# Machine-learning based noise characterization and correction on neutral
atoms NISQ devices

Ettore Canonici  Dept. of Physics and Astronomy & European Laboratory for Non-
Linear Spectroscopy (LENS), University of Florence, via G. Sansone 1, 50019
Sesto Fiorentino, Italy.  Stefano Martina  Dept. of Physics and Astronomy &
European Laboratory for Non-Linear Spectroscopy (LENS), University of
Florence, via G. Sansone 1, 50019 Sesto Fiorentino, Italy.  Riccardo Mengoni
CINECA, Via Magnanelli 6/3, 40033 Casalecchio di Reno, Bologna, Italy.
Daniele Ottaviani  CINECA, Via Magnanelli 6/3, 40033 Casalecchio di Reno,
Bologna, Italy.  Filippo Caruso<EMAIL_ADDRESS>Dept. of Physics and
Astronomy & European Laboratory for Non-Linear Spectroscopy (LENS), University
of Florence, via G. Sansone 1, 50019 Sesto Fiorentino, Italy.

###### Abstract

Neutral atoms devices represent a promising technology that uses optical
tweezers to geometrically arrange atoms and modulated laser pulses to control
the quantum states. A neutral atoms Noisy Intermediate Scale Quantum (NISQ)
device is developed by Pasqal with rubidium atoms that will allow to work with
up to 100 qubits. All NISQ devices are affected by noise that have an impact
on the computations results. Therefore it is important to better understand
and characterize the noise sources and possibly to correct them. Here, two
approaches are proposed to characterize and correct noise parameters on
neutral atoms NISQ devices. In particular the focus is on Pasqal devices and
Machine Learning (ML) techniques are adopted to pursue those objectives. To
characterize the noise parameters, several ML models are trained, using as
input only the measurements of the final quantum state of the atoms, to
predict laser intensity fluctuation and waist, temperature and false positive
and negative measurement rate. Moreover, an analysis is provided with the
scaling on the number of atoms in the system and on the number of measurements
used as input. Also, we compare on real data the values predicted with ML with
the a priori estimated parameters. Finally, a Reinforcement Learning (RL)
framework is employed to design a pulse in order to correct the effect of the
noise in the measurements. It is expected that the analysis performed in this
work will be useful for a better understanding of the quantum dynamic in
neutral atoms devices and for the widespread adoption of this class of NISQ
devices.

Machine Learning, Quantum Machine Learning, Quantum Noise, Quantum Noise
Spectroscopy, Quantum Noise Correction, Noisy Intermediate Scale Quantum
Devices, Neutral Atoms

##  I Introduction

In the last few years we are witnessing a revolution in the field of quantum
computing. The so called Noisy Intermediate Scale Quantum (NISQ) devices [1]
represent the state of the art in this field. The intermediate scale of such
devices refers to the fact that at the best of our technologies, we are still
capable of dealing with at most few hundreds of qubits. Several error
correction codes have been developed to deal with such noise [2, 3, 4], but
they require the adoption of auxiliary qubits further decreasing the resources
available for the computation. _Pasqal_ [5] has developed a NISQ device called
_Fresnel_ based on a neutral atom quantum processor capable of using up to 100
qubits [6] and provides a _Python_ library called _Pulser_ [7] that can be
used to prepare a setting either to run it on the real machines or to simulate
it on a built-in simulator.

Machine Learning (ML) is a field in the context of Artificial Intelligence
(AI) that deals with the study and realization of models that learn to make
predictions after being trained with data [8, 9]. Artificial Neural Networks
are ML methods organized in layers of artificial neurons that performs
calculations with weighted summation of the inputs followed by non-linear
activation functions. ML methods has already developed in the context of
quantum noise characterization [10, 11, 12, 13] and have already been adopted
in the context of error estimation. In [14] the authors train a recurrent
neural network to detect if certain errors happened in a quantum circuit and
use the model to enhance a surface error correction code. Surface error
correction codes allows an high error tolerance, however to be implemented
they need an high number of physical qubits [15]. By contrast, in our proposed
approach for noise mitigation, no additional qubits are needed for error
detection. In fact, our purpose is to learn how to modify the pulses in such a
way as to minimize the effect of noise without implementing error correction
codes. Moreover, we estimate the noise in devices with the analog interface
and not with the digital one. In fact, with neutral atoms devices it is
possible to take advantage of analog and digital modes. With the former, laser
pulses can be used to directly manipulate the Hamiltonian of the system:

$H=\frac{\hslash\Omega(t)}{2}\sum_{i}\sigma_{i}^{x}-\frac{\hslash\delta(t)}{2}\sum_{i}\sigma_{i}^{z}+\sum_{i<j}U_{ij}n_{i}n_{j}.$

With the digital mode, on the other hand, it is possible to evolve the state
of the system through quantum gates, thus creating quantum circuits. In [16]
the authors consider the noise to have the form of a Pauli channel and make
the assumption that the error rate is modeled with a Gibbs random field (GRF).
Those assumptions allows the authors to effectively learn the parameters of
the GRF to characterize the noise of a real IBM NISQ device. As discussed
below, in our work we use a different noise formalization, in fact we resort
on how the noise is implemented in the Pasqal simulator that we use to
generate the data to train the deep learning model.

RL is a ML methodology that requires the presence of a simulator of an
environment where an agent operates [17]. The agent is usually implemented as
a neural network that is trained to implement the policy that governs the
actions of the agent. Initially, for each episode (the elementary phase of
each RL algorithm that is repeated over time and is constituted of a series of
actions of the agent and reactions of the environment), the agent and the
environment are initialized in some initial state. Then, the agent perceive
some information about the environment and, based on that, the policy follows
a probability distribution of the possible next actions that the agent can
perform to change the state of the environment or the state of the agent
within the environment. The episode continue with the choosing of the best
action according to the policy and new steps until a predefined number of
steps or some episode-ending condition. RL have been already used in the
context of state preparation and circuit optimization [18, 19, 20]. In the
context of noise correction, RL have been adopted to correct the noise that
degrades a state over time [21] or to optimize existing quantum correction
codes [22]. In our work we instead focus on the task to correct the effects of
the noise of a defined quantum dynamics without modifying the base pulse.

Figure 1: Scheme of the noise estimation pipeline. A global pulse is defined
by the shapes of Rabi frequency $\Omega$ and detuning $\delta$ (a). A register
is prepared with the positions of a set of $n$ atoms (6 in the specific case)
that are irradiated by the laser pulse (b). When the pulse ends, the
excitation states of the atoms are measured and the process is repeated to
gather statistics on the occupation probabilities
$\bm{\mathcal{P}}=\mathcal{P}_{1},\dots,\mathcal{P}_{2^{n}}$ (c). The
probabilities are used as input to an Artificial Neural Network (ANN) that
predicts the noise parameters (d). The ANN is trained collecting a simulated
dataset of probabilities labelled with the corresponding values of noise. The
depicted setting is for the more general multiple parameters estimation. The
difference for the single parameter estimation is that the neural network have
only one output for $\sigma_{R}$ and the adopted pulses and atoms registers
are different.

##  II Noise benchmarking protocol

A setting consists in the topological arrangement of atoms and the description
of the laser pulses that interacts with them. Then, the computation on Quantum
Processing Units is structured in cycles of three phases: (i) the preparation
of the register, (ii) the quantum processing and (iii) the register readout.
In particular, on neutral atoms devices, the preparation of the register is
obtained using arrays of optical tweezers [23]. Initially the register is
initialized with atoms in random positions and afterward the single atoms are
moved in the desired positions. The quantum computation is performed
analogically using laser pulses that interact with the register atoms and can
excite them. The laser pulses are characterized by the values and shapes of
the Rabi frequency $\Omega(t)$ and detuning $\delta(t)$. Finally, the register
readout is performed by taking a fluorescence image to capture the energy
levels of the atoms. In Pasqal NISQ devices, it is possible to prepare
registers of maximum 100 atoms with a minimum distance of $4\mu m$ between
them arranged in bidimensional structures in an area of maximum radius $50\mu
m$.

NISQ devices, as the name suggests, are affected by several noise effects that
limit their applicability and the operations that can be reliably executed on
them. The devices used in the realization of a quantum computer are not ideal,
as they are affected by noise: for example, lasers are not exactly
monochromatic, and atoms are cooled by lasers to very low temperatures, but
still non-zero. These imperfections have an impact by introducing errors
during the preparation of the system, its evolution over time, and the
measurement. The effect is that the measured probabilities of occupation are
different from those we would have obtained from an ideal environment. In
general, there are different parameters that can be used to indicate different
sources of noise in the device [24]. In the present work we will focus on five
parameters that are considered predominant for their effects: the laser
intensity fluctuation $\sigma_{R}$ indicates the standard deviation of the
fluctuation of the desired Rabi frequency of the laser pulse; the laser waist
$w$ is the diameter of the gaussian laser beam; the false positive
measurements $\varepsilon$ represents the probability of wrongly measure as
excited an atom that was in the ground state; false negative measurements
$\varepsilon^{\prime}$ is the probability of measuring an excited atom in
ground state. Table 1 shows those sources of noise and their estimated values
provided informally by Pasqal.

Table 1: Summary of the main noise parameters with their respective values. We considered the parameters that are expected to have a predominant effect. Description | Parameter | Value
---|---|---
Laser Intensity fluctuation | $\sigma_{R}$ | $3\%$
Laser waist | $w$ | $68\mu m$
Temperature | $T$ | $30\mu K$
False positive measurement | $\varepsilon$ | $3\%$
False negative measurement | $\varepsilon^{\prime}$ | $8\%$

The objective of our work is the implementation of ML models to: (i) provide a
quantitative estimate of the noise; (ii) mitigate the effects of the noise. We
decided to formulate a supervised regression task to quantitatively estimate
the noise [16] and to use a Reinforcement Learning (RL) framework [17] to
mitigate the noise effect. Regarding the noise characterization, our aim is to
show that it is possible to estimate the noise parameters in the form of mean
values and error intervals. As depicted in fig. 1, the workflow begins with
the simulation of various executions, with different noise parameters, of a
quantum dynamic where a global pulse irradiates all the $n$ atoms of a
register. Afterward, the atoms occupation probabilities, that we call
$\bm{\mathcal{P}}=\mathcal{P}_{1},\dots,\mathcal{P}_{2^{n}}$, are collected
and used to train ANN models to predict the noise parameters that were used to
perturb the dynamics: _temperature_ , laser _waist_ , false positive
measurement rate _$\varepsilon$_ , false negative measurement rate
_$\varepsilon^{\prime}$_ and intensity fluctuation _$\sigma_{R}$_. At the end,
the trained models are used on prediction with the real data, obtaining an
estimation of the noise parameters. For the the simulations used in the
generation of the data and for the training of the models, we use our servers
with Nvidia TITAN RTX and GeForce RTX 3090 GPUs. Moreover we could also make
use of the CINECA Marconi100 supercomputer.

The rest of the paper is structured as in the following. First, in section
III.1 we consider the simpler problem of characterizing only a single noise
parameter, then in section III.2 we show the results of the characterization
of all the aforementioned parameters. In section IV we illustrate the RL error
correction protocol that we adopt.

##  III Noise characterization

###  III.1 Single parameter scenario

In this section we consider the estimation of a single noise parameter. After
preliminary analysis, we decided to focus on the noise effects that comes from
the laser intensity fluctuations $\sigma_{R}$.

Before describing the used methods, let us introduce the notation. We will
denote by $s_{i}$ the system composed of $i$ qubits. Globally, we consider
systems with a number of qubits from 2 to 5 and in the case of 4-qubit systems
we denote 6 different topologies with an extra alphanumeric index from $a$ to
$f$. Specifically, $s_{4a},s_{4b},\dots,s_{4f}$. Globally, we collected the
measurements of nine different runs on the real Pasqal NISQ devices (6
different topologies with 4 atoms and single topologies with 2,3 and 5 atoms)
characterized by a pulse with constant Rabi frequency $2\,\pi\,rad/\mu s$ of
duration $660\,ns$ and null detuning but with different number and positions
of the atoms.

In order to train the ML models to predict the values of $\sigma_{R}$, we
simulate the data for computation on the nine registers with different amount
of simulated noise effects. In detail, we preliminarily generate a sequence of
$10\,000$ $\sigma_{R}$ values extracted from a uniform distribution
$\mathcal{U}(0,0.15)$. These values are used to add noise in an equal number
of simulations, whose results are occupation probability vectors. Therefore,
in the end, $10\,000$ samples are obtained. This procedure is repeated for
each of the 9 quantum systems we mentioned above. The occupation probabilities
associated with the corresponding values of $\sigma_{R}$ for the 9 systems are
used to evaluate two different scalings: (i) in the quantum register size
comparing increasingly larger systems of 2,3,4 and 5 qubits and (ii) in the
number of measurements of multiple systems with 4 qubits where the occupation
probabilities of all the systems simulated with the same values of
$\sigma_{R}$ contributes to gather information on the noise effects during the
training of the ML models. In detail, we decided to use as input to the ML
models the concatenation of the probabilities of the systems and, for two
systems $s_{A}$ and $s_{B}$, we indicate the latter with the notation
$s_{A}\oplus
s_{B}=\mathcal{P}_{1,A},\dots,\mathcal{P}_{2^{n},A},\allowbreak\mathcal{P}_{1,B},\dots,\mathcal{P}_{2^{n},B}$.
In both scaling, the procedure is always the same: 20 models are trained on
each dataset through a 20-fold cross validation. From the 20 predicted
parameter values, the average value and the standard deviation can be obtained
to include the variability of the models’ predictions. Both analyses are
performed with linear regression as baseline model and with ANNs. Regarding
ANNs, they are trained for 150 epochs with the Adam optimizer and with
hyperparameter optimization. For a more in-depth discussion of the technical
details related to model design and hyperparameters optimization reefer to
section VI.1.

In the following, the ML models are trained and validated on the simulated
data, and subsequently they are also tested on the real measurements. Using
the simulated validation data, it is possible to monitor how the model is
capable of generalization to unseen measurements. In this regard, we report in
fig. 2 (the scaling (i) in fig. 2(a) and (ii) in fig. 2(b)) the Mean Absolute
Error (MAE), averaged for all the samples of the validation set, between the
predicted values of $\sigma_{R}$ and the ground truth that we recall is the
value of $\sigma_{R}$ used to perform the simulation. Again, having 20
estimates (one for each model), we calculate mean value and standard deviation
of the MAE to provide more robust results with associated uncertainty.
Regarding the estimation on the real data, we show in fig. 3 (fig. 3(a) for
the scaling (i) and fig. 3(b) for the scaling (ii)) the mean values and
standard deviations along the 20 models of the predicted values of
$\sigma_{R}$. In both fig. 2 and fig. 3, the result of the training of linear
regression models are depicted in black and the results of ANN in blue.
Additionally, in fig. 2(b) and fig. 3(b) we highlight in green for the linear
regression and in red for the ANN a specific case: the concatenation of the
measurements on two peculiar settings with four atoms, $s_{4a}$ and $s_{4b}$,
that have not only the same amount of atoms but also exactly the same
topology. Therefore, the latter can be seen as a special case of the scaling
(ii) where multiple measures of the same system are performed. Moreover, for
the real measurements we consider both orderings $s_{4a}\oplus s_{4b}$ and
$s_{4b}\oplus s_{4a}$ whose prediction results are reported with two couples
of green and red points in fig. 3(b) (not clearly visibles in the plot because
they are almost overlapping).

Figure 2: Scaling of single measurement for systems with an increasing number
of atoms (a) and scaling in the number of measurements for systems with four
atoms (b). We report the average absolute errors and standard deviations for
20 linear regression (in black and green) and 20 ANN (in blue and red) models
in the predictions of $\sigma_{R}$ on the synthetic validation set. The models
in (a) uses as input the measurements of $s_{2}$, $s_{3}$, $s_{4a}$ and
$s_{5}$. The models in (b) uses as input one or more concatenated measurements
of runs of the settings with four atoms (the fourth pair of points in (a) is
equal to the first pair in (b)). Indicating with $\cdot\oplus\cdot$ the
concatenation of the measurements of the settings, we report in (b) in black
and blue ${s_{4a},}\,{s_{4a}\oplus s_{4c},}\,{s_{4a}\oplus s_{4c}\oplus
s_{4d},}\,{s_{4a}\oplus s_{4c}\oplus s_{4d}\oplus s_{4e},}\,{s_{4a}\oplus
s_{4c}\oplus s_{4d}\oplus s_{4e}\oplus s_{4f}}$ and in green and red
${s_{4a}\oplus s_{4b}}$.

As expected, the prediction error is decreasing with the number of atoms in
the system because we get more information on the dynamic and thus on the
noise influencing it. In fig. 2 we can also observe that ANN are in general
more powerful respect to linear regression models (at the cost of more
resource-intensive computations). In fact, the errors for the ANN models are
always lower respect to the errors of linear regression models and the
difference is more pronounced increasing the number of atoms and measurements.
This can be explained with a better capacity of ANNs to model complex
dynamics.

Overall, comparing fig. 2(a) with 5 atoms and fig. 2(b) with number of
measurements equal to 2, seems to be more convenient to consider more
measurements respect to increase the number of atoms of the setting. Also,
comparing the green and red points with the black and blue ones for the same
number of measurements in fig. 2(b), can observe that can be slightly better
to consider multiple measurements of the same setting with the same topology
respect to collect measurements of a different setting with the same number of
atoms.

Figure 3: Predictions on real data of the value of $\sigma_{R}$ for the models
trained for the scaling in the number of atoms (a) and in the number of
measurements (b) reported in fig. 2. We report the average values and standard
deviations for the 20 linear regression (in black and green) and the 20 ANN
(in blue and red) models in the predictions of $\sigma_{R}$ using a set of
real measurements of the settings described in table 3 run on the Pasqal NISQ
devices. The models in (a) uses as input the measurements of $s_{2}$, $s_{3}$,
$s_{4a}$ and $s_{5}$. The models in (b) uses as input one or more concatenated
measurements of runs of the settings with four atoms (the fourth pair of
points in (a) is equal to the first pair in (b)). We report in (b) in black
and blue the incremental concatenation of $s_{4a}$, $s_{4c}$, $s_{4d}$,
$s_{4e}$ and $s_{4f}$. In green and red we report the concatenation of
$s_{4a}$ and $s_{4b}$. The order of the real measurements for the latter
concatenation is irrelevant, thus we report two green and two red points
(almost overlapping and not clearly discernible) to consider the two possible
concatenations. The horizontal red line indicates the value of $3\%$ for
$\sigma_{R}$ estimated by Pasqal.

We observe in fig. 3 that the values of $\sigma_{R}$ predicted for the
measurements of settings with $2$ and $5$ atoms are close to the estimated
value of $3\%$, however the prediction for the setting of $3$ atoms is lower
and the predictions for all the settings with $4$ atoms, and concatenation of
them, are around $7\%$. An explanation for this mismatch can be that the real
data used for the experiments was collected when the device was still under
development. Moreover the predictions consider only $\sigma_{R}$ as a variable
source for the noise, thus variations of the other noise parameters in the
real machine influence the predictions of $\sigma_{R}$. Nevertheless, it is
remarkable that the trained models have low standard deviations for the
predictions that, even if this does not exclude an high bias error, still
suggest a low variance error for the models. We can also observe that the
order of the measurement for the settings $s_{4a}$ and $s_{4b}$ do not
influence the predicted values – in fact the two green circles and the two red
circles in fig. 3 are almost overlapping.

To summarize, noise estimation based on supervised learning is possible. The
protocol we presented seems to suggest merging data from multiple similar
registers instead of larger registers directly. This may be useful because of
the difficulty in simulating larger systems. In addition, the estimates
obtained are derived by averaging estimates from 20 models. Moreover, the
associated standard deviation is small relative to the predicted value, so all
20 models converge to very similar values. Finally, we repeat that having
neglected several noise sources, the parameter values found could be effective
values.

###  III.2 Multiple parameters characterization

In this section we train a deep learning model in a multioutput regression
setting to estimate the values of all the noise parameters in table 1. We
simulated a dataset of $54\,000$ labelled samples for the 6-qubit system whose
topology can be observed in fig. 1(b). The used pulse sequence that defines
the dynamics is shown in fig. 1(a). Analogously to the scaling experiments in
the previous section, the measurement for each simulation is obtained sampling
500 runs. The values used in the simulations for each parameter are:
$\sigma_{R}=\mathcal{U}(0,0.15)$, $w(\mu m)=\mathcal{U}(0,200)$, $T(\mu
K)=\mathcal{U}(0,100)$, $\varepsilon=\mathcal{U}(0,0.15)$ and
$\varepsilon^{\prime}=\mathcal{U}(0,0.15)$.

After finding the best set of hyper-parameters, 20 models are trained using
the cross validation procedure to exploit the entire dataset and to obtain the
standard deviations of the predictions. Each one of the 20 models is trained
with early stopping for a maximum of 150 epochs. Further technical details
related to ANNs design and hyperparameter optimization can be found in section
VI.2.

Table 2: Predicted values on real data expressed as average and standard deviation of 20 models trained on cross validation. The last column report for practicity the same estimated values of table 1. Parameter | Predicted value | Estimated value
---|---|---
$\sigma_{R}$ | $0.079$ | $\pm$ | $0.005$ | $0.03$
$w$ | $122\mu m$ | $\pm$ | $6$ | $68\mu m$
$T$ | $56\mu K$ | $\pm$ | $4$ | $30\mu K$
$\varepsilon$ | $0.082$ | $\pm$ | $0.010$ | $0.03$
$\varepsilon^{\prime}$ | $0.078$ | $\pm$ | $0.005$ | $0.08$

In table 2 we show the resulting estimation of the main noise factors. Each
reported value is the average of the 20 models trained on different splits
with the corresponding standard deviation. We observe that the predicted
values do not match those estimated by Pasqal, although all 20 models always
converge to very similar values of the predictions. In this regard, the same
considerations expressed at the end of section III.1 are also valid for multi-
parameter estimation: ie, that the parameter predictions obtained could be
effective values that incorporate other neglected effects (noise sources,
influence of other neighboring atoms, etc.). Another possible factor could be
that the measurements came from a prototype NISQ, just as in the case of those
used in section III.1. Therefore, we can expect more agreement in the future
as a result of technical improvements. Moreover, it is worth noting that, even
if for the experiments in this section the setting and the pulse are different
to the ones used in section III.1, the predicted value for $\sigma_{R}$ is
comparable to the ones obtained for the estimation of the same parameter in
the settings with four atoms previously illustrated.

##  IV Error correction

Many techniques have been developed in the theory of classical error-
correcting codes [25, 26]. The key idea on which they are based is mainly
redundancy. Nonetheless, the addition of redundancy is not immediate in NISQ
devices because of the no cloning theorem [27]. However, some sort of
redundancy can be achieved in quantum devices by expanding the system to more
qubits [28]. In fact, all the most used quantum error correction techniques
require the use of more qubits than the ones strictly necessary for the
computation [29] but it is not feasible with NISQ devices. Therefore, we
propose to verify that it is possible to mitigate the effects of quantum noise
without extra qubits through the use of RL techniques. RL is a ML area where
an agent learns which actions to perform in order to maximize a reward [17].
Schematically, we can say that this is a closed-loop problem because the
actions of the learning system influence subsequent inputs. In addition, the
learner does not know a priori which action to perform and has to find out for
himself through trials and errors which actions lead to larger rewards.
Actions can influence not only the immediate reward but also future rewards.
RL, unlike Supervised Learning, does not require labelled input-output pairs,
but focuses on finding a balance between exploration of the actions space in
an environment and exploitation of the acquired knowledge. The agent must
exploit what it already knows in order to obtain reward, but it must also
explore in order to make better action selections in the future. The trade-off
is that neither exploration nor exploitation can be exclusively pursued
without failing in the task. The agent must try a variety of actions and
progressively favour those that seem to be the best. Any problem of learning
goal-oriented behaviour can be reduced to three signals that are exchanged
between an agent and its environment: a signal to represent the choices made
by the agent (the actions), a signal to represent the basis on which the
choices are made (the states) and a signal to define the agent’s goal (the
rewards). In detail, for each action of the agent at time $t$, its effects on
the environment are quantified by a reward $r_{t}$. Then the objective of the
training is to maximize the discounted cumulative reward
$R_{t_{0}}=\sum_{t=t_{0}}^{\infty}\gamma^{t-t_{0}}r_{t}$, where the discount
$\gamma\in(0,1)$ is an hyperparameter that controls the importance of rewards
far in the future respect to the ones immediately after $t_{0}$. This
objective is implemented with the idea that if we would have a function
$Q^{*}:State\times Action\rightarrow\mathbb{R}$ that given a state and an
action performed over that state, returns the cumulative discounted reward,
then the policy can be implemented with
$\pi^{*}(s)=\operatorname*{arg\,max}_{a}Q^{*}(s,a)$. In general, $Q^{*}$ is
unknown and is approximated by a neural network. For a defined policy $\pi$,
the $Q$ function obeys the Bellman equation $Q^{\pi}(s,a)=r+\gamma
Q^{\pi}(s^{\prime},\pi(s^{\prime}))$ where $r$ and $s^{\prime}$ are
respectively the reward and the next state obtained after the action $a$ on
the state $s$. The neural network that defines $Q$, and then the agent, is
trained minimizing over a batch of transitions the Huber loss
$\mathcal{L}(\delta)$ of the temporal difference error
$\delta=Q(s,a)-(r+\gamma\max_{a}Q(s^{\prime},a))$.

Figure 4: Standard pulse $P$ (a) to be corrected with a correction pulse
$P^{\prime}$ (b) to be added after $P$ to counteract the effects of the noise.
The Rabi frequency $\Omega$ is depicted in green and the detuning $\delta$ in
purple. $P$ is a pulse of duration $T=500ns$, Gaussian Rabi profile with area
equal to $\pi/2$ and detuning in the form of a ramp from $\delta_{0}=-20$
rad/$\mu s$ and $\delta_{T}=20$ rad/$\mu s$. $P^{\prime}$ is a pulse with the
same duration and characteristics of $P$ but with variable Rabi area $a$,
initial detuning $\delta_{i}$ and final detuning $\delta_{f}$.

We choose to correct the standard impulse $P$ depicted in fig. 4(a) applied to
a single qubit. $P$ has a Gaussian profile in the Rabi frequency $\Omega$ of
duration $T=500$ $ns$ and area $\pi/2$ and a ramp profile in detuning $\delta$
of duration $T=500$ $ns$ with $\delta_{0}=-20$ rad/$\mu s$ and $\delta_{T}=20$
rad/$\mu s$. The choosen approach to correct the noise is to apply the
correction pulse fig. 4(b) to be placed after the pulse to be corrected and
having the same characteristics and length of $T=500$ $ns$. In detail, we
choose a Gaussian profile in the Rabi frequency with variable area $a$ and a
ramp profile in detuning $\delta$ with variable initial $\delta_{i}$ and final
$\delta_{f}$. In such a way, the final atoms occupation probabilities with the
application of the corrected pulse $\bm{\mathcal{P}}_{P+P^{\prime}}^{noisy}$
and after the ideal pulse $\bm{\mathcal{P}}_{P}^{ideal}$ are _closer_ than
$\bm{\mathcal{P}}_{P}^{noisy}$ and $\bm{\mathcal{P}}_{P}^{ideal}$. By the
notation $\bm{\mathcal{P}}_{j}^{i}$ we denote the measurement
$\bm{\mathcal{P}}$ obtained after running a simulation with the pulse $j$ with
or without noise (respectively, $i=noisy$ or $i=ideal$). The training allows
to find the three optimal parameters $a$, $\delta_{i}$ and $\delta_{f}$ for
the correction impulse $P^{\prime}$.

In our RL framework, the state is represented by the occupation probabilities
that are estimated from the average of $10$ independent noisy simulations
whose probabilities are extracted from the amplitudes of 25 quantum states
uniformly sampled along the simulated dynamic. At the beginning of each
episode we choose $a=\pi/20$ and $\delta_{i}=\delta_{f}=0$ and they can have
values in the ranges $a\in[0,\pi/2]$ and $\delta_{i},\delta_{f}\in[-20,20]$.
The agent, implemented with an ANN that have an input layer of 50 units (2
basis for each one of the 25 intermediate states), two ReLU hidden layer of
128 neurons and an output layer of 6 neurons, selects one among four possible
actions: $a^{t}=a^{t-1}+\Delta a$, $a^{t}=a^{t-1}-\Delta a$,
$\delta_{i}^{t}=\delta_{i}^{t-1}+\Delta\delta_{i}$,
$\delta_{i}^{t}=\delta_{i}^{t-1}-\Delta\delta_{i}$,
$\delta_{f}^{t}=\delta_{f}^{t-1}+\Delta\delta_{f}$,
$\delta_{f}^{t}=\delta_{f}^{t-1}-\Delta\delta_{f}$. We choose fixed values for
$\Delta a=\pi/200$ and $\Delta\delta_{i}=\Delta\delta_{f}=0.2$. Each episode
is constituted of a series of steps at increasing values of $t$. For each
step, the chosen action is applied, a correction impulse $P_{t}^{\prime}$
characterized by $a^{t}$, $\delta_{0}^{t}$ and $\delta_{f}^{t}$ is generated
and used in a new simulation obtaining a new probability vector
$\bm{\mathcal{P}}_{P+P_{t}^{\prime}}^{noisy}$ for the final quantum state of
the corrected noisy simulation and the reward $r(t)$ before proceeding with
the next step. The episode ends when the action causes $a$, $\delta_{0}$ or
$\delta_{f}$ to go out of boundaries or after 100 steps. The reward is defined
as:

$r(t)=\begin{cases}1&\text{if
}\quad\left|\bm{\mathcal{P}}_{P+P_{t}^{\prime}}^{noisy}-\bm{\mathcal{P}}_{P}^{ideal}\right|_{1}\
<\
\left|\bm{\mathcal{P}}_{P+P_{t-1}^{\prime}}^{noisy}-\bm{\mathcal{P}}_{P}^{ideal}\right|_{1},\\\
0&\text{otherwise},\end{cases}$ (1)

where $|\cdot|_{1}$ is the $\ell_{1}$ norm. Specifically, the reward is $1$ if
the last action at step $t$ makes the corrected noisy simulation closer to the
ideal one respect to the previous step $t-1$ and $0$ otherwise. During the
training we monitor the Kullback–Leibler (KL) divergence between
$\bm{\mathcal{P}}_{P+P_{t}^{\prime}}^{noisy}$ and
$\bm{\mathcal{P}}_{P}^{ideal}$:

$D_{KL}(\bm{\mathcal{P}}_{P+P_{t}^{\prime}}^{noisy},\bm{\mathcal{P}}_{P}^{ideal})=\sum_{i=1}^{2}\left(\bm{\mathcal{P}}_{P+P_{t}^{\prime}}^{noisy}\right)_{i}\log\left(\frac{\left(\bm{\mathcal{P}}_{P+P_{t}^{\prime}}^{noisy}\right)_{i}}{\left(\bm{\mathcal{P}}_{P}^{ideal}\right)_{i}}\right),$
(2)

averaged for all the steps $t$ within each episode. The evolution of the
averaged KL divergence for the $1\,000$ training episodes is reported in fig.
5 where we can observe that it effectively decreases below the reference value
of $D_{KL}(\bm{\mathcal{P}}_{P}^{noisy},\bm{\mathcal{P}}_{P}^{ideal})=0.0011$
reported with the red line and calculated with the average for $100$ noisy
simulations without the correction pulse.

Figure 5: Evolution of the KL divergence between the corrected noisy
simulation and the ideal one averaged for each episode. The red line is the
reference value of $0.0011$ for the KL divergence between the uncorrected
noisy simulation and the ideal one averaged over $100$ simulations.

##  V Conclusions and outlooks

We presented two applications of ML to the context of quantum noise
characterization and correction. To characterize the noise we collected a
dataset of multiple simulated noisy measurement of different settings in
Pasqal quantum machines to train ML models and we test them on real data. For
the noise correction we trained a RL model to find a correction pulse to
counteract the effects of the noise affecting a simulated test setting.
Regarding the noise characterization, we compared ANN with linear regression
models in predicting the value of the laser intensity fluctuation
$\sigma_{R}$, scaling the number of qubit in the register and the number of
measurements of the system. We found that ANN perform better than linear
regression and that the model accuracies increases both with the number of
qubits and with the number of measurements. Moreover, we have insights that in
order to better characterize the noise parameters it is more effective to
increase the number of measurements respect to the number of qubits. When we
tried to predict the noise parameters on real NISQ devices we found that, for
every set of measurement, 40 different models (ANN and linear regression
trained independently in a 20 fold cross validation setting) agree on the
predictions and therefore the variance error is low. Finally, we trained 20
ANN models in a multiregression setting to predict five different noise
parameter values and also in this case the models agree between them when
tested on real data. Regarding the noise correction, the proposed approach
successfully learns to correct a simulated noisy pulse and to make the
measured probabilities closer to the ideal ones.

We believe that the results presented in this work can be used to better
quantify the effects of the noise affecting the Pasqal, and in general neutral
atoms, NISQ devices and to counteract those effects. The presented tecniques
are dependent on the atoms topology and the pulse shape. Thus, the ML models
can be trained to characterize and correct the noise of single quantum gates
that compose more complex Hamiltonians.

The accuracy of the predicted noise parameters depends on the accuracy of the
simulation and in particular on the accuracy of the simulator noise model.

In previous works [11, 30, 13, 31] and in preliminary experiments using Pasqal
simulator, there is an evidence of the improvement of the noise
characterization when more temporal statistics are collected. We adopted this
strategy in this paper for the noise correction, where the occupation
probabilities are obtained from the amplitudes of the intermediate quantum
states sampled at regular steps within the simulated dynamic. However, in real
NISQ devices, intermediate measurements of the dynamic are less
straightforward because of the impossibility of observing a system without
changing it. We can obtain the same effect independently measuring incremental
subdynamics from $t=0$ to subsequent time steps of the full dynamic. To
implement this approach on Pasqal machines, we can design a full pulse that is
subsequently split in sub-pulses at times
$[t_{0},t_{1}],[t_{1},t_{2}],\dots,[t_{n-1},t_{n}]$. The measurements at time
$t_{k}$ for $k=1,\dots,n$ can be obtained initialising the register always to
the same initial setting and performing the computation considering the
effects of all the sub-pulses spanning the times $[t_{0},t_{k}]$ from the
first to the one before $t_{k}$. The ML models can then process all the
measurements obtained at times $t_{1},\dots,t_{k}$ and in that way we expect
to obtain better results for the characterization of the noise. Moreover, we
can also use ANN more suitable for data organized in temporal sequences, i.e.
Recurrent Neural Network (RNN).

Finally, in the context of Quantum Machine Learning (QML) [32, 33] our work is
framed as a classical ML approach to process quantum data. Future research
lines may include the design of QML models for the noise characterization and
correction implemented directly within the quantum dynamic of neutral atoms
devices or of other NISQ devices. For instance pattern matching QML techniques
[34] can be adapted for the identification of noise patterns [13]
characteristics to the neutral atoms dynamics.

Acknowledgements

This work was financially supported by the European Union’s Horizon 2020
research and innovation programme under FET-OPEN GA n. 828946–PATHOS. We
acknowledge the CINECA award under the ISCRA initiative, for the availability
of high performance computing resources, as _Marconi100_ supercomputer, and
their support. S.M. acknowledges financial support from PNRR MUR project
PE0000023-NQSTI. Finally, we are also thankful to Pasqal for the provided data
that we have used to test our protocol.

## References

  * Preskill [2018] J. Preskill, Quantum computing in the nisq era and beyond, Quantum 2, 79 (2018).
  * Cai  _et al._ [2021] W. Cai, Y. Ma, W. Wang, C.-L. Zou, and L. Sun, Bosonic quantum error correction codes in superconducting quantum circuits, Fundamental Research 1, 50 (2021).
  * Chen  _et al._ [2021] Z. Chen, K. Satzinger, J. Atalaya, A. Korotkov, A. Dunsworth, D. Sank, C. Quintana, M. McEwen, R. Barends, P. Klimov, S. Hong, C. Jones, A. Petukhov, D. Kafri, S. Demura, B. Burkett, C. Gidney, A. Fowler, A. Paler, and J. Kelly, Exponential suppression of bit or phase errors with cyclic error correction, Nature 595, 383 (2021).
  * Lidar and Brun [2013] D. Lidar and T. Brun,  _Quantum Error Correction_ (Cambridge University Press, 2013).
  * [5] Programmable atomic arrays - PASQAL, https://www.pasqal.com/, [Online; accessed 14-2-2023].
  * Henriet  _et al._ [2020] L. Henriet, L. Beguin, A. Signoles, T. Lahaye, A. Browaeys, G.-O. Reymond, and C. Jurczak, Quantum computing with neutral atoms, Quantum 4, 327 (2020).
  * Silvério  _et al._ [2022] H. Silvério, S. Grijalva, C. Dalyac, L. Leclerc, P. J. Karalekas, N. Shammah, M. Beji, L.-P. Henry, and L. Henriet, Pulser: An open-source package for the design of pulse sequences in programmable neutral-atom arrays, Quantum 6, 629 (2022).
  * Shalev-Shwartz and Ben-David [2014] S. Shalev-Shwartz and S. Ben-David,  _Understanding machine learning: From theory to algorithms_ (Cambridge university press, 2014).
  * Ian Goodfellow and Courville [2016] Y. B. Ian Goodfellow and A. Courville,  _Deep learning_ (MIT press, 2016).
  * Youssry  _et al._ [2020] A. Youssry, G. A. Paz-Silva, and C. Ferrie, Characterization and control of open quantum systems beyond quantum noise spectroscopy, npj Quantum Information 6, 1 (2020).
  * Martina  _et al._ [2023a] S. Martina, S. Gherardini, and F. Caruso, Machine learning classification of non-markovian noise disturbing quantum dynamics, Physica Scripta 98, 035104 (2023a).
  * Wise  _et al._ [2021] D. F. Wise, J. J. Morton, and S. Dhomkar, Using deep learning to understand and mitigate the qubit noise environment, PRX Quantum 2, 010316 (2021).
  * Martina  _et al._ [2022] S. Martina, L. Buffoni, S. Gherardini, and F. Caruso, Learning the noise fingerprint of quantum devices, Quantum Machine Intelligence 4, 1 (2022).
  * Baireuther  _et al._ [2018] P. Baireuther, T. E. O’Brien, B. Tarasinski, and C. W. Beenakker, Machine-learning-assisted correction of correlated qubit errors in a topological code, Quantum 2, 48 (2018).
  * Fowler  _et al._ [2012] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface codes: Towards practical large-scale quantum computation, Physical Review A 86, 032324 (2012).
  * Harper  _et al._ [2020] R. Harper, S. Flammia, and J. Wallman, Efficient learning of quantum noise, Nature Physics 16, 1 (2020).
  * Sutton and Barto [2018] R. S. Sutton and A. G. Barto,  _Reinforcement learning: An introduction_ (MIT press, 2018).
  * Baum  _et al._ [2021] Y. Baum, M. Amico, S. Howell, M. Hush, M. Liuzzi, P. Mundada, T. Merkh, A. Carvalho, and M. Biercuk, Experimental deep reinforcement learning for error-robust gate-set design on a superconducting quantum computer, PRX Quantum 2 (2021).
  * Goldschmidt  _et al._ [2022] A. J. Goldschmidt, J. L. DuBois, S. L. Brunton, and J. N. Kutz, Model predictive control for robust quantum state preparation, Quantum 6, 837 (2022).
  * Porotti  _et al._ [2022] R. Porotti, A. Essig, B. Huard, and F. Marquardt, Deep Reinforcement Learning for Quantum State Preparation with Weak Nonlinear Measurements, Quantum 6, 747 (2022).
  * Sweke  _et al._ [2020] R. Sweke, M. S. Kesselring, E. P. van Nieuwenburg, and J. Eisert, Reinforcement learning decoders for fault-tolerant quantum computation, Machine Learning: Science and Technology 2, 025005 (2020).
  * Colomer  _et al._ [2020] L. D. Colomer, M. Skotiniotis, and R. Muñoz-Tapia, Reinforcement learning for optimal error correction of toric codes, Physics Letters A 384, 126353 (2020).
  * Schymik  _et al._ [2020] K.-N. Schymik, V. Lienhard, D. Barredo, P. Scholl, H. Williams, A. Browaeys, and T. Lahaye, Enhanced atom-by-atom assembly of arbitrary tweezer arrays, Physical Review A 102, 063107 (2020).
  * De Léséleuc  _et al._ [2018] S. De Léséleuc, D. Barredo, V. Lienhard, A. Browaeys, and T. Lahaye, Analysis of imperfections in the coherent optical excitation of single atoms to rydberg states, Physical Review A 97, 053803 (2018).
  * Hamming [1950] R. W. Hamming, Error detecting and error correcting codes, The Bell system technical journal 29, 147 (1950).
  * Peterson  _et al._ [1972] W. W. Peterson, W. Peterson, E. J. Weldon, and E. J. Weldon,  _Error-correcting codes_ (MIT press, 1972).
  * Wootters and Zurek [1982] W. K. Wootters and W. H. Zurek, A single quantum cannot be cloned, Nature 299, 802 (1982).
  * Shor [1995] P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Physical review A 52, R2493 (1995).
  * Roffe [2019] J. Roffe, Quantum error correction: an introductory guide, Contemporary Physics 60, 226 (2019).
  * Martina  _et al._ [2023b] S. Martina, S. Hernández-Gómez, S. Gherardini, F. Caruso, and N. Fabbri, Deep learning enhanced noise spectroscopy of a spin qubit environment, Machine Learning: Science and Technology 4, 02LT01 (2023b).
  * Fanchini  _et al._ [2021] F. F. Fanchini, G. Karpat, D. Z. Rossatto, A. Norambuena, and R. Coto, Estimating the degree of non-markovianity using machine learning, Physical Review A 103, 022425 (2021).
  * Dunjko  _et al._ [2016] V. Dunjko, J. M. Taylor, and H. J. Briegel, Quantum-enhanced machine learning, Phys. Rev. Lett. 117, 130501 (2016).
  * Aïmeur  _et al._ [2006] E. Aïmeur, G. Brassard, and S. Gambs, Machine learning in a quantum world, in  _Advances in Artificial Intelligence_ , edited by L. Lamontagne and M. Marchand (Springer Berlin Heidelberg, Berlin, Heidelberg, 2006) pp. 431–442.
  * Das  _et al._ [2023] S. Das, J. Zhang, S. Martina, D. Suter, and F. Caruso, Quantum pattern recognition on real quantum processing units, Quantum Machine Intelligence 5, 16 (2023).
  * Paszke  _et al._ [2017] A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer, Automatic differentiation in pytorch, in  _NIPS-W_ (2017).
  * Paszke  _et al._ [2019] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, _et al._ , Pytorch: An imperative style, high-performance deep learning library, Advances in neural information processing systems 32 (2019).
  * Kingma and Ba [2014] D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, arXiv preprint arXiv:1412.6980 (2014).
  * Liaw  _et al._ [2018] R. Liaw, E. Liang, R. Nishihara, P. Moritz, J. E. Gonzalez, and I. Stoica, Tune: A research platform for distributed model selection and training, arXiv preprint arXiv:1807.05118 (2018).
  * Li  _et al._ [2020] L. Li, K. Jamieson, A. Rostamizadeh, E. Gonina, J. Ben-Tzur, M. Hardt, B. Recht, and A. Talwalkar, A system for massively parallel hyperparameter tuning, Proceedings of Machine Learning and Systems 2, 230 (2020).
  * Bergstra  _et al._ [2013] J. Bergstra, D. Yamins, D. D. Cox, _et al._ , Hyperopt: A python library for optimizing the hyperparameters of machine learning algorithms, in  _Proceedings of the 12th Python in science conference_ , Vol. 13 (Citeseer, 2013) p. 20.

##  VI Methods

###  VI.1 Single parameter characterization

In this section we describe with more details the topologies of the analyzed
quantum systems and the ANN models used in section III.1.

Table 3:  Quantum systems used for single parameter estimation $\sigma_{R}$. By the notation $s_{i}$ we denote the system formed by $i$ atoms. In the case of 4 atoms, having used 6 different systems for the spatial arrangement of atoms, we use an additional subscript $s_{4j}$, with $j=\\{a,b,c,d,e,f\\}$. Also, when a quantum register of a system $s_{k}$ is entirely contained in the quantum register of a larger system $s_{k}^{\prime}$, with $k^{\prime}>k$, we use the notation $s_{k}\subset s_{k^{\prime}}$. Name | $s_{2}$ | $s_{3}$ | $s_{4a}$ | $s_{5}$ | $s_{4b}$ | $s_{4c}$ | $s_{4d}$ | $s_{4e}$ | $s_{4f}$
---|---|---|---|---|---|---|---|---|---
Atoms | $2$ | $3$ | $4$ | $5$ | $4$ | $4$ | $4$ | $4$ | $4$
Properties | $\subset s_{4a}$ | $\subset s_{5}$ | $\subset s_{5}$ |  | $=s_{4a}$ |  |  |  |

The registers summarized in table 3 have an incremental number of atoms from 2
to 5 and some of them are chosen in a way such that the positions of the atoms
of every register are included in the subsequent ones as far as possible.

To be precise, with the notation $s_{k}\subset s_{k^{\prime}}$ and
$k<k^{\prime}$, we indicate that the quantum register of $s_{k^{\prime}}$ is
the same as that of $s_{k}$ with the addition of an atom and that therefore
the coordinates of the atoms in common are the same. In detail, the setting
with five atoms ($s_{5}$) have atoms in the same positions of the ones of the
settings of dimensionality four ($s_{4a}$) and three ($s_{3}$) plus extra
atoms in other positions. Moreover, $s_{4a}$ contains all the two atoms of the
setting $s_{2}$, but $s_{3}$ includes only one of the two atoms of $s_{2}$ and
$s_{4a}$ only two of the three atoms of $s_{3}$. We denote the latter
properties with the notation $s_{2}\subset s_{4a}\subset s_{5}$ and
$s_{3}\subset s_{5}$. In addition, we collected also further measurements of
settings with 4 atoms. In detail, we run a second setting $s_{4b}$ with the
atoms in the same position of $s_{4a}$ and other four settings with different
positions for the atoms: $s_{4c}$, $s_{4d}$, $s_{4e}$ and $s_{4f}$. We choose
those specific settings because we want to evaluate the two different scaling:
(i) in the number of atoms, (ii) in the number of measurements of different
settings with the same number of atoms. Specifically, we consider for (i)
$s_{2}$, $s_{3}$, $s_{4a}$ and $s_{5}$ and for (ii) $s_{4a}$, $s_{4b}$,
$s_{4c}$, $s_{4d}$, $s_{4e}$ and $s_{4f}$.

The trained ANNs are composed by a single hidden layer of 100 neurons with
ReLU activation function and the output layer with a single neuron with
sigmoid activation function. The targets are normalized between 0 and 1 before
the training and the inverse transformation is applied to calculate the
prediction error. The models are developed in _PyTorch_ [35, 36] and trained
with mini batch gradient descent to minimize the L1 loss using the Adam
optimizer [37] with learning rate 0.001 and batch size 512. All models are
trained with early stopping for a maximum of 150 epochs.

To perform the scaling (i) we trained four different models using as inputs
the $2^{2}$, $2^{3}$, $2^{4}$ and $2^{5}$ measurements of respectively the
settings $s_{2}$, $s_{3}$, $s_{4a}$ and $s_{5}$. To perform the scaling (ii)
we consider the measurement coming from the following systems:

  * •

$s_{4a}$ ($2^{4}=16$ measurements)

  * •

$s_{4a}\oplus s_{4b}$ ($2^{4}+16=2^{5}$ measurements)

  * •

$s_{4a}\oplus s_{4b}\oplus s_{4c}$ ($2^{4}+32=48$ measurements)

  * •

$s_{4a}\oplus s_{4b}\oplus s_{4c}\oplus s_{4d}$ ($2^{4}+48=64$ measurements)

  * •

$s_{4a}\oplus s_{4b}\oplus s_{4c}\oplus s_{4d}\oplus s_{4e}$ ($2^{4}+64=80$
measurements)

  * •

$s_{4a}\oplus s_{4b}\oplus s_{4c}\oplus s_{4d}\oplus s_{4e}\oplus s_{4f}$
($2^{4}+80=96$ measurements).

In all the cases, the datasets are split in 20 equal parts to perform a
20-fold cross validation and we report the resulting average mean absolute
error and its standard deviation for the 20 models.

As a remark, the Pulser simulator allows to specify the number of samples per
run to speedup the computation. In that case, for each run the final quantum
state is preliminary calculated, then the specified number of measurements is
obtained from such state. Even if this expedient is useful to spare resources,
we found in preliminary experiments that it is counter-productive in the
context of noise estimation. In fact, for all the samples of one run, the
Hamiltonian defining the evolution is always the same and also the noise that
influences it. For this reason, in our work we keep the number of samples per
run equal to 1 forcing the resampling of the noise at each single measurement.

Moreover, in this context is better to consider more measurements respect to
increase the number of atoms of the setting. In detail, considering both
subfigures, the number of data points for the measurement of the setting with
5 atoms in fig. 2(a), i.e. $2^{5}=32$, is equal to the ones for two
concatenated measurements of settings with 4 atoms in fig. 2(b), i.e.
$2^{4}+2^{4}=32$, but the error in the latter case is lower than the former.

###  VI.2 Multiple parameters characterization

Before training the models, the noise parameters are normalised between 0 and
1 to avoid uneven prediction error during the loss calculation. The models are
implemented in Pytorch [35, 36] and are trained with minibatch gradient
descent to minimise the L1 loss using Adam [37]. Regarding the architecture of
the models, the ANNs is a Multi Layer Perceptron (MLP) with the ReLU
activation function for all the hidden layers and the sigmoid activation
function for the last layer. The best combination of number of neuron layers,
number of neurons in each layer, batch size and learning rate is chosen with
an hyper-parameter optimization procedure. The latter is implemented using the
python library Ray Tune [38] with the ASHA scheduler [39] and the HyperOpt
search algorithm [40]. The ASHA scheduler allows multiple models to be trained
in parallel, iteratively interrupting the training of the least promising one
and thus reducing the duration of the hyper-parameter optimization. In our
case at each epoch it halved the models by discarding those with the highest
calculated loss on the validation set. HyperOpt search algorithm, on the other
hand, chooses the most probable best combinations of hyper-parameters based on
the previously trained and/or stopped models. By this procedure, the model
with the most promising set of hyper-parameters is chosen from 1000 models
trained with the Adam optimizer. The hyper-parameters are sampled in the
following ranges: number of hidden layers from 1 to 100, number of neurons in
each layer from 5 to 200, batch size in $\\{2,4,8,16,32\\}$ and learning rate
from $log-uniform(10^{-4},10^{-1})$. At the end, the best hyper-parameters
combination is: 1 hidden layer of 117 neurons, batch size 16, initial learning
rate $\approx 0.069$, dropout probability $\approx 0.044$ and L2
regularization $\approx 0.0002$.

After finding the best set of hyper-parameters, 20 models are trained using
the cross validation procedure to exploit the entire dataset and to obtain the
standard deviations of the predictions. In detail, for the cross validation
the dataset is divided into 20 equal parts, 18 are used for training, one for
validation and one for testing. The advantage of using the cross validation
procedure is that a different block is used for the test of each model and
also, in this way all the samples of the dataset are exploited for the
training. Each one of the 20 models is trained with early stopping for a
maximum of 150 epochs.

  *[NISQ]: Noisy Intermediate Scale Quantum
  *[ML]: Machine Learning
  *[RL]: Reinforcement Learning
  *[AI]: Artificial Intelligence
  *[GRF]: Gibbs random field
  *[ANN]: Artificial Neural Network
  *[MAE]: Mean Absolute Error
  *[ANNs]: Artificial Neural Network
  *[KL]: Kullback–Leibler
  *[RNN]: Recurrent Neural Network
  *[QML]: Quantum Machine Learning
  *[MLP]: Multi Layer Perceptron


# Studies with Onia at LHCb

L. Li Gioi - for the LHCb collaboration  CNRS - Laboratoire de Physique
Corpusculaire de Clermont-Ferrand, France

###### Abstract

The production of $c\bar{c}$ and $b\bar{b}$ is studied in $pp$ collisions at
$\sqrt{s}=7$ TeV with the LHCb detector. The results of these measurements are
compared with different theoretical models. Results and prospects are also
shown for exotics: the mass measurement of the $X(3872)$ and the search of the
$X(4140)$. Most of the presented results are based on the 2010 LHCb dataset
(35 pb-1).

##  I Introduction

Heavy quarkonium production remains a challenging problem for the
understanding of Quantum Chromodynamics (QCD). At the centre-of mass energies
for proton-proton collisions at the Large Hadron Collider, $c\bar{c}$ pairs
are expected to be produced predominantly via Leading Order gluon gluon
interactions, which can be computed using perturbative QCD, followed by the
formation of the bound charmonium states described by non perturbative models.
Recent approaches make use of non-relativistic QCD factorization (NRQCD) which
assumes a combination of the colour-singlet and colour-octet $c\bar{c}$ as it
evolves towards the final bound state via the exchange of soft gluons
ref:intr1 . Next-to-Leading Order (NLO) QCD corrections in charmonium and
bottomonium production are also essential for the description of the
experimental data ref:intr2 ; ref:intr3 . Studies of $\psi(2S)$,
$\Upsilon(1S)$ and $\chi_{c}$ production cross sections are presented.

The $X(3872)$ meson is an exotic meson discovered in 2003 by the Belle
collaboration in the $B^{\pm}\to X(3872)K^{\pm}$, $X(3872)\to
J/\psi\pi^{+}\pi^{-}$ decay chain ref:intr4 . Its existence was confirmed by
the CDF ref:intr5 , D0 ref:intr6  and BaBar ref:intr7  collaborations. The
$X(3872)$ particle, together with several other new states subsequently
observed in the mass range 3.9-4.7 GeV/c2, has led to a resurgence of interest
in exotic meson spectroscopy ref:intr8 . Several properties of the $X(3872)$
have been measured; however, its nature is still uncertain and several models
have been proposed. First, it is not excluded that the $X(3872)$ is a
conventional charmonium state with one candidate being the $\eta_{c2}(1D)$
meson ref:intr8 . However, the mass of this state is predicted to be far below
the observed $X(3872)$ mass. Given the proximity of the $X(3872)$ mass to the
$D^{*0}\bar{D^{0}}$ threshold, one possibility is that the $X(3872)$ is a
loosely bound deuteron-like $D^{*0}\bar{D^{0}}$ molecule, i.e. a ((uc) - (cu))
system ref:intr8 . Another more exotic possibility is that the $X(3872)$ is a
tetraquark state ref:intr9 .

The CDF experiment has reported $>5\ \sigma$ evidence for $X(4140)$ state
(also referred to as $Y(4140)$ in the literature) in 6.0 fb-1 of $p\bar{p}$
data collected at Tevatron ref:intr10 . The relative rate was measured to be
${B(B^{+}\to X(4140)K^{+})\times B(X(4140)\to J/\psi\phi)}/{B(B^{+}\to
J/\psi\phi K^{+})}=0.149\pm 0.039\pm 0.024$. This observation has triggered
wide interest among model builders of exotic hadronic states. It has been
suggested that the $X(4140)$ could be a molecular state ref:intr11 ;
ref:intr12 , a tetraquark state ref:intr13 ; ref:intr14 , a hybrid state
ref:intr15 ; ref:intr16  or a re-scattering effect ref:intr17 ; ref:intr18 .

##  II $\psi(2S)$ Production Cross Section

Two decay modes of the $\psi(2S)$ meson have been studied ref:confPsi2S :
$\psi(2S)\to\mu^{+}\mu^{-}$ and $\psi(2S)\to J/\psi(\mu\mu)\pi^{+}\pi^{-}$.
The differential cross-section for the inclusive $\psi(2S)$ production is
computed as

$\frac{d\sigma}{dp_{T}}(p_{T})=\frac{N_{\psi(2S)}(p_{T})}{L_{int}\epsilon(p_{T})B(\psi(2S)\to
f)\Delta p_{T}}$ (1)

where $N_{\psi(2S)}(p_{T})$ is the number of observed $\psi(2S)$ decays,
$L_{int}$ is the integrated luminosity, $\epsilon(p_{T})$ is the total
detection efficiency including acceptance effects, $B(\psi(2S)\to f)$ is
$B(\psi(2S)\to J/\psi\pi^{+}\pi^{-})B(J/\psi\to\mu^{+}\mu^{-})$ for
$\psi(2S)\to J/\psi(\mu\mu)\pi^{+}\pi^{-}$ and the dielectron branching ratio
($B(\psi(2S)\to e^{+}e^{-})$) for $\psi(2S)\to\mu^{+}\mu^{-}$ and $\Delta
p_{T}=$ 1 GeV/c is the bin size. The dielectron branching ratio is used,
assuming lepton universality, since it has a much smaller error than the
dimuon one (2.2% and 10% respectively). In order to estimate the number of
$\psi(2S)$ signal events, a fit is performed independently in each $(p_{T})$
bin.

Figure 1: Comparison of the LHCb results for the differential production
cross-section of $\psi(2S)$ with the predictions for prompt production by a
NLO NRQCD model ref:Psi2S_1 . LHCb data include also $\psi(2S)$ from b.

In Fig. 1 a comparison between the measurements presented of the differential
$\psi(2S)$ production cross-section multiplied by the
$\psi(2S)\to\mu^{+}\mu^{-}$ branching ratio and a recent theory prediction
ref:Psi2S_1  for prompt $\psi(2S)$ in the rapidity range $2\leq
y=\frac{1}{2}ln\frac{E+p_{z}}{E-p_{z}}\leq 4.5$ is shown. Here $E$ and $p_{z}$
are the $\psi(2S)$ energy and momentum in the $z$ direction measured in the pp
centre-of-mass frame; the z-axis is defined along the beam axis in the LHCb
frame, oriented from the VELO to the Muon Detector. The differential cross-
section in ref:Psi2S_1  has been evaluated for the $\psi(2S)$ prompt
production at the LHC at next-to-leading order in non-relativistic QCD,
including color-singlet and color-octet contributions. The integrated cross-
section in the full range of $p_{T}$ and $y$ respectively to
$\psi(2S)\to\mu^{+}\mu^{-}$ and $\psi(2S)\to J/\psi\pi^{+}\pi^{-}$ is found to
be:

$\displaystyle\sigma(0<p_{T}\leq 12\ \mbox{GeV/c},2<y\leq 4.5)$
$\displaystyle=$ $\displaystyle 1.88\pm 0.02\pm 0.31^{+0.25}_{-0.48}\
\mbox{$\mu$b}$ $\displaystyle\sigma(3<p_{T}\leq 16\ \mbox{GeV/c},2<y\leq 4.5)$
$\displaystyle=$ $\displaystyle 0.62\pm 0.04\pm 0.12^{+0.07}_{-0.14}\
\mbox{$\mu$b}$ (2)

where the first uncertainty is statistical, the second is systematic and the
third is the uncertainty due to the unknown polarization.

##  III $\Upsilon(1S)$ Production Cross Section

$\Upsilon(1S)$ meson has been studied in the decay mode
$\Upsilon(1S)\to\mu^{+}\mu^{-}$ ref:confY1S . The double differential cross-
section for the inclusive $\Upsilon(1S)$ production is computed as

$\frac{d^{2}\sigma}{dp_{T}dy}(p_{T},y)=\frac{N_{\Upsilon(1S)}(p_{T},y)}{L_{int}\epsilon(p_{T},y)B(\Upsilon(1S)\to\mu^{+}\mu^{-})\Delta
p_{T}\Delta y}$ (3)

where $N_{\Upsilon(1S)}(p_{T},y)$ is the number of observed
$\Upsilon(1S)\to\mu^{+}\mu^{-}$ decays, $\epsilon(p_{T},y)$ is the total
detection efficiency including acceptance effects, $L_{int}$ is the integrated
luminosity, $B(\Upsilon(1S)\to\mu^{+}\mu^{-})$ is the branching fraction, and
$\Delta p_{T}\Delta y=1\times 0.5$ GeV/c is the rapidity and $p_{T}$ bin
sizes. In order to estimate the number of $\Upsilon(1S)$ signal events, a fit
is performed independently in each of the 15 $p_{T}$ times 5 $y$ bins. The
double differential cross-section as a function of $p_{T}$ and $y$ is shown in
Fig. 2. The integrated cross-section in the full range of $y$ and $p_{T}$ is
found to be

$\sigma(pp\to\Upsilon(1S)X;p_{T}(\Upsilon(1S))<15\
\mbox{GeV/c};2<y(\Upsilon(1S))<4.5)=108.3\pm 0.7^{+30.9}_{-25.8}\ \mbox{nb}$
(4)

where the first uncertainty is statistical, and the second systematic. The
latter includes ${}^{+18.8}_{-7.9}$ nb from the unknown polarization, $\pm
10.8$ nb from the luminosity determination and $\pm 22.0$ nb from other
sources. The integrated cross-section is about a factor 100 smaller than the
integrated $J/\psi$ cross-section in the identical $y$ and $p_{T}$ region
ref:Y1S_1 , and a factor three smaller than the integrated $\Upsilon(1S)$
cross-section in the central region $|y|<2$ as measured by CMS ref:Y1S_2 .

Figure 2: Differential $\Upsilon(1S)$ production cross-section as a function
of $p_{T}$ in bins of rapidity.

The CMS and the LHCb measurements agree; the difference in the integrated
cross section results is due to the different rapidity ranges. Fig. 3 shows
the comparison of the LHCb cross-section measurement in bins of $y$ integrated
over $p_{T}$ with the same measurement of CMS, in the $y$ ranges covered by
the two experiments.

Figure 3: Differential $\Upsilon(1S)$ production cross-section as a function
of $y$ integrated over $p_{T}$ as measured by the CMS ref:Y1S_2  and LHCb
experiments.

##  IV $\sigma(\chi_{c2})/\sigma(\chi_{c1})$ Production Cross Sections Ratio

$\chi_{c1}$ and $\chi_{c2}$ have been reconstructed in the final state
$J/\psi\gamma$. The production cross-section ratio of the $\chi_{c2}$ and
$\chi_{c1}$ states is measured ref:confChic  using

$\frac{\sigma(\chi_{c2})}{\sigma(\chi_{c1})}=\frac{N_{\chi_{c2}}}{N_{\chi_{c1}}}\cdot\frac{\epsilon^{\chi_{c1}}_{J/\psi}\epsilon^{\chi_{c1}}_{\gamma}\epsilon^{\chi_{c1}}_{sel}}{\epsilon^{\chi_{c2}}_{J/\psi}\epsilon^{\chi_{c2}}_{\gamma}\epsilon^{\chi_{c2}}_{sel}}\cdot\frac{B(\chi_{c1}\rightarrow
J/\psi\gamma)}{B(\chi_{c2}\rightarrow J/\psi\gamma)}$ (5)

where $B(\chi_{c1}\rightarrow J/\psi\gamma)$ and $B(\chi_{c2}\rightarrow
J/\psi\gamma)$ are the $\chi_{c1}$ and $\chi_{c2}$ branching ratios to the
final state $J/\psi\gamma$, $\epsilon^{\chi_{c2}}_{J/\psi}$
($\epsilon^{\chi_{c2}}_{\gamma}$) is the efficiency to reconstruct and select
a $J/\psi$ ($\gamma$) from $\chi_{c}$ decay and $\epsilon^{\chi_{c}}_{sel}$ is
the efficiency to select the $\chi_{c}$ candidate. The measurement method
consists of extracting the two $N_{\chi_{ci}}$ yields (for $\chi_{c1}$ and
$\chi_{c2}$) from an unbinned maximum likelihood fit to $\Delta
M=M(\chi_{c})-M(J/\psi)$ mass difference distribution. With the ratio of the
resolution parameters and the mass differences fixed, a fit is then performed
to the data in the full $J/\psi\ p_{T}$ range $J/\psi\ p_{T}\in[3,15]$ GeV/c
in order to extract the resolution scale $\sigma_{res}(\chi_{c1})$. Here, the
sample is subdivided into candidates with converted (after the magnet) and
non-converted photons, in order to account for the different calorimeter
resolution in the two cases. Converted photons are identified by the presence
of activity in the scintillator pad detector at the entrance to the
calorimeter system. Photons that convert before the magnet have a low
probability to be reconstructed, because either one or both electrons are
swept out of the detector acceptance by the magnetic field, and are not
considered here. The resolution scales, $\sigma_{res}(\chi_{c1})$, are
measured to be $22.8\pm 1.1$ MeV/c2 and $18.4\pm 0.4$ MeV/c2 for converted and
non converted candidates, respectively. The fit is then performed in bins of
$J/\psi\ p_{T}$. For each bin the value of $\sigma_{res}(\chi_{c1})$ is fixed
to the value extracted from the fit to the full range $J/\psi\ p_{T}$. The
presence of $\chi_{c}$ polarized states would modify the efficiencies
calculated from the Monte Carlo. In order to take into account possible
polarization scenarios, the relevant combination of weights has to be taken
into account. The results from the not-converted and converted samples are
combined by sampling the combination of the statistical and uncorrelated
systematic uncertainties using a toy Monte Carlo. Finally, the correlated
systematic uncertainty from the branching ratio is calculated using the
combined central values for $\sigma(\chi_{c2})/\sigma(\chi_{c1})$. The
preliminary result for the ratio of the prompt $\chi_{c2}$ to $\chi_{c1}$
production cross-sections as a function of $J/\psi\ p_{T}$ is given in Fig. 4.
Comparisons to the theory predictions from the ChiGen MC generator ref:Chic_1
and from the NLO NRQCD calculations ref:Chic_2 , in the rapidity range [2;
4.5], are also shown in the figure. Fig. 4 also shows the maximum effect of
the unknown $\chi_{c}$ polarization on the result, shown as a black shaded
area around the data points. The upper limit of the shaded area corresponds to
the spin state ( $\chi_{c1}$ : $m_{J}$ = 1; $\chi_{c2}$ : $m_{J}$ = 2) and the
lower limit corresponds to the spin state ( $\chi_{c1}$ : $m_{J}$ = 0;
$\chi_{c2}$ : $m_{J}$ = 0). The results are broadly in agreement at high
$J/\psi\ p_{T}$ with the colour singlet model, however, they are not yet
precise enough to rule out sizeable colour octet terms. There are indications
of a discrepancy in the mid to low $J/\psi\ p_{T}$ region. This may be
explained by a more complete modeling of the transition from the high to low
$J/\psi\ p_{T}$ regions, which is sensitive to non-perturbative effects and/or
sizeable higher-order perturbative corrections.

Figure 4: The ratio $\sigma(\chi_{c2})/\sigma(\chi_{c1})$ in bins of in the
range of $J/\psi\ p_{T}$ : [3; 15] GeV/c. The internal error bars correspond
to the statistical error on the $\chi_{c1}$ and $\chi_{c2}$ yields; the
external error bars include the contribution from the systematic uncertainties
(apart from the polarization). The shaded area around the data points (black)
shows the maximum effect of the unknown $\chi_{c}$ polarization on the result.
The two other bands correspond to the ChiGen MC generator theoretical
prediction ref:Chic_1  (in blue) and NLO NRQCD ref:Chic_2  (in red).

##  V $X(3872)$ Mass Measurement

Inclusively produced $X(3872)$ mesons are reconstructed and selected in the
$X(3872)\to J/\psi\pi^{+}\pi^{-}$, $J/\psi\to\mu^{+}\mu^{-}$ decay mode
ref:confXmass . A momentum scale calibration is performed to account for a
mixture of effects related to imperfections in the knowledge of the magnetic
field map and of the alignment of the tracking system.

###  V.1 Momentum Scale Calibration

The momentum scale is calibrated using a large sample of
$J/\psi\to\mu^{+}\mu^{-}$ decays. This calibration, gives an overall scale
factor that is applied to all measurements of particle momenta. The
calibration is checked to be also valid for two-body decays of the
$\Upsilon(1S)$, $D^{0}$ and $K^{0}$. In addition, its effect on $\psi(2S)\to
J/\psi\pi^{+}\pi^{-}$, which have kinematics similar to the $X(3872)\to
J/\psi\pi^{+}\pi^{-}$, is studied. The uncalibrated value of the $\psi(2S)$
mass in data is $3685.94\pm 0.06(stat)$ MeV/c2. After the momentum scale
calibration it becomes $3686.12\pm 0.06(stat)$ MeV/c2, in good agreement with
the PDG value of $3686.09\pm 0.04$ MeV/c2. The measured $J/\psi$ mass after
alignment and calibration is checked to be stable over the whole 2010 data-
taking period.

###  V.2 Results

The masses of the $\psi(2S)$ and $X(3872)$ are determined from an extended
unbinned maximum likelihood fit of the reconstructed $J/\psi\pi^{+}\pi^{-}$
mass in the interval $3.6<M_{J/\psi\pi^{+}\pi^{-}}<3.95$ GeV/c2. The
functional form of the background is studied using the same-sign pion events.
The $\psi(2S)$ and $X(3872)$ signals are each described with a Voigt function
defined as the convolution of a non-relativistic Breit-Wigner with a Gaussian
function. The intrinsic width of the $\psi(2S)$ is fixed to the PDG value,
$\Gamma_{\psi(2S)}=0.317$ MeV/c2. The $X(3872)$ intrinsic width is poorly
known. The BaBar ref:Xmass_1  and Belle ref:intr4  collaborations have
published $90\%$ confidence level limits of $\Gamma_{X(3872)}<3.3$ MeV/c2 and
$\Gamma_{X(3872)}<2.3$ MeV/c2, respectively. The results of the Belle and
BaBar analyses have been combined by CDF to yield $\Gamma_{X(3872)}=1.3\pm
0.6$ MeV/c2 ref:Xmass_2 . For these studies, the procedure adopted in
ref:Xmass_2  has been adopted and the natural width has been fixed to this
value in the default fit. Fig. 5 shows the invariant $J/\psi\pi^{+}\pi^{-}$
mass distributions for opposite-sign (black points) and same-sign (blue filled
histogram) candidates. Clear signals for both the $\psi(2S)$ and the $X(3872)$
can be seen. From a comparison of fits with and without the $X(3872)$
component the statistical significance of the $X(3872)$ signal is estimated to
be $9\ \sigma$. The uncertainties on the parameters reported by the fit are in
good agreement with expectations based on toy Monte Carlo studies. The
preliminary LHCb result

$M_{X(3872)}=3871.96\pm 0.46\pm 0.10\ \mbox{MeV/c${}^{2}$}$ (6)

is in good agreement with the published measurements ref:intr4 ; ref:Xmass_1 ;
ref:Xmass_2 ; ref:intr5  and with their average, $3871.56\pm 0.22$ MeV/c2. The
new world average value including LHCb measurement, $3871.63\pm 0.20$ MeV/c2,
is consistent within uncertainties with the sum of the $D^{0}$ and $D^{*0}$
masses, $3871.79\pm 0.29$ MeV/c2, computed from the results of the global PDG
fit of the charm meson masses.

Figure 5: Invariant mass distribution of $J/\psi\pi^{+}\pi^{-}$ (black points
with statistical error bars) and same-sign $J/\psi\pi^{\pm}\pi^{\pm}$ (blue
filled histogram) candidates. The red curve is the result of the fit described
in the text. The insert shows a zoom of the region around the $X(3872)$ mass.

##  VI $X(3872)$ Production Cross Section

The product of the inclusive production cross-section $\sigma(pp\to
X(3872)+...)$ by the $X(3872)\to J/\psi\pi^{+}\pi^{-}$ branching fraction is
computed as ref:confXCrSec

$\sigma(pp\to X(3872)+...)\times B(X(3872)\to
J/\psi\pi^{+}\pi^{-})=\frac{N^{corr}_{X(3872)}}{\eta_{tot}\times L_{int}\times
B(J/\psi\to\mu^{+}\mu^{-})}$ (7)

where $N^{corr}_{X(3872)}$ is the efficiency-corrected yield of $X(3872)\to
J/\psi(\mu^{+}\mu^{-})\pi^{+}\pi^{-}$ signal decays, $\eta_{tot}$ is a
multiplicative factor to the efficiency that accounts for known differences
between the data and simulation, $L_{int}$ is the integrated luminosity, and
the notation $B(...)$ is used for branching fractions. The $X(3872)$ signal
yield is determined from an extended, unbinned maximum likelihood fit of the
reconstructed $J/\psi\pi^{+}\pi^{-}$ mass in the interval
$3.82<M_{J/\psi\pi^{+}\pi^{-}}<3.95$ GeV/c2. The fitting function is the same
used for the $X(3872)$ mass measurement. The mass resolution is fixed in this
analysis to

$\sigma(X(3872))=\sigma(\psi(2S))\frac{\sigma^{MC}_{X(3872)}}{\sigma^{MC}_{\psi(2S)}}=3.26\pm
0.10\ \mbox{MeV/c${}^{2}$}$ (8)

where $\sigma(\psi(2S))=2.48\pm 0.08$ MeV/c2 is the mass resolution fitted in
data for the kinematically similar decay $\psi(2S)\to J/\psi\pi^{+}\pi^{-}$,
and where $\sigma^{MC}_{X(3872)}$ and $\sigma^{MC}_{\psi(2S)}$ are the mass
resolutions determined from fully simulated Monte Carlo events. As in section
V.2, the natural width is fixed to 1.3 MeV/c2. The effect of fixing parameters
that are not perfectly known is investigated as part of the systematic
studies. The $X(3872)$ signal yield is approximately $68\%$ of that reported
in ref:confXmass  due to the additional trigger requirements and fiducial cuts
on $y$ and $p_{T}$. Applying the procedure discussed above, and using the
$p_{T}-$ and $y-$dependent efficiency from simulation, the efficiency-
corrected signal yield is $N^{corr}_{X(3872)}=9597\pm 2217$, where the quoted
uncertainty is statistical. Two factors enter into the determination of the
correction factor $\eta_{tot}$. The first is a factor $1.024\pm 0.011$ which
accounts for differences in the efficiency of the muon identification in the
data and simulation. The second is a factor of $0.96\pm 0.02$ that accounts
for observed differences in the efficiency of global event cuts applied in the
trigger. Multiplying these values together gives $\eta_{tot}=0.983\pm 0.023$.
The uncertainty on this number is taken into account in the estimation of the
systematic uncertainty. The preliminary measurement is

$\sigma(pp\to X(3872)+...)\times B(X(3872)\to J/\psi\pi^{+}\pi^{-})=4.74\pm
1.10\pm 1.01\ \mbox{nb}$ (9)

where $\sigma(pp\to X(3872)+...)$ is the cross-section for producing an
$X(3872)$ particle in $pp$ collisions at $\sqrt{s}=7$ TeV (either promptly or
from the decay of other particles) with a transverse momentum between 5 and 20
GeV/c and a rapidity between 2.5 and 4.5.

##  VII Search of the $X(4140)$

The $X(4140)$ has been studied using a data sample of approximately 0.376 fb-1
ref:confX4140  selecting $B^{+}\to J/\psi\phi K^{+}$, $\phi\to K^{+}K^{-}$. No
narrow structure has been seen near the threshold in $M(J/\psi\phi)-M(J/\psi)$
distribution as shown in Fig. 6. In the CDF analysis, they fit their data with
spin 0 relativistic Breit-Wigner function on top of three-body phase-space,
all smeared with the detector resolution ref:intr10 . To quantify the
disagreement with CDF the same function has been used. The efficiency
dependence is extracted from the MC simulations and applied as a correction to
the three-body phase-space function. Mass and width of $X(4140)$ peak are
fixed to the values obtained by the CDF collaboration. The mass difference
resolution is determined from the $B^{+}\to X(4140)K^{+}$ MC simulation. The
fit to $M(J/\psi\phi)-M(J/\psi)$ distribution gives a $X(4140)$ amplitude of
$6.9\pm 4.7$ events (Fig. 6a) and has a confidence level (CL) of $3\%$. When
using a quadratic polynomial instead of three-body phase-space function for
the background, the preferred value of the $X(4140)$ amplitude is zero (it is
restricted not to go below zero), with a positive error of 3 events. This fit
is shown in Fig. 6b and has a confidence level of $11\%$. Using the $B^{+}\to
J/\psi\phi K^{+}$ yield multiplied by this efficiency ratio ($B^{+}\to
X(4140)K^{+}$, $X(4140)\to J/\psi\phi$)/($B^{+}\to J/\psi\phi K^{+}$) and
multiplied by the CDF value for $B(B^{+}\to X(4140)K^{+})/B(B^{+}\to
J/\psi\phi K^{+})$ ref:intr10 , leads to a prediction of observed $35\pm 9\pm
6$ events, where the first uncertainty is statistical from the CDF data and
the second is systematic including both CDF and LHCb contributions. The
central value of this estimate is illustrated in Fig. 6. The CDF result
disagrees by three standard deviations with the fit to LHCb data using the
polynomial background. The disagreement is 2.4 $\sigma$ when the efficiency-
corrected three-body phase-space background shape is used.

Figure 6: Fit of $X(4140)$ signal on top of smooth background to
$M(J/\psi\phi)-M(J/\psi)$ for the $B^{+}\to J/\psi\phi K^{+}$ data. The solid
line represents the result of the fit to LHCb data. The dashed line on top
illustrates the expected signal amplitude from the CDF results as explained in
the text. The top and bottom plots differ by the type of the background
function used in the fit: a) efficiency-shaped three-body phase-spaced; b)
quadratic polynomial.

##  VIII Exclusive Dimuon Production

Exclusive particle production in proton-proton collisions are elastic
processes in which the protons remain intact, $p+p\to p+X+p$, and the
additional particles are created through photon and/or gluon propagators. In
the case of purely photon propagators, this is an electromagnetic process
which can be theoretically calculated with high accuracy. When gluons are
involved, these processes constitute an important testing ground for QCD,
since the object that couples to the proton must be colourless. Thus the
pomeron (two gluon states) or possibly an odderon (three gluons), predicted in
QCD but never unambiguously observed, can be studied in a clean experimental
environment. The cleanest experimental manifestation of these occur in final
states containing two protons and two muons or in events containing two
protons, two muons and a photon. The former can be produced in the diphoton
process giving a continuous dimuon invariant mass spectrum, or in the photon-
pomeron process which can produce $\phi$, $J/\psi$, $\psi(2S)$,
$\Upsilon(1S)$, $\Upsilon(2S)$ or $\Upsilon(3S)$ which decay to two muons. The
latter is a signal for double pomeron exchange which produces
$\chi_{c}$($\chi_{b}$) that decay to $J/\psi$ ($\Upsilon$) plus a photon. The
final state protons are only marginally deflected, go down the beam-pipe, and
remain undetected. The experimental signal therefore in LHCb is a completely
empty event except for two muons and possibly a photon. However, because LHCb
is not hermetic, there will be sizeable backgrounds from non-elastic processes
where the other particles travel outside the detector acceptance.

The cross-section $\sigma$, is calculated ref:confExcl  from the number, $N$,
of selected events having corrected for efficiency, $\epsilon$, and purity,
$p$, and dividing by the luminosity, $L$, via $\sigma=(pN)/(\epsilon L)$. The
efficiency for selecting the events has been determined from simulation. The
measured cross-sections are:

$\displaystyle\sigma_{J/\psi\to\mu^{+}\mu^{−}}(2<\eta_{\mu^{+}},\eta{\mu^{−}}<4.5)$
$\displaystyle=$ $\displaystyle 474\pm 12\pm 51\pm 92\ \mbox{pb}$
$\displaystyle\sigma_{\psi(2S)\to\mu^{+}\mu^{−}}(2<\eta_{\mu^{+}},\eta{\mu^{−}}<4.5)$
$\displaystyle=$ $\displaystyle 12.2\pm 1.8\pm 1.3\pm 2.4\ \mbox{pb}$
$\displaystyle\sigma_{\chi_{c0}\to
J/\psi\gamma\to\mu^{+}\mu^{−}\gamma}(2<\eta_{\mu^{+}},\eta{\mu^{−}},\eta_{\gamma}<4.5)$
$\displaystyle=$ $\displaystyle 9.3\pm 2.2\pm 3.5\pm 1.8\ \mbox{pb}$ (10)
$\displaystyle\sigma_{\chi_{c1}\to
J/\psi\gamma\to\mu^{+}\mu^{−}\gamma}(2<\eta_{\mu^{+}},\eta{\mu^{−}},\eta_{\gamma}<4.5)$
$\displaystyle=$ $\displaystyle 16.4\pm 5.3\pm 5.8\pm 3.2\ \mbox{pb}$
$\displaystyle\sigma_{\chi_{c2}\to
J/\psi\gamma\to\mu^{+}\mu^{−}\gamma}(2<\eta_{\mu^{+}},\eta{\mu^{−}},\eta_{\gamma}<4.5)$
$\displaystyle=$ $\displaystyle 28.0\pm 5.4\pm 9.7\pm 5.4\ \mbox{pb}$
$\displaystyle\sigma_{pp\to
p\mu^{+}\mu^{-}p}(2<\eta_{\mu^{+}},\eta{\mu^{−}}<4.5,M_{\mu^{+}\mu^{-}}>2.5\
\mbox{GeV/c${}^{2}$})$ $\displaystyle=$ $\displaystyle 67\pm 10\pm 7\pm 15\
\mbox{pb}$

where the first uncertainty is statistical, the second is systematic, and the
third comes from the estimate of the luminosity. Note that these numbers are
cross-section times the branching ratio into the final state of interest, and
all final state particles are required to be between pseudorapidities of 2 and
4.5. The cross sections are quoted in a limited pseudorapidity range as the
models under consideration have pseudorapidity dependence.

##  IX Summary

LHCb performed many analysis of the quarkonium states using 2010 collected
data (35 pb-1). The measurement of the production cross sections of charmonium
and bottomonium states ($\psi(2S)$, $\Upsilon(1S)$, $\chi_{c}$) are useful to
test theoretical models. For the exotic states, the measurement of the
$X(3872)$ mass and cross section has been performed and the CDF narrow
$X(4140)$ has been studied using a dataset of 376 pb-1 and its existence not
confirmed. LHCb has a very high $J/\psi$ statistic in 2011 data that will
allow to have a lot of new results in the near future.

## References

  * (1) G. T. Bodwin, et al. Phys.Rev., D51:1125–1171, 1995.
  * (2) J. M. Campbell, et al. Phys.Rev.Lett., 98:252002, 2007.
  * (3) Y.-Q. Ma, et al. Phys.Rev., D83:111503, 2011.
  * (4) S.K. Choi et al. [Belle collaboration], Phys. Rev. Lett. 91 (2003) 262001, arXiv:hep-ex/0309032.
  * (5) D. Acosta et al. [CDF collaboration], Phys. Rev. Lett. 93 (2004) 072001, arXiv:hep-ex/0312021.
  * (6) V. M. Abazov et al. [D0 collaboration], Phys. Rev. Lett. 93 (2004) 162002, arXiv:hep-ex/0405004.
  * (7) B. Aubert et al. [Babar collaboration], Phys. Rev. D 71 (2005) 071103, arXiv:hep-ex/0406022.
  * (8) E. Swanson, Physics Reports 429 (2006) 243, arXiv:hep-ph/0601110.
  * (9) L. Maiani, F. Piccinini, A. D. Polosa and V. Riquer, Phys. Rev. D 71 (2005) 014028, arXiv:hep-ph/0412098.
  * (10) The CDF Collaboration, T. Aaltonen et al., arXiv:1101.6058.
  * (11) X. Liu and S.-L. Zhu, Phys.Rev. D80 (2009) 017502, [arXiv:0903.2529].
  * (12) Z.-G. Wang, Z.-C. Liu, and X.-H. Zhang, Eur. Phys. J. C64 (2009) 373–386, [arXiv:0907.1467].
  * (13) F. Stancu, J. Phys. G37 (2010) 075017, [arXiv:0906.2485].
  * (14) N. V. Drenska, R. Faccini, and A. D. Polosa, Phys. Rev. D79 (2009) 077502, [arXiv:0902.2803].
  * (15) N. 130 Mahajan, Phys. Lett. B679 (2009) 228–230, [arXiv:0903.3107].
  * (16) Z.-G. Wang, Eur. Phys. J. C63 (2009) 115–122, [arXiv:0903.5200].
  * (17) X. Liu, Phys. Lett. B680 (2009) 137–140, [arXiv:0904.0136].
  * (18) D. V. Bugg, arXiv:1103.5363.
  * (19) LHCb-CONF-2011-026.
  * (20) Y.-Q.Ma et al., Phys. Rev. Lett. 106, 042002, (2011) and private communication.
  * (21) LHCb-CONF-2011-016.
  * (22) The LHCb Collaboration, R. Aaij et al., Eur.Phys.J.C71:1645,2011.
  * (23) The CMS Collaboration, CMSBPH-10-003, CERN-PH-EP-2010-055; arXiv:1012.5545 [hep-ex].
  * (24) LHCb-CONF-2011-020.
  * (25) L. A. Harland-Lang andW. J. Stirling, http://projects.hepforge.org/superchic/chigen.html.
  * (26) Y. Ma, K. Wang, K. Chao, arXiv:1002.3987 (hep-ph).
  * (27) LHCb-CONF-2011-021.
  * (28) B. Aubert et al. [Babar collaboration], Phys. Rev. D 77 (2008) 111101, arXiv:0803.2838 [hep-ex].
  * (29) T. Aaltonen et al. [CDF collaboration], Phys. Rev. Lett. 103 (2009) 152001, arXiv:0906.5218 [hep-ex].
  * (30) LHCb-CONF-2011-043.
  * (31) M. Pivk and F.R. Le Diberder , Nucl. Instrum. Meth. A 555 (2005) 356–369, arXiv:physics/0402083v3.
  * (32) S.-K. Choi et al. [Belle collaboration], arXiv:1107.0163v1 [hep-ex], submitted to Phys. Rev. D.
  * (33) LHCb-CONF-2011-045.
  * (34) LHCb-CONF-2011-022.