantum dynamics.

##  S-9 Construction of multiple qubits and two-qubit operations

In our experiment, we have performed single qubit control based on the orbital
state of a BEC. To engineer scalable quantum circuits with the orbital qubit
setup, we need to introduce single-site controllablility, which is
experimentally accessible with quantum microscope techniques Wang  _et al._
(2016); Qiu  _et al._ (2020). To selectively control the orbital transition on
one targeting site, say at $x_{0}$, we propose to maintain the optical
potential approximately homogeneous elsewhere, and tune the gap between $s$
and $d$ orbitals at site $x_{0}$ far off-resonant from other sites by
manipulating the local potential using quantum microscope techniques. The
overall optical potential is now denoted as $\tilde{V}(x)$, and the gap
between $s$ and $d$ orbitals at the targeting site is $\tilde{\Delta}_{0}$.
The orbital transition at $x_{0}$ can then be controlled following the same
way as demonstrated experimentally in this work, namely by introducing a
modulating lattice potential,

$\Delta\tilde{V}(x,t)=A\sin(\omega t+\varphi)\tilde{V}(x)$ (S9)

The modulation frequency $\omega$ should be chosen to match the gap
$\tilde{\Delta}_{0}$. Working in the limit of one particle per lattice site,
the interaction effects during the control process of the single-qubit
rotation are negligible. The integration of quantum microscope techniques to
the present holonomic orbital qubit control would make an orbital-based
quantum computing platform with single-qubit addressibility.

The two-qubit gate can be achieved by adapting the atom collision scheme De
Chiara  _et al._ (2008); Jensen  _et al._ (2019); Mamaev  _et al._ (2020);
Hartke  _et al._ (2021). Considering two nearby sites, denoted as $a$ and $b$,
the orbital states are $|s_{a}s_{b}\rangle$, $|d_{a}d_{b}\rangle$,
$|s_{a}d_{b}\rangle$, and $|d_{a}s_{b}\rangle$. The local interactions
projected to this subspace give a vanishing contribution for each site only
contains one particle. The leading order interaction that contributes to the
dynamics in the two-qubit computation subspace is thus between neighboring
sites. Given the presence of a large gap between $s$ and $d$ orbitals, the
interaction processes that do not respect the conservation of single-particle
energy are negligible. The relevant interactions between neighboring sites are
then

$H_{\rm
int,nn}=U_{1}\hat{s}_{a}^{\dagger}\hat{s}_{b}^{\dagger}\hat{s}_{b}\hat{s}_{a}+U_{2}\hat{d}_{a}^{\dagger}\hat{d}_{b}^{\dagger}\hat{d}_{b}\hat{d}_{a}+U_{3}\left(\hat{s}_{a}^{\dagger}\hat{d}_{b}^{\dagger}\hat{d}_{b}\hat{s}_{a}+\hat{d}_{a}^{\dagger}\hat{s}_{b}^{\dagger}\hat{s}_{b}\hat{d}_{a}\right)+U_{4}\left(\hat{s}_{a}^{\dagger}\hat{d}_{b}^{\dagger}\hat{s}_{b}\hat{d}_{a}+\hat{d}_{a}^{\dagger}\hat{s}_{b}^{\dagger}\hat{d}_{b}\hat{s}_{a}\right)$
(S10)

where we have introduced orbital annihilation operators $\hat{s}_{a}$,
$\hat{s}_{b}$, $\hat{d}_{a}$, $\hat{d}_{b}$ on the two sites. For the
orthogonality between different wannier functions on each site, $U_{4}$ is
much smaller than $U_{3}$, and can be neglected. The interactions $U_{1}$ and
$U_{2}$ only contribute to the diagonal terms in the two-qubit computation
subspace. The $U_{3}$ process is more nontrivial, causing a swap operation
between $|s_{a}d_{b}\rangle$ and $|d_{a}s_{b}\rangle$. The swap operation is
completed with the interaction turned on for a time duration

$\tau_{\rm swap}=\pi\hbar/(2U_{3}).$ (S11)

A $\sqrt{\rm swap}$ gate is then reached by letting atoms interact for a time
duration

$\tau_{\sqrt{\rm swap}}=\tau_{\rm swap}/2.$ (S12)

This makes an entangling two-qubit gate, which is sufficient to make a
universal gate set by combining with single-qubit operations Nielsen and
Chuang (2002). In order to make an efficient $\sqrt{\rm swap}$ gate, it is
required to move the two sites closer than the lattice spacing by adiabatic
time evolution. This requires quantum microscope or sublattice techniques.