# A Plan for Electron Ion Collider in China

Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, 730000,
China
E-mail

###### Abstract:

One of the frontier research areas common to both nuclear and particle physics
is the study of hadron structure and the strong interaction. In this
contribution we will discuss a plan for a polarized electron-ion collider in
China (EicC) and its physics goals.

##  1 Introduction

Electron Ion Colliders (EICs) [1], regarded as a “super electron microscope”,
can provide the clearest image inside the nucleon. It is the most ideal tool
for us to understand the deep structure of the nuclear matter, especially the
quark-gluon structure of the nucleon and nuclei. With high luminosity, broad
kinematic range and various polarized ion beams, EIC will be superior to any
facility of the past and present time. Polarized EICs are the next generation
“multi-dimensional electron microscopes” that are most effective in studying
the deep structure and strong interactions of particles. Currently there are
several EIC plans over the world, such as electron Relativistic Heavy Ion
Collider (eRHIC) [2], JLab Electron Ion Collider (JLEIC) [3] in USA, and LHeC
[4] at CERN (unpolarized). The EIC plans have been discussed in many details
in the last decade which are summarized in the INT report [5] and the EIC
whitepaper [1]. Currently, there are no large-scale accelerator experimental
facilities for the study of nucleon structure physics in China.

##  2 EicC Plan and Its Physics Goals

Based on the HIAF (the Heavy Ion High Intensity Accelerator Facility, approved
in 2015 in China), the IMP is proposing to build a high luminosity polarized
EIC facility in China, EicC, to carry out the frontier research on nucleon
structure studies.

The EicC will be constructed in two stages, i.e. EicC-I and EicC-II, as
summarized in Table 1.

Accelerator | Electron beam | Proton beam | $\sqrt{S}$ | Luminosity ($\mathrm{cm}^{-2}\mathrm{s}^{-1}$)
---|---|---|---|---
EicC I | $3\sim 5$ | $12\sim 30$ | 12 $\sim 24$ | $4\times 10^{33}$
EicC II | $5\sim 10$ | $60\sim 100$ | 35 $\sim 63$ | $1\times 10^{35}$
Table 1: EicC-I and EicC-II accelerator design parameters. Both electron and
ion beams are polarized. Energy unit in GeV.

We compare the EicC with the existing or planned ep facilities in Fig. 1.

Figure 1: Compare the EicC with non-polarized and polarized electron-nucleon
machines. The values of x are also listed in the top of the figure. Figure 2:
Compare the kinematic ranges of EicC with JLab 12 GeV, for DIS process. Figure
3: Conceptual layout of EicC-I and EicC-II, based on HIAF.

In Fig. 2 we show the kinematic ranges for the EicC and JLab 12 GeV facility.
Here we can see that the EicC-I is the best window for sea quark and valence
quarks, and the EicC-II can explore gluons and sea quarks structures. The
center of mass energy $\sqrt{S}$ is about 16 GeV for the EicC-I. This will
provide a clean environment for studying the bottom-hadron, especially the
exoctic bottom-quark baryons productions within 2 $\times$ 10 -2 $<$ x $<$ 0.5
and 1 $<$ Q2 $<$ 100 GeV2. Unlike other EIC proposals [1, 2, 3, 4] where the
focus is at high energy and small-x in order to probe the gluon’s role to the
inner structure of nucleons, EicC will focus on valance- and sea-quark
contributions to nucleon structure and inter-nucleon effects.

The conceptual layout of EicC-I and EicC-II based on HIAF is presented in Fig.
3.

###  2.1 EicC-I Main Physics Goals

The EicC-I covers the range of sea quark, so it will offer the best chance to
measure precisely the sea quark 1D and 3D distributions. The EicC-I main
physics goals are:

  1. 1.

High precision quantitative measurements of sea quarks 1D structure

  2. 2.

High precision quantitative measurements of sea quark TMD

  3. 3.

High precision quantitative measurements of sea quark GPD

  4. 4.

Exploring hadronization mechanisms

  5. 5.

Systematic studies of baryon states with heavy flavor

####  2.1.1 Study of nucleon structure

EicC will perform extensive study on 1D and 3D structure of nucleon and
explore the fundamental questions the previous experiments have not answered
yet. High-precision measurements of the distribution functions of valence
quarks, sea quarks and gluons at EicC will uncover the internal structure of
nucleons and ultimately solve the puzzles about nucleon properties. EicC will
provide great opportunities to great discoveries in the area of nucleon
structure. Fig. 4 shows EicC-I (3.5 $\times$20 GeV) simulation of TMD Colins
function asymmetry uncertainty which is pretty small in both valence and sea
quark region.

Besides the physics listed above, EicC-I also has the potential to make
important contributions to other areas, such as the structure functions of
pion and kaon, EMC-SRC effects, etc.

Figure 4: Simulation of TMD Colins asymmetry uncertainty in 4D(x, z, PT and
Q2) for one year runing of EicC-I with energy 3.5 $\times$20 GeV.

####  2.1.2 Systematic Studies of Baryon States with Heavy Flavor

Study of the nucleon and its resonance is an important subject of hadron
physics. Tens of exotic states have been observed in the current machines, but
their nature is still unclear; some of them are only observed in a specific
process. Hence, further studies on the experiment side, are needed. Because of
its high energy, high luminosity and low background, especially, the polarized
EicC can be used to pin down the quantum number of the observed particles ,
EicC-I will be an excellent place to study those predicted penta-quark states
with hidden charm and hidden beauty baryons [7, 8, 9, 10, 11]. Fig. 5 shows
the total cross section of ep $\rightarrow$ ep$\Upsilon$ via the expected
hidden beauty baryon Pb (i.e. N${}^{*+}_{b\bar{b}}$) [7]. Meanwhile, EicC-I
can also be used to study baryon states with open heavy flavor and other
exotic particles.

Figure 5: Total cross section of ep $\rightarrow$ eN${}^{*+}_{b\bar{b}}$
$\rightarrow$ ep$\Upsilon$ [7]. The center mass of energy of EicC-I is greater
than 16 GeV.

Another topic is to study non-perturbative gluon. As we know, about 22$\%$ of
proton mass comes from trace anomaly, but so far we know very little about it.
JLab 12 [6] will measure the J/$\psi$ production near threshold (at small Q2).
With high center of mass energy, the EicC will offer precision measurement of
$\Upsilon$ production near threshold. the heavier mass of the bottom should
help to suppress the theoretical systematic uncertainties. This measurement
will shed light on the proton mass study. The third topic is that we can image
gluon GPD structures through electro-production of$J/\psi$ and $\Upsilon$ at
large value of invariant mass W [10, 12].

###  2.2  EicC-II Main Physics Goals

We plan to build EicC-II based on EicC-I by improving the energy and
luminosity. Besides higher-precision measurements of nucleon structures and
hadron spectroscopy, there are good prospects for searching for new physics
beyond standard model on EicC-II due to its high energy, high luminosity and
polarization of electron beam. Physics goals of EicC-II:

  1. 1.

Precise measurements of the multi-dimensional spin structure of gluons and sea
quarks in the nucleon.

  2. 2.

Search for new physics beyond standard model through precise measurements of
charged lepton flavor violation processes and the weak mixing angle.

##  3 Conclusion

In summary, EicC will provide an excellent infrastructure for study both sea-
quark and gluon contributions to the proton structure. EicC is complementary
to JLab 12 GeV and EIC physics programs in U.S. and Europe. The highly
polarized EicC plus the advanced high-tech detecting system will allow a
series of world-class precision measurements for nucleon structure related
physics. In addition to the advancement of the science, the EicC will also
greatly boost the technology development for the detector system. In the mean
time, a team of young scientists for both physics and technology will also be
trained for the future.

Acknowledgments: We are grateful for many enlightening discussions with
physics,detector and accelerator working group. Especially thanks to J. P.
Chen, N. Xu, Z. Yang and B. S. Zou for fruitful discussion and suggestion.

## References

  * [1] EIC-White Paper, Electron-Ion Collider: Next QCD Frontier,e-Print: arXiv:1212.1701 [nucl-ex].
  * [2] eRHIC Homepage, http://www.phenix.bnl.gov/WWW/publish/abhay/Home- of- EIC/.
  * [3] JLab EIC Homepage (https://eic.jlab.org/wiki/index.php/Main-Page).
  * [4] J. L. Abelleira Fernandez et al., A Large Hadron Electron Collider at CERN: Report on the Physics and Design Concepts for Machine and Detector, http://arxiv.org/abs/1206.2913.
  * [5] INT-Write-Up, Gluons and the quark sea at high energies: distributions, polarization, tomography, e-Print: arXiv:1108.1713 [nucl-th].
  * [6] JLab 12 GeV Upgrade Homepage: http://www.jlab.org/12GeV/; J.P. Chen, H. Gao, T.K. Hemmick, Z.-E. Meziani, P.A. Souder, the SoLID Collaboration, A White Paper on SoLID (Solenoidal Large Intensity Device), : arXiv:1409.7741 (2014).
  * [7] Jia-Jun Wu, Lu Zhao, B. S. Zou, Physics Letters B 709 (2012) 70.
  * [8] Y. Huang, J. He, H. F. Zhang and X. R. Chen, J. Physics G 41, 115004 (2014).
  * [9] Y. Huang, J.-J. Xie, J. He, X. Chen and H.-F. Zhang, Photoproduction of hidden-charm states in $\gamma$ p $\rightarrow$ $\bar{D^{*0}}$ $\lambda_{c}^{+}$ reaction near threshold, arXiv:1604.05969.
  * [10] S. Joosten, Z.-E. Meziani, Heavy Quarkonium Production at Threshold: from JLab to EIC，PoS QCDEV2017 (2018) 017, arXiv:1802.02616v2 [hep-ex].
  * [11] S. Joosten, Probing the gluonic structure of the nucleon through quarkonium production, arXiv:1803.08615v1 [hep-ph].
  * [12] M. Diehl, Proceedings of DIS 2012, arXiv:1206.0844.


# Running the Dual-PQC GAN on noisy simulators and real quantum hardware

Su Yeon Chang1,2 Edwin Agnew3 Elías F. Combarro4 Michele Grossi1 Steven
Herbert3,5 and Sofia Vallecorsa1 1 CERN Openlab, 1211 Geneva 23, Switzerland
2 Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), 1015
Lausanne, Switzerland  3 Cambridge Quantum, Cambridge, Terrington House, 13–15
Hills Road, CB2 1NL, United Kingdom  4 Department of Computer Science,
University of Oviedo, 33003 Oviedo, Asturias, Spain  5 Department of Computer
Science and Technology, University of Cambridge, Cambridge CB2 1TN, United
Kingdom<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In an earlier work [1], we introduced dual-Parameterized Quantum Circuit (PQC)
Generative Adversarial Networks (GAN), an advanced prototype of a quantum GAN.
We applied the model on a realistic High-Energy Physics (HEP) use case: the
exact theoretical simulation of a calorimeter response with a reduced problem
size. This paper explores the dual-PQC GAN for a more practical usage by
testing its performance in the presence of different types of quantum noise,
which are the major obstacles to overcome for successful deployment using
near-term quantum devices. The results propose the possibility of running the
model on current real hardware, but improvements are still required in some
areas.

##  1 Introduction

Quantum computing emerges as a promising technique to complement the
traditional ‘classical’ computing, thanks to its potential to speed up
computations or solve problems that classical algorithms cannot address [2].
However, current Noisy Intermediate-Scale Quantum (NISQ) devices suffer from
intrinsic noise from different sources which yield biased results. Quantum
Machine Learning (QML) via Variation Quantum Algorithms (VQA) is one of the
algorithms which can be successfully simulated with a reasonable circuit depth
in the presence of noise [3]. Still, the influence of noise is not negligible,
which is compounded by the vanishing gradient problem [4], therefore further
research is required to improve the performance of algorithms.

In order to understand the impact of quantum noise in QML, our work
specifically focuses on Dual-Parameterized Quantum Circuit (PQC) Generative
Adversarial Networks (GAN), which are a new prototype of quantum GANs
characterized by a classical discriminator and two quantum generators that
take the form of PQCs. In our earlier work [1], we have demonstrated that the
dual-PQC GAN is able to imitate reduced size pixelated images of calorimeter
outputs in High-Energy Physics (HEP), but only in the absence of noise.

This paper investigates the training of dual-PQC GAN in the presence of
quantum noise. We start by briefly summarizing the architecture of dual-PQC
GAN in Section 2. In Section 3, we test the dual-PQC GAN on noise models with
two-qubit gate errors only, which are the dominating factor in current
architectures. In particular, the impact of different training hyperparameters
is investigated within a wider range of errors with respect to that of current
real hardware. Finally, Section 4 presents the results obtained both on
superconducting and trapped-ion quantum hardware for the optimal
hyperparameters found in noisy simulations. Ultimately, our work aims to
provide a global overview of the effect of different types of noise in the
training of dual-PQC GAN and suggests realistic solutions to provide model
convergence.

##  2 Dual-PQC GAN

This section summarizes the architecture of dual-PQC GAN model, which is a new
type of quantum GAN that we introduced in a previous study [1]. It is a hybrid
qGAN architecture which has one classical discriminator and two parameterized
quantum circuits, PQC1 and PQC2, sharing the role of the generator.

Consider a training set $X\subset\mathbb{R}^{2^{n}}$ of $N=2^{n}$ pixel images
following a certain distribution. PQC1, with $n_{1}$ qubits, learns the
probability distribution over image samples by measuring its output state to
produce $n_{1}$ bits. This bit string is then used to initialise PQC2 (of
$n_{2}$ qubits) with the corresponding computational basis,
$\left|i\right\rangle\in\mathbb{R}^{2^{n_{1}}}$. By repeatedly measuring $n$
output qubits of PQC2, a probability distribution is constructed over $2^{n}$
computational basis states and interpreted as an image, $\mathcal{I}_{i}$, of
size $2^{n}$. The classical discriminator takes the training set and the
images generated by PQC2, and it classifies them into real and fake.
Ultimately, the dual-PQC GAN model can generate $2^{n_{1}}$ images of size
$2^{n}$.

input Fake Data by Sampling Classical Discriminator input Y1.0e0]ptReal Data
Predicted Labels Discard
${\left|0\right\rangle,...,\left|2^{n_{1}}-1\right\rangle}~{}~{}~{}~{}$
$\left|0\right\rangle^{\otimes n_{2}-n_{1}}~{}~{}~{}~{}$ PQC1 $n_{1}$ qubits
depth $d_{g,1}$ PQC2 $n_{2}$ qubits depth $d_{g,2}$ $n_{2}-n$ $n$
$n_{2}-n_{1}$ $n_{1}$ Classical OptimizationA

Y1.0e0]pt

A

Y1.0e0]pt

A

Y1.0e0]pt

Figure 1: Schematic diagram of dual-PQC GAN to reproduce images of $2^{n}$
pixels.

In Ref [1], we apply Dual-PQC GAN on a realistic HEP use-case to reproduce
calorimeter outputs of a reduced size, which are interpreted as pixelated
images. In order to facilitate the comparison between real and fake images, we
classify the training set into 4 classes via K-means clustering and average
over each class. Note that this classification is simply for the comparison,
and the whole set of images is used for the training.

In the simulations, the performance of the model is quantified with two
metrics:

  1. 1.

Relative entropy, $D_{KL}(\mathcal{I}_{mean}\|\tilde{\mathcal{I}}_{mean})$,
between the average of the real images, $\tilde{\mathcal{I}}_{mean}$, and the
generated images, $\mathcal{I}_{mean}$, with
$D_{KL}(p\|q)=\sum_{j}p(j)\log\frac{p(j)}{q(j)}$

  2. 2.

Individual relative entropy, the mean of the minimum relative entropy for each
of the generated images with respect to the real images:
$D_{KL,ind}=\frac{1}{2^{n_{1}}}\sum_{i=0}^{2^{n_{1}}-1}\min_{j}D_{KL}(\tilde{\mathcal{I}}_{j}\|\mathcal{I}_{i})$.

##  3 Hyperparameter scan on noise model

Before running the model on real hardware, we search for the optimal
hyperparameters using grid search over different factors: PQC1 and PQC2
learning rate, discriminator learning rate, and decay rate. The hyperparameter
scan was executed with the Qiskit noise model which mimics the noise of a real
quantum device, using the dual-PQC model with $n_{1}=2$, $n_{2}=4$, $d_{1}=2$
and $d_{2}=5$. As the readout error is negligible compared to the two-qubit
gate error in this configuration, only the latter is included in this noise
model.

(a) $p=0$

(b) $p=0.02$

(c) $p=0.04$

Figure 2: The relative entropy obtained from the hyperparameter scan with
different two-qubit gate error, $p$. The results with the lowest $D_{KL}$ and
$D_{KL,ind}$ at the end of the training are also displayed with two plots
overlapping on (c). Note that all the tests where the losses diverge are
excluded from the plot.

(a) $p=0$

(b) $p=0.02$

(c) $p=0.04$

(d) $p=0$

(e) $p=0.02$

(f) $p=0.04$

Figure 3: The mean image (a,b,c) and individual images (d,e,f) obtained at the
end of the Dual-PQC GAN training with the optimal hyperparameters, which give
the lowest $D_{KL,ind}$ for different two-qubit gate errors, $p$.

As shown in Figure 2, the standard deviation of the relative entropy increases
as the two-qubit gate error $p$ increases. But at the same time, we
consistently get hyperparmeters which make the training converge in all cases,
proving that the model can be trained even in the presence of noise under the
optimal choice of hyperparameters. The convergence in mean and individual
images obtained in the presence of different two-qubit gate errors shown in
Figure 3 also emphasizes this fact.

##  4 Results on real hardware

###  4.1 Inference on real hardware

Before training the dual-PQC GAN on real quantum hardware, we start by testing
the inference of the pre-trained model in order to check the impact of real
quantum noise, including two-qubit gate errors and readout errors, on the
generation of images. This also allows us to compare the performance of
currently accessible quantum device technologies: superconducting chips from
IBM [5], and ion-trap machines from IONQ [6]. Regarding IBM chips, qubits with
the lowest error rates are explicitly chosen from the IBM Quantum Lab while
the IONQ machines were accessed via Amazon Web Services (AWS) Braket.

(a)

(b)

(c)

(d)

Figure 4: Mean (a,c) and individual images (b,d) obtained by inference test on
ibmq_jakarta (a,b) and IONQ (c,d).

Figure 4 presents the mean and individual images obtained from inference tests
on real quantum devices. The pre-trained weights are taken from the test which
reproduced Figure 3(b) and 3(e). Looking at the low standard deviation suggest
the feasibility of dual-PQC training for superconducting chips, while more
studies are required with currently accessible trapped-ion devices. This
prediction is also confirmed in Table 1 which summarizes the error rates of
each devices and the result of inference tests.

Table 1: $D_{KL}$ and $D_{KL,ind}$ averaged over 20 inference tests on
different quantum hardware and their error rates [5]. The results obtained
with the two-qubit gate error only noise model is also displayed for
comparison.

Device | Readout error | CX error |  $D_{KL}$ ($\times 10^{-2}$) |  $D_{KL,ind}$ ($\times 10^{-2}$)
---|---|---|---|---
Noise Model | NULL | $2.00\cdot 10^{-2}$ | $0.07\pm 0.04$ | $5.54\pm 0.04$
ibmq_jakarta | $2.80\cdot 10^{-2}$ | $1.37\cdot 10^{-2}$ | $0.14\pm 0.14$ | $6.49\pm 0.54$
ibm_lagos | $1.15\cdot 10^{-2}$ | $5.58\cdot 10^{-3}$ | $0.26\pm 0.11$ | $6.92\pm 0.71$
ibmq_casablanca | $2.61\cdot 10^{-2}$ | $4.58\cdot 10^{-2}$ | $4.03\pm 1.08$ | $6.58\pm 0.81$
ibm_perth | $2.34\cdot 10^{-2}$ | $1.68\cdot 10^{-2}$ | $0.71\pm 0.19$ | $10.03\pm 0.65$
IONQ | NULL | $1.59\cdot 10^{-2}$ | $1.24\pm 0.74$ | $10.10\pm 5.62$
|  |  |  |

###  4.2 Training on real hardware

We conclude the study with the training on real quantum hardware. Figure 5
displays the result of dual-PQC GAN training on two real IBMQ machines,
ibmq_lagos and ibm_perth. The mean images and the progress in relative entropy
prove that the dual-PQC model can be successfully trained on average on both
devices with final $D_{KL}$ of $7.64\times 10^{-3}$ and $2.74\times 10^{-3}$,
respectively. Unfortunately, despite low $D_{KL,ind}$ of $1.29\times 10^{-2}$
and $3.56\times 10^{-3}$, respectively, the current training with real quantum
hardware falls into a mode collapse, one of the most common failures in
classical GAN [7], where the model only reproduces a low variety of samples as
shown on Figure 5(b) and 5(e).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5: The results of dual-PQC GAN training on ibm_lagos (a,b,c) and
ibm_perth (d,e,f). The convergence in mean image (a,d) and the decrease in
$D_{KL}$ show that the training is successful on average. However, the mode
collapse failure is observed for the individual images with an overlap between
$\mathcal{I}_{2}$, $\mathcal{I}_{3}$ on Figure (b) and $\mathcal{I}_{0}$,
$\mathcal{I}_{1}$ on Figure (e).

##  5 Conclusions

In this paper, we studied the impact of quantum noise on dual-PQC GAN
training. The results showed that with an appropriate choice of
hyperparameters, the dual-PQC GAN converges under real working conditions –
specifically in the presence of two-qubit gate noise – producing images close
to the real ones. Moreover, the convergence in mean image is observed on the
real quantum hardware training, yet the mode collapse failure emerges as an
issue to solve.

Future research will target methods to avoid the presence of mode collapse,
for example by adding an additional term to the loss function. Furthermore, an
increase in the size of training set in order to reproduce higher number of
images with more pixels will be investigated. This project is supported by
CERN’s Quantum Technology Initiative. We acknowledge the use of IBM Quantum,
and Amazon Web Services’ quantum computing service Amazon Braket for
implementing work described in this article. The views expressed are only
those of the authors, and do not reflect the official policy or position of
IBM, the IBM Quantum team, Amazon team or Amazon Web Services. One of the
authors was supported in part under grant PID2020-119082RB-C22 funded by
MCIN/AEI/10.13039/501100011033 and grant AYUD/2021/50994 funded by Gobierno
del Principado de Asturias (Spain).

## References

## References

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