Abdoulaye Kane

Latifa Chafiq

Sadou Dalhatou

Pierre Bonnet

Maryline Nasr

Nathalie Gaillard

Jean Marie

Dangwang Dikdim

Guillaume Monier

Aymen Amine Assadi

Hicham Zeghioud
email: hicham.zeghioud@unilasalle.fr

g-C3N4/TiO2 S-scheme heterojunction photocatalyst with enhanced photocatalytic Carbamazepine degradation and mineralization

Keywords: photocatalysis, g-C3N4/TiO2, UVA, Carbamazepine, mineralization

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Hospitals consumed water about 750 liters per bed per day, which is 10 times higher than the daily consumption per citizen. This big consumption implies a significant amount of wastewater. In fact, more than 10% of drug residues found in the surface water in France come from hospitals [START_REF] Brunel | Elimination des effluents liquides des établissements hospitaliers[END_REF][START_REF] Emmanuel | Evaluation Des Risques Sanitaires Ecotoxicologiques Liés Aux Effluents Hospitaliers[END_REF]. The specificity of hospital effluents is their complex compositions. They contain 1000 times more detergents than drugs; they also contain pathogenic microorganisms, antibiotic-resistant bacteria, endocrine disruptors, metals, radioisotopes... [START_REF] Dremont | La Gestion des Effluents Liquides en Milieu Hospitalier[END_REF]. Despite their complexity, hospital effluents are considered to be of the same pollutant load as urban wastewaters and are discharged into public sewer networks [START_REF] Verlicchi | Hospital effluents as a source of emerging pollutants: An overview of micropollutants and sustainable treatment options[END_REF]. For all these reasons, the development of processes that will allow the pre-treatment of wastewater at health care institutions is necessary in order to reduce the risks of contamination of the public sanitation network, and therefore human and animal health.

Nowadays, Advanced Oxidation Processes (AOP) such as photocatalysis are being studied more and more as a hydrogen production technique [START_REF] Jiang | Step-scheme ZnO@ZnS hollow microspheres for improved photocatalytic H2 production performance[END_REF][START_REF] Liao | Hierarchical porous NiO as a noble-metal-free cocatalyst for enhanced photocatalytic H2 production of nitrogen-deficient g-C3N4[END_REF], fuel generation method [START_REF] Wang | Molecularfunctionalized engineering of porous carbon nitride nanosheets for widespectrum responsive solar fuel generation[END_REF] and alternative methods for the complete elimination of pollution in wastewater, and it's considered one of the most promising green chemistry technologies [START_REF] Senthil | Synthesis and characterization of low-cost g-C3N4/TiO2 composite with enhanced photocatalytic performance under visible-light irradiation[END_REF][START_REF] Miranda | Improved photocatalytic activity of g-C3N4/TiO2 composites prepared by a simple impregnation method[END_REF]. This process is based on the formation of a strong oxidant; the hydroxyl radical OH° which are characterized by an oxidation potential of +2.80V/ENH [START_REF] Carre | Compréhension des mécanismes lors de la photocatalyse appliquée à la dégradation des microorganismes[END_REF]. Hence, its high reactivity and its low selectivity allow the mineralization of many organic and inorganic molecules, thus it can ultimately lead to their mineralization, with the formation of CO2, H2O and inorganic ions [START_REF] Carre | Compréhension des mécanismes lors de la photocatalyse appliquée à la dégradation des microorganismes[END_REF][START_REF] Zaviska | Procédés d'oxydation avancée dans le traitement des eaux et des effluents industriels: Application à la dégradation des polluants réfractaires[END_REF]. Titanium dioxide (TiO2) is the most widely studied, and used for environmental remediation, hydrogen energy production, and solar cells due to its environmentalfriendly properties, good chemical stability, high abundance, low cost, and good photocatalytic activity [START_REF] Senthil | Synthesis and characterization of low-cost g-C3N4/TiO2 composite with enhanced photocatalytic performance under visible-light irradiation[END_REF][START_REF] Jiang | Construction of TiO2 hollow nanosphere/g-C3N4 composites with superior visible-light photocatalytic activity and mechanism insight[END_REF] Recently, graphite-like carbon nitride has attracted great interest due to its wide potential for photocatalytic application under visible light, owing to its band gap of 2.70 eV, good physicochemical stability, abundance, low cost and can be easily prepared [START_REF] Jiang | Construction of TiO2 hollow nanosphere/g-C3N4 composites with superior visible-light photocatalytic activity and mechanism insight[END_REF][START_REF] Wang | Recent Advances of Graphitic Carbon Nitride-Based Structures and Applications in Catalyst[END_REF]. However, many researchers confirm that the photocatalytic performances of TiO2 and g-C3N4 taken separately are still unsatisfying for heterogeneous photocatalysis, Owning to the low solar energy conversion efficiency and the high recombination of photogenerated electron hole pairs [START_REF] Jiang | Doping of graphitic carbon nitride for photocatalysis: A reveiw[END_REF][START_REF] Camposeco | Synthesis, characterization and photocatalytic activity of TiO2 nanostructures: Nanotubes, nanofibers, nanowires and nanoparticles[END_REF].

In order to overcome the drawbacks of this catalysts many strategies are studied such as doping [START_REF] Jiang | Doping of graphitic carbon nitride for photocatalysis: A reveiw[END_REF][START_REF] Chen | Large scale photochemical synthesis of M@TiO2 nanocomposites (M = Ag, Pd, Au, Pt) and their optical properties, CO oxidation performance, and antibacterial effect[END_REF], coupling with other semiconductor materials [START_REF] Askari | Synthesis of TiO2 nanoparticles and decorated multi-wall carbon nanotube (MWCNT) with anatase TiO2 nanoparticles and study of optical properties and structural characterization of TiO2/MWCNT nanocomposite[END_REF][START_REF] Zarrin | Photocatalytic activity of TiO2/Nb2O5/PANI and TiO2/Nb2O5/RGO as new nanocomposites for degradation of organic pollutants[END_REF], the coupling of the UV and visible light, the addition of strong oxidants such as H2O2, SO4 •-, Fe 2+ , and O3 in the reaction medium are used for improving the photocatalytic activity [START_REF] Ali | Carbamazepine Degradation by UV and UV-assisted AOPs: Kinetics, Mechanism and Toxicity Investigations[END_REF][START_REF] Rajeswari | A study on synergistic effect of photocatalytic ozonation for carbaryl degradation[END_REF]. However, coupling two semiconductors to form a heterojuntion structure, in other words composite is proven to be a very effective method to improve the photocatalytic efficiency, there are already many researchers focusing on the composite structure of TiO2 and g-C3N4, and the resultant catalysts prove higher photocatalytic activity [START_REF] Jiang | Construction of TiO2 hollow nanosphere/g-C3N4 composites with superior visible-light photocatalytic activity and mechanism insight[END_REF][START_REF] Chen | Construction of Heterostructured g-C3N4/Ag/TiO2 Microspheres with Enhanced Photocatalysis Performance under Visible-Light Irradiation[END_REF].

The improvement of g-C3N4/TiO2 catalyst performance is attributed to the formation of a good heterojunction between the materials leading to a synergetic effect. This includes the increasing of surface area, reduction of band gap, improved charge transfer at the interface as well as the inhibition of their recombination.

However, knowing the mechanism involved in the g-C3N4/TiO2 heterostructure is very important to explain the enhancement of the catalytic efficiency. Several techniques were employed for this purpose including quenching tests, electron paramagnetic resonance (EPR) analyses, spectroscopic analyses (XPS) and density functional theory (DFT) calculations. The combination of these methods has significantly contributed on identification of the real type of constructed heterostructure. For many years, the mechanisms of charge transfer were mainly explained by II-type heterojunction and Z-scheme [START_REF] Liu | Construction of direct Zscheme g-C3N4/TiO2 nanorod composites for promoting photocatalytic activity[END_REF][START_REF] Yan | Theoretical study on transport-scheme conversion of g-C3N4/TiO2 heterojunctions by oxygen vacancies[END_REF].

Unfortunately, these mechanisms presented some limitations from thermodynamically and kinetically plans particularly, thus reduce both oxidant and reductant power of the photocatalyst [START_REF] Wang | Rapid toxicity elimination of organic pollutants by the photocatalysis of environment-friendly and magnetically recoverable step-scheme SnFe2O4/ZnFe2O4 nano-heterojunctions[END_REF][START_REF] Fu | Ultrathin 2D/2D WO3/g-C3N4 step-scheme H2-production photocatalyst[END_REF]. Consequently, a new heterojunction called step-scheme "s-scheme" has been recently proposed by [START_REF] Fu | Ultrathin 2D/2D WO3/g-C3N4 step-scheme H2-production photocatalyst[END_REF] research group to overcome these drawbacks [START_REF] Fu | Ultrathin 2D/2D WO3/g-C3N4 step-scheme H2-production photocatalyst[END_REF]. In this mechanism, not only the migration of electrons is accelerated, but their reduction performance is effectively exploited [START_REF] Wu | Study on TiO2/g-C3N4 S-Scheme heterojunction photocatalyst for enhanced formaldehyde decomposition[END_REF].

Briefly, S-scheme heterojunction involves two n-type semiconductors where the one with high conduction band (CB) energy is considered as reduction photocatalyst and the other as oxidation photocatalyst [START_REF] Xu | S-scheme heterojunction photocatalyst[END_REF]. When irradiated, the electrons in the CB of OP were moved to VB of RP for recombination. Meanwhile, both endowed with high redox potential, electrons accumulated in CB of RP and holes contained in VB of OP were largely available for redox reactions [START_REF] Wang | Rapid toxicity elimination of organic pollutants by the photocatalysis of environment-friendly and magnetically recoverable step-scheme SnFe2O4/ZnFe2O4 nano-heterojunctions[END_REF][START_REF] Xu | Novel g-C3N4/g-C3N4 Sscheme isotype heterojunction for improved photocatalytic hydrogen generation[END_REF]. As a result, recent advances on designing g-C3N4 heterostructures based on s-scheme mechanism have significantly contributed to the improvement of photocatalytic degradation of pollutants [START_REF] Barzegar | S-scheme heterojunction g-C3N4/TiO2 with enhanced photocatalytic activity for degradation of a binary mixture of cationic dyes using solar parabolic trough reactor[END_REF][START_REF] Guo | S-scheme Ti0.7Sn0.3O2/g-C3N4 heterojunction composite for enhanced photocatalytic pollutants degradation[END_REF][START_REF] Nguyen | g-C3N4nanosheet/ZnCr2O4 S-scheme heterojunction photocatalyst with enhanced visible-light photocatalytic activity for degradation of phenol and tetracycline[END_REF][START_REF] Wang | Build-in electric field induced step-scheme TiO2/W18O49 heterojunction for enhanced photocatalytic activity under visible-light irradiation[END_REF][START_REF] Wang | Sulfur-doped g-C3N4/TiO2. Sscheme heterojunction photocatalyst for Congo Red photodegradation[END_REF].

In this present paper we report the elaboration of g-C3N4/TiO2 composites with different amount of g-C3N4 and starting from the synthesis of basic materials based on ecological criterion. The photocatalytic performance of as-prepared materials was evaluated through the photodegradation of Carbamazepine under UVA. The degradation mechanism was also discussed. We choose to study the degradation of Carbamazepine because it's one of the most frequently used drugs for the treatment of epilepsy, bipolar disorder and as mood stabilizing [START_REF] Pharmacomedicale | Carbamazépine (sauf comme régulateur de l'humeur)[END_REF]. The large-scale use and its resistance to conventional and biological water treatment explain its presence and its persistence in the surface waters, furthermore CBZ is categorized as toxic to aquatic life including bacteria, invertebrates, algae and fish [START_REF] Ali | Carbamazepine Degradation by UV and UV-assisted AOPs: Kinetics, Mechanism and Toxicity Investigations[END_REF][START_REF] Wang | Carbamazepine degradation by gamma irradiation coupled to biological treatment[END_REF].

Experiment

Chemical reagents

Titanium isopropoxyde (TTiP, 99%) was purchased from Sigma Aldrich, acetic acid (CH3COOH), dicyandiamide (C2H4N4), methanol (CH3OH) and carbamazepine (C15H12N2O) were purchased from VWR. All reagents were of analytical grade and were used without further purification. Ultra-pure water was used throughout the experiments carried out in this study to avoid any external contamination. The physicochemical properties of CBZ are presented in table 1. The TiO2 nanoparticles were prepared by the sol-gel method, based on the previous literature [START_REF] Askari | Synthesis of TiO2 nanoparticles and decorated multi-wall carbon nanotube (MWCNT) with anatase TiO2 nanoparticles and study of optical properties and structural characterization of TiO2/MWCNT nanocomposite[END_REF]. Typically, 20 mL of titanium isopropoxide were dissolved in 40 mL of acetic acid. To homogenize, the solution was set under constant stirring for 15 min, the hydrolysis was accomplished by adding 120 mL of deionized water drop by drop, and then the solution was maintained under stirring for 2 h. Afterwards, the gel was placed in the oven at 90°C for 12 h. Sallow-white crystals were obtained by drying the gel at 200°C for 2 h. The sample was pulverized in a porcelain mortar, and then annealed in air at 400 °C for 4 h.

Synthesis of graphitic carbon nitride (g-C3N4)

The bulk g-C3N4 was prepared by thermal polymerization based on previous literature [START_REF] Emmanuel | Evaluation Des Risques Sanitaires Ecotoxicologiques Liés Aux Effluents Hospitaliers[END_REF][START_REF] Dremont | La Gestion des Effluents Liquides en Milieu Hospitalier[END_REF]. Typically, 10 g of dicyandiamide was put into an alumina crucible, and then it was heated at 550°C in a muffle furnace for 4 hours at a heating rate of 20°C/min.

After cooled to room temperature, the yellow sample was pulverized into powders for further use.

Synthesis of g-C3N4/TiO2 composite

The g-C3N4/TiO2 composite were achieved by wet-impregnation method [START_REF] Senthil | Synthesis and characterization of low-cost g-C3N4/TiO2 composite with enhanced photocatalytic performance under visible-light irradiation[END_REF]. In a typical procedure, the appropriate amount of g-C3N4 and TiO2 were dissolved separately by methanol in beakers. Then, the beakers were placed in an ultrasonic bath for 30 min. Then the tow solutions were mixed and stirred at room temperature for 24 h. The resulting powder was maintained under room temperature for 12 h to evaporate the solvent. The obtained photocatalysts were denoted as X %CN/TO (X is the g-C3N4 percentage mass ratio).

The photo-catalytique treatment setup (Figure1S), its description and experimental conditions were reported in detail in supporting information.

Sample characterization

• The X-ray diffraction (XRD) patterns of as-prepared catalysts were made using a D8 Bruker spectrometer with λ=0.15418 nm as a wavelength of the beam, the incident angle of 2θ, 5-130° using 0,017° each step, and the acceleration tension is 40 kV and current emission equals 30 Ma.

• Nitrogen adsorption-desorption isotherm were obtained using a Quantachrome instrument at 77k. The specific surface area measures are based on Brunauer-Emmett-Teller's equation (BET).

• Raman spectra were acquired by using Raman spectrometer of JobinYvon company model T64000. The wavelength of laser was 514.5 nm (2.41 eV) and the power was set at 100 mWatt. The measurement was carried out in solid state by dispersing the sample powder upon glass slide under air at room temperature.

• The scanning electron microscopy (SEM) images of the photocatalysts were obtained using an (JEOL 5910 LV) apparatus with EDS elementary analysis using SDD detector (Bruker)

• The UV-vis diffuse reflectance spectra (UV-vis DRS) of the photocatalysts were recorded by Cary 300 instrument with scan Rate of 600 (nm/min) in shifting range of 80 to 500 cm -1 .

• The chemical states of wt%g-C3N4/TiO2 composites were analyzed using Xray photoelectron spectroscopy (XPS) in an ultrahigh vacuum photoelectron spectrometer equipped with an Omicron DAR 400 X-ray source (Mg Kα source at 1253.6 eV) and an Omicron EA 125 hemispherical analyzer.

Results and discussion

Crystal structure

The XRD patterns of the prepared samples are shown in Fig. 1. For TiO2 (figure 1a), the diffraction picks are : 2θ= 25,32°, 37,82°, 47,07°, 53,97°, 55,15°, 62,75°, 68,72°, 70,30° and 75,12° the projections of these values on a crystallographic plane gives the following (hkl) coordinates : (101), ( 004), ( 200), ( 105), ( 211), ( 204), ( 116), ( 220) and ( 215). Based on these results we conclude that the crystallographic plane corresponds to Anatase phase (JCPDS 21-1272), which confirms that our TiO2 is mostly Anatase [START_REF] Pragathiswaran | Synthesis and characterization of TiO2/ZnO-Ag nanocomposite for photocatalytic degradation of dyes and anti-microbial activity[END_REF]. The g-C3N4 analysis results (figure 1b),

shows that the compound is pure with a pic at 2θ = 13,1° that corresponds to the plan (100) of tri-s-triazine, and the pic at 27,4° relative to crystallographic plane (002) which is generated by an inter-layer stack [START_REF] Li | the enhanced photo-catalytic CO2 reduction performance of g-C3N4 with high selectivity by coupling CoNiSx[END_REF][START_REF] Wang | In situ micro wave assisted synthesis of porous N TiO2/g-C3N4 heterojunctions with enhanced visiblelight photocatalytic properties[END_REF].

Regarding to composite materials (fig. 1c), we notice that TiO2 and g-C3N4 coexist in composites, the peaks appeared on the XRD patterns of both g-C3N4 and TiO2 can be observed without obvious change. The effect of thermal treatment of g-C3N4 under Argon was also discussed in fig 2S and3S. The crystallinity of the composites 10% and 30% of g-C3N4 was measured by Raman spectroscopy (Fig. 1d). The figures display the presence of TiO2 anatase phase, which is correlated with diffraction analysis (The main bands of interest are 143 cm -1 , 195 cm -1 , 395 cm -1 , 515 cm -1 , and 640 cm -1 ) [START_REF] Al-Hajjia | Construction of mesoporous g-C3N4/TiO2 nanocrystals with enhanced photonic efficiency[END_REF], while the g-C3N4 does not appear on the Raman spectrum because of its fluorescence.

Specific Surface Area (SSA)

The N2 adsorption-desorption isotherm of TiO2, g-C3N4, A10 and A30 are shown in table 2. According to the Brunauer-Emmett-Teller (BET) theory, pure TiO2 had a high surface area of 125.57 m 2 /g, while g-C3N4 had a relatively low surface area of 10.51 m 2 /g. This means that the elaborated TiO2 would be able to better fix the molecule to be degraded than g-C3N4. Moreover, the initial surface area of TiO2 gradually diminish as carbon nitride presence rises, the value goes from 80.65 m 2 /g to 59.67 m 2 /g for 10% and 30% of carbon nitride respectively. This is due to the effect of the incorporation of the carbon nitride into the TiO2 pores.

Morphology and elemental analysis

The morphology and microstructure of as-prepared samples were further investigated by SEM images (Fig. 2). The SEM studies of the composite 10%wt of carbon nitride (Fig. 3a) showed agglomerated TiO2 nanoparticles attached on to the sheet-like structure of g-C3N4. Thus, we can conclude that the composite A10 present a good homogeneity. From Fig. 3c, it can be seen that the composite with 30%wt of carbon nitride present less good homogeneity comparing to A10, but steel have a good morphology, and A30 could present a good ability to ensure a good separation of the charges. Thus, this will allow decreasing the rate of recombination.

Moreover, the EDX analysis clearly confirms, the purity of the all our samples and also confirms the presence of g-C3N4 in lower amount with respect to TiO2 in both composites. The good homogeneity between TiO2 and g-C3N4 should facilitate the efficient removal of photo-generated electrons accumulate on (101) facets of TiO2, leaving holes left on (001) facets. This efficient special separation of photogenerated electron-hole pairs in the composite should benefit the photocatalytic activity. 1

1 Effect of contact interface between TiO2 and g-C3N4 on the photoreactivity of g-C3N4/TiO2 photocatalyst: (0 0 1) vs (1 0 1) facets of TiO2. The XPS spectra of the Ti 2p, C 1s, O1s and N1s core levels are presented in Fig. 4 for A10 and A30. Slight differences between the two wt%g-C3N4/TiO2 composite spectra are detected. For A30, a pure TiO2 phase is shown by the Ti2p (Ti2p3/2 at 458.3 eV) and O1s peak decompositions, and the gC3N4 phase is observed in the C1s (N-C=N at 288.5 eV) and N1s (C-N=C at 398.4 eV) peak decompositions [START_REF] Li | Efficient and Stable Photoelectrochemical Seawater Splitting with TiO2@g-C3N4 Nanorod Arrays Decorated by Co-Pi[END_REF].

These results point out the fact that there are very weak interactions between the TiO2 and the gC3N4 particles. For A10, the TiO2 phase is also revealed in the Ti2p and O1s peak decompositions but a slight widening of the Ti2p peak is observed which may be due to the formation of Ti 3+ or to the possible interaction with gC3N4.

This last interpretation is strengthened by the energy shift observed in the C1s (N-C=N at 289.7 eV) and N1s (C-N=C at 397.8 eV) peaks. Moreover, the XPS results are consistent with EDX analysis showing no contamination.

Optical absorption studies (UV-vis Diffuse Reflectance)

We carried out the diffuse spectral analysis (DRS) of the calcined TiO2, the pure g-C3N4 and the two composite A10 and A30, in order to study the absorption spectra and the change of the band gap of our samples caused by the heat treatment and the addition of 10 and 30%wt of g-C3N4. According to the obtained results of the DRS as displayed in Fig. 5 (a) the absorption sharp edges of TiO2, the pure g-C3N4

and the two composites A10 and A30 were found to be around 399. al [START_REF] Liao | Nitrogen vacancy induced in situ g-C3N4 p-n homojunction for boosting visible light-driven hydrogen evolution[END_REF]. It is obvious that the TiO2 and the two composites will absorb in visible region, the band gap was decreased because of the addition of the g-C3N4 and because of the creation of the oxygen vacancies due to the thermal treatment. Indeed, comparing to the pure TiO2, the composites exhibited enhanced absorption at higher wavelengths due to the optimal heterojunction created between the two basic materials. From Fig 6(a), it can be seen that the doping of TiO2 increase the CBZ removal. After 6 h of UV light irradiation, the degradation percentages were 5.97, 59.94, 71.41 and 64.39% for g-C3N4, TiO2, 10% g-C3N4/TiO2 and 30% g-C3N4/TiO2, respectively. It is obvious g-C3N4 exhibited the lowest photocatalytic effect. This is because of its poor photo-response under UV area. The 10% g-C3N4/TiO2 sample shows the best catalytic performance because of the low band gap values (2.78 eV) compared to the other materials (TiO2 (3.11 eV) and 30% g-C3N4/TiO2, (2.99 eV)) evaluated from Diffuse Reflectance Spectra, which can lead to an enhancement of the absorption of UV light. Both composite materials possess higher degradation activity than pure TiO2, which highlight the synergetic effect between the two raw materials through the electron-hole separation [START_REF] Liu | Synthesis of g-C3N4/TiO2 nanostructures for enhanced photocatalytic reduction of U(VI) in water[END_REF][START_REF] Boonprakob | Enhanced visible-light photocatalytic activity of g-C3N4/TiO2 films[END_REF]. In addition, the performance of composite material revealed that A10 is better than A30, meaning an optimal value of g-C3N4 loading is 10%. This behavior could be explained by two reasons: in one hand, the loading of excess amount of g-C3N4 particles in TiO2 may reduce the availability of catalytic sites for CBZ degradation by a formation of a barrier layer coating the surface of TiO2 [START_REF] Li | Synthesis of carbon-doped g-C3N4 composites with enhanced visible-light photocatalytic activity[END_REF]. This clearly is in consistent with the less homogeneity structure of A30 compare to A10 observed in SEM images (Fig 2 ) as well as the BET surface area of the samples (Table 2). Secondly, the superior light absorption for A30 means it could be more sensitive in higher wavelengths according to UV-Vis absorption spectra and then it might be have good catalytic activity in this light area compare to the one used in this study.

The kinetic degradation of CBZ has been checked according to the first-order model which the equation ( 1) is expressed below:

𝑙𝑛 𝐶 𝐶 0 = -𝑘𝑡 (1)
Where: 𝐶 0 and 𝐶 are the initial concentration of CBZ and after a given treatment time.

𝑘 is the apparent rate constant (min -1 ) and 𝑡 the time (min).

The Fig 6(b) illustrates the plot of ln C0/C vs t. It could be observed that the degradation of CBZ by g-C3N4/TiO2 composite fits well the pseudo-first order model which can be explained by the straight line and good regression coefficient (R 2 > 0.98). Such as for the degradation percentage, the reaction rate is fastest in the case of A10 material (k=0.0034 min -1 ) compare to the other materials. The apparent kinetics constants were 0.0025 min -1 and 0.0029 min -1 for A30 and pure TiO2, respectively. This finding indicates that the photocatalytic performance has been promoted through the loading of g-C3N4 onto TiO2.

The present results are compared with those reported in literature (Table 3). Even some works reported relatively higher photodegradation with less time taken for the photocatalytic treatment, the initial concentration of Carbamazepine treated in the present study is relatively higher than those reported studies. The developed catalyst presented a good specific area and an interesting band gap.

Study

The effect of some operating parameters such as calcination of catalyst and H2O2 addition on the photo-catalytique degradation of CBZ was also investigated and discussed in the supporting information (Fig 7S).

Mechanism of CBZ degradation

For better understanding the mechanism of the augmentation of the degradation efficiency by 10% g-C3N4/TiO2 composite material, the charge separation at the interface of the two semi-conductors was investigated and the possible degradation mechanism of CBZ under 10% g-C3N4/TiO2/UV system was illustrated in Fig 7. The conduction (𝐸 𝐶𝐵 ) and valence (𝐸 𝑉𝐵 ) bands energy were calculated through the following equation 2 and 3 [START_REF] Senthil | Synthesis and characterization of low-cost g-C3N4/TiO2 composite with enhanced photocatalytic performance under visible-light irradiation[END_REF]:

𝐸 𝐶𝐵 = 𝜒 -𝐸 0 - 1 2 𝐸 𝑔 (2)
𝐸 𝑔 = 𝐸 𝑉𝐵 -𝐸 𝐶𝐵 (3)
Where: 𝜒 is the electronegativity of the semiconductor and the value were 4.73 and 5.81eV for g-C3N4 and TiO2, respectively [START_REF] Chen | Construction of heterostructured g-C3N4/Ag/TiO2 microspheres with enhanced photocatalysis performance under visible-light irradiation[END_REF]. 𝐸 0 is the energy of free electrons vs NHE (4.5 eV) [START_REF] Xu | The absolute energy positions of conduction and valence bands of selected semiconducting minerals[END_REF]. From UV-Vis absorption spectra the band gap energy (𝐸 𝑔 )

obtained in this study were 3.11 and 2.7 eV for pure TiO2 and g-C3N4, respectively.

Then, the 𝐸 𝐶𝐵 and 𝐸 𝑉𝐵 were calculated and the values were of -0.25 and +2.86 eV for TiO2, whereas the values were -1.12 and +1.58 eV for g-C3N4.

Additionally, it is widely reported that the energy level of Fermi for pure TiO2 is less than pure g-C3N4 [START_REF] Liu | A new understanding of the photocatalytic mechanism of the direct Z-scheme g-C3N4/TiO2 heterostructure[END_REF][START_REF] Wang | Construction of NiO/g-C3N4 p-n heterojunctions for enhanced photocatalytic CO2 reduction[END_REF]. However, when they are in contact such as the present case, the electrons migrate continuously from g-C3N4 to TiO2 up to reaching the same Fermi energy. As a result, those movements of electrons provoke the upward and downward band edge of TiO2 and g-C3N4, respectively. This migration of electron also leads to the creation of partial charges on each semiconductor and a built of an internal electric field with TiO2 positively charged whereas g-C3N4 is negatively charged. The obtained 𝐸 𝐶𝐵 value of TiO2 (-0.25 eV) superior than redox potential of O2/O2 •-(-0.33 V vs. NHE, pH = 7) [START_REF] Wang | Build-in electric field induced step-scheme TiO2/W18O49 heterojunction for enhanced photocatalytic activity under visible-light irradiation[END_REF], obviously suppose that reaction of reduction of adsorbed O2 to form O2 •-cannot occur on CB of TiO2. In the same way, the H2O oxidation to form HO • at g-C3N4 VB could not be possible due to fact that the 𝐸 𝑉𝐵 value (+1.58 eV) is less positive than potential of HO • /H2O (2.27 V vs. NHE, pH = 7) [START_REF] Li | In situ growing Bi2MoO6 on gC3N4 nanosheets with enhanced photocatalytic hydrogen evolution and disinfection of bacteria under visible light irradiation[END_REF]. Consequently, the charge migration exhibited in this work obeys also to the typical S-scheme heterojunction mechanism [START_REF] Xu | S-scheme heterojunction photocatalyst[END_REF][START_REF] Barzegar | S-scheme heterojunction g-C3N4/TiO2 with enhanced photocatalytic activity for degradation of a binary mixture of cationic dyes using solar parabolic trough reactor[END_REF][START_REF] Guo | S-scheme Ti0.7Sn0.3O2/g-C3N4 heterojunction composite for enhanced photocatalytic pollutants degradation[END_REF][START_REF] Wang | Sulfur-doped g-C3N4/TiO2. Sscheme heterojunction photocatalyst for Congo Red photodegradation[END_REF]. The photocatalytic mechanism of CBZ degradation could be explained as follows: when the composite material is irradiated by UV light, the electrons were moved from VB to CB of TiO2 (δ-) .

Subsequently, these electrons were transferred to the VB of g-C3N4 (δ+) for recombination with holes. Therefore, free electrons in CB of g-C3N4 and holes in VB of TiO2 were effectively used in oxidizing reactions for CBZ removal while owing in the same time the long life of useful photo-generated species. This inhibition of electron-holes pair recombination in this S-scheme heterojunction is responsible in the improvement of photocatalytic activity.

Mineralization studies

In order to test the catalytic ability of the composite for mineralization of CBZ, the total organic carbon (TOC) has been measured after 6 h of irradiation. The result is displayed at the Fig 8.

Conclusion

The new catalytic materials g-C3N4/TiO2 were prepared by wet-impregnation method.

Different amounts of g-C3N4 were successfully loaded onto TiO2 according to the characterization analyses. The thermal treatment contributes to create the oxygen vacancies. The A10 material showed well distribution of g-C3N4 on TiO2 and the morphology reveals a quite homogeneity structure. The materials were tested for the photodegradation of CBZ under UV irradiation. The 10% g-C3N4/TiO2 showed remarkable catalytic activity with the best degradation percentage of 71.41% and the highest reaction rate (0.0035 min -1 ). The enhancement of catalytic efficiency of the composite was attributed to the good heterojunction between the two semiconductors leading to a synergetic effect. This, through an increasing of the surface area, diminution of band gap and the efficient electron-hole pairs separation. This study demonstrates an efficient approach to improve the performance of heterogeneous photocatalysis applied in the field of wastewater treatment.

Figure 1 :

 1 Figure 1: DRX of; (a) TiO2 with A for Anatase (b); g-C3N4, (c) the composite

Figure 2 :Figure 3 :

 23 Figure 2: SEM images of A10 and A30

Figure 4 :

 4 Figure 4: the XPS spectra for A10 and A30.

Figure 5 : 3 . 5 . Evaluation of photocatalytic activity of samples 3 . 5 . 1 .

 535351 Figure 5: (a) Diffuse Reflectance Spectra of pure g-C3N4, TiO2 and the tow

Figure 6 :

 6 Figure 6: (a) Degradation of CBZ by photocatalysis under UV irradiation, (b) First-

Figure 7 :

 7 Figure 7: Possible mechanism of g-C3N4/TiO2 heterostructure under UV irradiation

Figure 8 :

 8 Figure 8: Mineralization of CBZ by photocalysis under UV irradiation. Conditions:

Table 1 :

 1 Physicochemical properties of CBZ.

	Propriety	Carbamazepine
	Empirical formula	C15H12N2O
	Molecular weight (g/mol)	236,27 g/mol
	Abbreviation	CBZ
	λmax (nm)	265 nm
	Chemical structure
	2

.2. Synthesis: Catalyst preparation 2.2.1. Synthesis of titanium dioxide (TiO2)

Table 2 :

 2 The specific surface area of the samples

	Materials	g-C3N4	TiO2	A10	A30
	Specific surface area (m 2 /g)	10.51	125.57	80.65	59.67

Table 3 :

 3 Comparison of present degradation efficiency with literature data.

	Photocatalyst Bande	Surface	Pollutant		Treatment	Degradation	Referenc
		Gap	area	(Initial.con)	duration	efficiency	es
		(eV)	(SBET)
			(m 2 /g)
	g-C3N4/TiO2	2.8	176	-		-	-	[45]
	(1:4)
	g-C3N4/TiO2	2.83	70.2	Rhodium		80 min	100 %	[46]
				Boride	(4.79
				mg/l)
	g-C3N4/TiO2	-	-	Methyl orange	180 min	80.33 %	[47]
				(5 mg/l)
	g-C3N4/TiO2	-	-	2-Chlorophen	180 min	96.6%	[48]
	nanotube			(30 mg/l)

Acknowledgments

The authors would like to thank Ivane LELIEVRE (Unilasalle Rennes) for her technical help and Lawrence Frezet for BET measurements (ICCF). The authors also thank the Labex IMOBS3 for the financial support (Maryline NASR grant for N-MAFEN² project).


Siyuan Niu
email: siyuan.niu@lirmm.fr

Aida Todri-Sanial

Analyzing Strategies for Dynamical Decoupling Insertion on IBM Quantum Computer

Keywords: Dynamical decoupling, Error mitigation, NISQ era

Near-term quantum devices are subject to errors and decoherence error is one of the non-negligible sources. Dynamical decoupling (DD) is a well-known technique to protect idle qubits from decoherence error. However, the optimal approach to inserting DD sequences still remains unclear. In this paper, we identify different conditions that lead to idle qubits and evaluate strategies for DD insertion under these specific conditions. Specifically, we divide the idle qubit into crosstalkidle or idle-idle qubit depending on its coupling with other qubits and report the DD insertion strategies for the two types of idle qubits. We also perform Ramsey experiment to understand the reasons behind the strategy choice. Finally, we provide design guidelines for DD insertion for small circuits and insights for large-scale circuit design.

I. INTRODUCTION

Quantum computing has become an active research field in recent years. It is expected to address some classically intractable computational problems, such as integer factorization [START_REF] Peter | Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer[END_REF], optimization [START_REF] Farhi | A quantum approximate optimization algorithm[END_REF], quantum simulation [START_REF] Iulia | Quantum simulation[END_REF], etc. Quantum supremacy has been achieved for superconducting and photonic device [START_REF] Arute | Quantum supremacy using a programmable superconducting processor[END_REF], [START_REF] Zhong | Quantum computational advantage using photons[END_REF]. IBM released the largest universal quantum machine with 127 superconducting qubits. However, the near-term quantum hardware is still regarded as Noisy Intermediate-Scaled Quantum (NISQ) machine [START_REF] Preskill | Quantum computing in the nisq era and beyond[END_REF] with hardware limitations depending on the quantum technology used. Qubits have unavoidable error rates, which makes finding "real" quantum advantage difficult. It is predicted to require more than one thousand physical qubits to encode an errorfree logical qubit for error correction [START_REF] Austin G Fowler | Surface codes: Towards practical large-scale quantum computation[END_REF], which is beyond the current quantum technology. Instead, quantum error mitigation was proposed to reduce the noise impact and improve the fidelity of NISQ applications [START_REF] Endo | Practical quantum error mitigation for near-future applications[END_REF].

There are different error sources on NISQ devices, such as gate error, readout error, decoherence error and crosstalk. In our paper, we focus on the decoherence error which occurs on the idle qubits, where errors accumulate over time even if there is no operation, also called idling error. In superconducting quantum devices, quantum operations are performed in parallel if applied to different qubits. But these parallel operations lead to a high probability of idle qubits introducing decoherence This work is funded by the QuantUM Initiative of the Region Occitanie, University of Montpellier and IBM Montpellier. errors, mainly for two main reasons: (1) Gate latencies vary across different qubits and gate types. For example, for IBM Q 27 Montreal, the average gate duration for two-qubit gates is 2.6x larger than for single-qubit gates and the longest twoqubit gate duration is 2.3x larger than the shortest one. [START_REF] Bravyi | Mitigating measurement errors in multiqubit experiments[END_REF] The nearest-neighbor connectivity of superconducting device needs SWAP gate to realize non-adjacent connections, which is composed of three CNOTs and requires long duration to execute.

Dynamical Decoupling (DD) is one of the simplest methods for error mitigation without any extra circuit overhead, and is dedicated to the decoherence error suppression [START_REF] Alexandre | Robust dynamical decoupling[END_REF]. The basic idea of DD is to repetitively insert DD sequences to the idle qubit and make sure to return back to the qubit's original state after insertion. There are various DD sequences, such as CPMG [START_REF] Meiboom | Modified spin-echo method for measuring nuclear relaxation times[END_REF], XY4 [START_REF] Viola | Dynamical decoupling of open quantum systems[END_REF], KDD [START_REF] Alexandre | Robust dynamical decoupling for quantum computing and quantum memory[END_REF], etc. It has been demonstrated to improve circuit fidelity on IBM and Rigetti platforms using XY4 decoupling sequence [START_REF] Pokharel | Demonstration of fidelity improvement using dynamical decoupling with superconducting qubits[END_REF]. Also, DD has been proved to have the capability of suppressing ZZ-crosstalk in superconducting qubits [START_REF] Vinay Tripathi | Suppression of crosstalk in superconducting qubits using dynamical decoupling[END_REF]. However, DD insertion is not trivial such that it cannot always improve application fidelity. Whether one can obtain favorable DD insertion depends on many factors such as the given quantum application, the choice of DD strategy, the number of repetitions of DD sequences, etc. Several quantum software tools were developed to help design DD insertion strategies to obtain better circuit fidelity. Das et al. proposed ADAPT [START_REF] Das | Adapt: Mitigating idling errors in qubits via adaptive dynamical decoupling[END_REF], an adaptive DD framework to insert DD pulses to a subset of idle qubits. Moreover, Ravi et al. introduced VAQEM [START_REF] Subramanian | Vaqem: A variational approach to quantum error mitigation[END_REF], which dynamically selects the length of DD sequence for variational quantum algorithms. These two techniques need a large overhead of tuning DD sequences with extra circuits. Recently, Niu et al. reported the impact of different DD strategies on well-known quantum applications to provide insights for application-specific error mitigation using DD technique [START_REF] Niu | Pulse-level noise mitigation on quantum applications[END_REF]. However, these previous methods only concentrate on application-level DD insertion instead of understanding and analyzing the conditions that lead to idle qubits, hence decoherence errors.

In order to find the optimal DD insertion strategy, one needs first to understand when the idle qubits usually occur in quantum circuits. In our paper, we address this issue by investigating the conditions that cause idle qubits and evaluating the strategies for DD insertion under specific configurations to improve the circuit fidelity. We create a three-qubit circuit to discuss the DD insertion strategy and perform the experiments on IBM Q 7 Lagos. Our major contributions can be listed as follows:

• We identify several conditions leading to idle qubits:

CNOT-induced idle, SWAP-induced idle (multiple CNOTs), and combine them with additional delays that might be caused by other operations. • For each condition, we assess different DD insertion strategies and report their impact on idle-idle qubit and crosstalk-idle qubit. • We perform the Ramsey experiment to explain the results and provide guidelines for DD insertion technique on small-scale quantum circuits as well as insights for largescale circuits.

II. BACKGROUND

A. NISQ hardware

Today's quantum device is in the NISQ era with hardware limitations and erroneous operations. For superconducting quantum devices, one of the currently leading quantum technologies and the focus of this paper, the connectivity can only be realized between neighbor qubits. A quantum circuit needs to be transpiled to satisfy the nearest-neighbor connectivity by inserting SWAP gates before executing on the real hardware, known as the qubit mapping problem [START_REF] Li | Tackling the qubit mapping problem for nisq-era quantum devices[END_REF], [START_REF] Niu | A hardware-aware heuristic for the qubit mapping problem in the nisq era[END_REF]. Moreover, the near-term quantum machine is not error corrected and we broadly classify the error sources as follows: (1) Operational error. It includes gate error and SPAM error. (2) Decoherence error. A qubit can only maintain its state for a limited amount of time due to its fragile nature. (3) Crosstalk. The state of a qubit might be corrupted by the simultaneous operations occurring on its neighbor qubits. All of these errors prevent a quantum circuit to obtain reliable results. Several quantum error mitigation techniques have been proposed, such as zero noise extrapolation [START_REF] Temme | Error mitigation for short-depth quantum circuits[END_REF], Pauli twirling [START_REF] Cai | Constructing smaller pauli twirling sets for arbitrary error channels[END_REF], measurement error mitigation [START_REF] Bravyi | Mitigating measurement errors in multiqubit experiments[END_REF], dynamical decoupling [START_REF] Alexandre | Robust dynamical decoupling[END_REF], etc.

B. Dynamical decoupling

DD is widely used to suppress the impact of decoherence errors caused by the interaction between the system and the environment. Instead of keeping a qubit in the idle state, DD continuously inserts pulses during the idle time and makes the qubit return back to its original state after DD insertion such that the overall DD sequences act like an identity gate. DD has shown benefits in improving circuit fidelity on IBM, Rigetti, and Google quantum devices [START_REF] Chen | Exponential suppression of bit or phase errors with cyclic error correction[END_REF], [START_REF] Pokharel | Demonstration of fidelity improvement using dynamical decoupling with superconducting qubits[END_REF].

Different DD strategies have been proposed, such as CPMG [START_REF] Meiboom | Modified spin-echo method for measuring nuclear relaxation times[END_REF], XY4 [START_REF] Viola | Dynamical decoupling of open quantum systems[END_REF], UDD [START_REF] Götz | Keeping a quantum bit alive by optimized π-pulse sequences[END_REF], KDD [START_REF] Alexandre | Robust dynamical decoupling for quantum computing and quantum memory[END_REF], etc. In our paper, we evaluate one of the most frequently used DD strategies: CPMG. It was introduced to reduce the inhomogeneous effects of the environment. The basis DD cycle to constitute the CPMG sequence is X -X.

C. Motivational example

The idle qubit can be divided into two types: (1) idle-idle qubit, where no operation is applied to its neighbor qubits in parallel. (2) crosstalk-idle qubit, where simultaneous operations are occurring on its neighbor qubits such that the target qubit has a probability of being influenced by crosstalk. It has been demonstrated that two-qubit operation is the dominant crosstalk source for simulataneous operations [START_REF] Murali | Software mitigation of crosstalk on noisy intermediate-scale quantum computers[END_REF]. Therefore, we ignore the case when crosstalk comes from parallel singlequbit operations and only consider the crosstalk due to CNOT gate.

H H + Delay q 0 q 1 q 2 (a) (b) H H + DD q 0 q 1 q 2
Fig. 1: A motivational example to show the DD impact on circuit fidelity where the main qubit q 0 is under (a) free evolution and (b) DD-inserted evolution.

Here, we show a motivational example in Fig. 1 to characterize the impact of DD sequence on idle-idle qubit and crosstalk-idle qubit. We create a 3-qubit circuit, where q 0 is the main qubit to measure, q 1 and q 2 are two spectator qubits to apply CNOT gate. First, the main qubit is under a free evolution where the duration of the delay (idle time) is the same as the CNOT gate applied to the spectator qubits (Fig. 1(a)). Second, the main qubit is under a DD-inserted evolution during the idle time (Fig. 1(b)). Two Hadamard gates are inserted at the beginning and the end of the circuit, respectively. The first H gate is used to put the main qubit in a superposition state. The second one is to undo the initial H gate such that the ideal state of the main qubit should be |0 if no errors occur during the idle time. The middle part inside the barriers is repeated from 10 to 50 times with a step of 10 to show the qubit state evolution across time. In our case, we evaluate the performance of CPMG so that a X -X sequence is inserted and only repeated once during the overall idle time. Note that, the number of

� � � � � � � � � � � � � � � � � � ����������������������������������������� ����� �� (a) �� �� �� �� �� ��������������� ��� ��� ��� ��� � � � � � � � � � � � � � � � � � � ������������������������������������ ����� �� (b)
Fig. 3: The average probability of obtaining |0 when the main qubit q 0 is considered as (a) crosstalk-idle qubit. (b) idle-idle qubit.

DD sequences inserted during the idle time can influence the circuit fidelity [START_REF] Subramanian | Vaqem: A variational approach to quantum error mitigation[END_REF], but this is not in the scope of this paper. We map the main qubit of the circuit to each physical qubit on the quantum hardware and the spectator qubits to two physically connected qubits. Note that, we always choose the hardware-native CNOT so that no additional single-qubit gate is required when executed on real quantum hardware. For example, if we map the main qubit q 0 to Q 0 in IBM Q 7 Lagos shown in Fig. 2, there are five possibilities for spectator qubits:

{Q 1 , Q 2 },{Q 1 , Q 3 },{Q 3 , Q 5 },{Q 4 , Q 5 },{Q 5 , Q 6 }.
Based on the impact of spectator qubits on the main qubit, the main qubit can be categorized into two types of idle qubit: (1) crosstalk-idle qubit, where one of the spectator qubit is a neighbor qubit to the target main qubit and the spectator qubits is one pair from

{Q 1 , Q 2 },{Q 1 , Q 3 }.
(2) idle-idle qubit, where the spectator qubits are further away from the target main qubit (the distance between spectator qubits and main qubit is more than one on hardware topology), and they can be from

{Q 3 , Q 5 },{Q 4 , Q 5 },{Q 5 , Q 6 }.
The results of the motivational example are shown in Fig. 3. The crosstalk impact is significant such that the probability of obtaining |0 for crosstalk-idle qubit decreases much faster than idle-idle qubit. When the quantum coherence is completely lost, the state of the main qubit becomes ρ = 1 2 (|0 0 + |1 1 ) so that the probability of obtaining |0 is approximately 0.5 in the end. DD is favorable to increase circuit fidelity for the two types of idle qubits.

III. RELATED WORKS

Even though DD has shown to play an important role in reducing decoherence error on different quantum machines, there is still no clear answer to the optimal way to insert DD pulses. Here are some interesting questions: (1) Which DD strategy performs the best for a specific platform or application? (2) Should we insert DD sequences to all the idle qubits? (3) What is the optimal number of the DD pulses during the idle time? (4) What conditions lead to idle qubits and what is the strategy to insert DD to idle qubits under different conditions? The first question was discussed in [START_REF] Niu | Pulse-level noise mitigation on quantum applications[END_REF], where the authors comprehensively evaluated the performance [START_REF] Das | Adapt: Mitigating idling errors in qubits via adaptive dynamical decoupling[END_REF]. They demonstrated that DD performs better when applied to a subset of qubits rather than all. However, this method requires executing a decoy circuit (a circuit that is structurally similar to the given circuit) with DD insertions on different subsets of qubits to find the best way to insert DD for the input circuit, which introduces a large overhead. Moreover, Ravi et al. introduced VAQEM to solve the third problem [START_REF] Subramanian | Vaqem: A variational approach to quantum error mitigation[END_REF], an error mitigation technique that tunes the number of DD sequences inserted in the idle time and selects the best one that boosts the circuit fidelity, which also requires additional circuit measurement overhead. In our paper, we focus on the fourth problem. We determine different conditions that lead to idle qubits and evaluate various strategies of inserting DD pulses to find the best solution that benefits the circuit fidelity.

q 0 q 1 q 2 (a) q 0 q 1 q 2 (b)

IV. CONTEXT-INDUCED IDLE QUBIT

For superconducting quantum devices, quantum operations are able to be executed in parallel if applied to different qubits. Parallel scheduling is also preferable since it can reduce the circuit depth so that the decoherence error is decreased as well. However, the gate latency varies across two-qubit gates and their mean latency is much longer than one-qubit gates. Moreover, due to the hardware limitation of the nearestneighbor connectivity for superconducting quantum devices, SWAP gates are generally needed to execute the two-qubit gates applied to qubits that are not physically connected. Therefore, idle qubits occur frequently in a quantum circuit. In general, there are two conditions leading to idle qubits: (1) CNOT-induced idle qubit caused by CNOT gates executed in parallel, shown in Fig. 4(a). ( 2) SWAP-induced idle qubit caused by simultaneous SWAP gates, shown in Fig. 4(b). Moreover, the idle time can become longer if some other gates are running in the middle that evolves q 1 or q 2 .

In this section, we discuss the impact of different DD insertion strategies on these two configurations for two types of idle qubits: idle-idle qubit and crosstalk-idle qubit. We create a 3-qubit circuit similarly to Fig. 1 and discuss the two conditions in the following sections.

A. CNOT-induced Idle Qubit

The basic configuration of purely CNOT-induced idle qubit is shown in section II-C, where inserting DD sequences to idle qubits can significantly improve the circuit fidelity for both idle-idle and crosstalk-idle qubits. Here, we discuss the case when the idle qubit is induced by a CNOT gate as well as some other gates, which are indicated by a Delay operation. Overall, there are four DD insertion strategies: (1) Insert DD pulses during the overall idle time (labeled as DD), shown in Fig. 5(b). ( 2) First insert DD sequence only during the CNOT gate operation and then a Delay operation, which has the same length as the Delay in spectator qubits (labeled as DD Delay), shown in Fig. 5(c). ( 3) First insert Delay operation during the parallel CNOT operation and then DD sequence during the Delay in spectator qubits (labeled as Delay DD), shown in Fig. 5(d). ( 4) Insert two independent DD pulses for CNOT and Delay, respectively (labeled as DD DD), shown in Fig. 5(e). The length of the Delay operation is the same as the CNOT duration to represent the extra idle time induced by other operations. We compare them with the baseline (Fig. 5(a)), where the main qubit q 0 is under a free evolution marked as Delay all, which is the overall duration of the spectator qubits. We repeat the part between the barriers from 10 to 50 times with a step of 10 to evaluate the fidelity change over time.

�� �� �� �� �� ��������������� ��� ��� ��� ��� ��� � � � � � � � � � � � � � � � � � � �������������������������������������������� ����� �� �������� �������� ����� (a) �� �� �� �� �� ��������������� ��� ��� ��� ��� ��� ��� ��� � � � � � � � � � � � � � � � � � � ��������������������������������������� ����� �� �������� �������� ����� (b)
Fig. 6: Results of CNOT and other delay induced idle qubit when main qubit q 0 is considered as (a) crosstalk-idle qubit. (b) idle-idle qubit.

The results are shown in Fig. 6. For crosstalk-idle qubit, double DD (DD DD) gives the best result and improves the fidelity by 15.7% compared to single DD. DD Delay and Delay DD obtain unsatisfactory results. Delay DD has the worst results and is even sometimes worse than the baseline. Whereas for idle-idle qubit, double DD and single DD perform approximately the same and also have the best results. DD Delay and Delay DD obtain very close results and are better than the baseline.

B. SWAP-induced Idle Qubit

For SWAP-induced idle qubit, we mainly focus on two cases:

(1) Purely SWAP-induced idle qubit. (2) SWAP and other delay induced idle qubit. Note that, since SWAP gate is composed of three CNOT gates, exploring the first case also indicates the DD insertion strategy for successive CNOTs.

For purely SWAP-induced idle time, there are two strategies: (1) Insert a sequence of DD pulses for the overall idle time (labeled as DD). ( 2) Insert DD pulses for each CNOT of the SWAP gate independently (labeled as DD*3). In other words, three sequences of DD pulses are inserted. The results are shown in Fig. 7. For crosstalk-idle qubit, inserting DD only once during the SWAP duration performs slightly better than inserting it three times. Both cases outperform the baseline. Whereas for idle-idle qubit, DD*3 or single DD gives similar results. It shows that the increase of DD sequences cannot always bring benefits for the circuit fidelity.

While for SWAP and other delay induced idle qubit, there are also four DD insertion strategies, similar as CNOT and other delay induced idle qubit as explained in section IV-A but replacing CNOT by SWAP. Since inserting DD pulses for the total SWAP duration performs better than inserting three independent DD pulses based on the previous experimental results, we only insert DD pulses once for SWAP gate. The results are shown in Fig. 8. Similarly as the experiments for CNOT plus delay induced idle qubit, double DD performs the best for crosstalk-idle qubit and the fidelity is improved by 20.4% compared with single DD. While both single DD and double DD achieve the best performance for idle-idle qubit. Delay DD keeps being the worst case for idle-idle and crosstalk-idle qubit.

V. RAMSEY EXPERIMENT

In order to explore the reasons why double DD performs better than single DD for crosstalk-idle qubit, we characterize the crosstalk impact on idle qubit by performing Ramsey experiments to evaluate the frequency changes.

There are two types of crosstalk. The first one is an alwayson-ZZ interaction between coupled qubits [START_REF] Kandala | Demonstration of a high-fidelity cnot gate for fixed-frequency transmons with engineered z z suppression[END_REF]. The second is due to the simultaneous operations and the most significant crosstalk is caused by the two-qubit cross-resonance gate [START_REF] Magesan | Effective hamiltonian models of the cross-resonance gate[END_REF]. We perform four types of Ramsey experiments to measure the detuning frequency of the main qubit: (1) a two-qubit circuit with q 1 in |0 (Fig. 9 (3) a three-qubit circuit where q 1 is initialized in |0 and CNOT gate is applied between q 1 and q 2 (Fig. 9(c)). (4) a three-qubit circuit where q 1 is initialized in |1 and CNOT gate is applied between q 1 and q 2 (Fig. 9(d)). The main qubit q 0 is first set to a superposition state and then performs a free evolution marked as a Delay. The duration of the Delay is set to match the simultaneous CNOT duration. After the delay, the main qubit adds a phase gate to better measure the detuning frequency and another H gate in the end. We construct a list of circuits consisting of 1 to 50 repetitions of the middle part of the circuit inside the barriers to evaluate the main qubit decoherence across time.

� � � � � �� �� �� �� ������������������ �� ��� ��� ��� ��� � � � � � � � � � � � � � � � � � � � � � � ���������������������������� ��������������������������� ��������������������������� ������������������������������ ������������������������������ (a) � � � � � �� �� �� �� ������������������ �� ��� ��� ��� ��� ��� ��� � � � � � � � � � � � � � � � � � � � � � � ���������������������������� ��������������������������� ��������������������������� ������������������������������ ������������������������������ (b)
Fig. 10: Results of Ramsey experiment by mapping the main qubit q 0 to Q 6 and mapping q 1 , q 2 to (a)

Q 5 , Q 4 . (b) Q 1 , Q 3 .
We show the results of mapping the main qubit q 0 to Q 6 on IBM Q 7 Lagos. We first map q 1 and q 2 to Q 5 and Q 4 on the hardware to physically couple q 0 and q 1 . We can see a frequency difference of around 14.6 kHz between the first two cases, which is due to the always-on-ZZ interaction. A frequency difference of 14.2 kHz exists between the first and the third case, which illustrates the impact of cross-resonance induced crosstalk on changing the adjacent qubit frequency. Whereas when we decouple the qubits by mapping q 1 and q 2 to Q 1 and Q 3 , in other words, no qubit is physically connected with the main qubit in the circuit, the frequency of the main qubit keeps almost the same as shown in Fig. 10(b). We have results with the same trend by mapping the main qubit to each physical qubit and the spectator qubits to all the connected qubits. Note that, for each experiment, the four types of circuits are executed in one job so that the calibration data keep unchanged.

Since DD is composed of π pulses, if there is a frequency shift on the DD-target qubit, it might introduce deviations in π rotations such that the idle qubit cannot come back to its original state after DD insertion, which explains the reason why double DD performs better than single DD for crosstalkidle qubit.

VI. DISCUSSION

As idle qubits frequently appear in superconducting circuits, as well as the severe impact of decoherence error, especially for large circuits, it is important to assess the optimal strategy to insert DD sequences under different conditions to mitigate the decoherence error. Here, we provide some guidelines that can help the community better insert DD pulses to build circuits with high fidelity:

• For crosstalk-idle qubit, if it is induced by CNOT or SWAP along with some other delay, it is recommended to insert DD sequences during CNOT/SWAP and the other delay separately.

• If the crosstalk-idle qubit is induced by a sequence of CNOT or SWAP, only applying DD once during the overall idle time can achieve good results.

• For idle-idle qubits, inserting DD only once during the total idle time is enough to improve the circuit fidelity significantly. In our paper, we use small circuits to understand the impact of different DD insertion strategies. Based on our observations and guidelines, our next step is to explore the DD insertion strategies for large circuits and build a general DD-insertion compiler to mitigate decoherence error without the overhead of executing additional circuits. Also, we only explore CPMG pulses due to its simplicity and single-axis π rotation. It is interesting to analyze other more complex DD techniques with extra phase shift such as XY 4. These techniques can further contribute to the pulse-level circuit optimization [START_REF] Liang | Variational quantum pulse learning[END_REF], [START_REF] Shi | Optimized compilation of aggregated instructions for realistic quantum computers[END_REF].

VII. CONCLUSION

Today's quantum computer performs error-prone quantum operations with decoherence error being one of the primary error sources. In this paper, we focus on the decoherence error mitigation and explore the generally used method -dynamical decoupling. The idle qubit is caused under different conditions and sometimes influenced by crosstalk depending on the qubit coupling. First, we analyze CNOT and SWAP induced idle qubit and further divide it into crosstalk-idle and idle-idle. Second, we examine DD insertion strategies for the two types of idle qubits under the analyzed configurations. Finally, we provide guildelines of DD insertion strategies and help the community design DD-protected circuits to achieve reliable results. Our work can be regarded as the first step to developing a general DD-insertion compiler for decoherence error mitigation.

Fig. 2 :

 2 Fig.2:The main qubit q 0 mapped to Q 0 in red color performs as (a) crosstalk-idle qubit, or (b) idle-idle qubit, when the spectator qubits are mapped to one of the qubit pairs whose link is highlighted in blue color in (a) (resp. (b)).

Fig. 4 :

 4 Fig. 4: (a) CNOT-induced idle qubit. (b) SWAP-induced idle qubit. The idle time is highlighted in the red frame.

Fig. 5 :

 5 Fig. 5: Experiments of CNOT and other delay induced idle qubit. (a) Baseline, where q 0 is under free evolution during the overall idle time. (b) Inserting DD during the overall idle time. (c) Inserting DD during the CNOT duration and then Delay. (d) Inserting Delay during CNOT duration and then DD. (d) Inserting double DD separately during CNOT and Delay.

Fig. 7 :Fig. 8 :Fig. 9 :

 789 Fig.7: Results of purely SWAP induced idle qubit when main qubit q 0 is considered as (a) crosstalk-idle qubit. (b) idle-idle qubit.

ACKNOWLEDGMENT

This work is funded by the QuantUM Initiative of the Region Occitanie, University of Montpellier and IBM Montpellier. The authors sincerely appreciate the discussions with Dr. Nick Bronn and are very grateful to Adrien Suau for the helpful suggestions.


Luis Gustavo

Giacon Villani

Samuel Da Silva
email: samuel.silva13@unesp.br

Americo Cunha
email: americo.cunha@uerj.br

L G G Villani
email: luis.villani@ufes.br

An optimizationless stochastic Volterra series approach for nonlinear model identification

Keywords: Stochastic Volterra series, Uncertain systems, Nonlinear systems, Stochastic models

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Systems identification is an essential topic of research related to dynamics control since the prediction of complex systems behavior, subjected to nonlinearities and uncertainties, is an everyday task in engineering sciences [START_REF] Ljung | Guest editorial: Special issue on system identification[END_REF]. This problem is relatively simple and well solved when the linearity hypothesis is applicable. However, a lot of real systems may exhibit nonlinear phenomena, depending on factors such as (i) operational conditions, (ii) constitutive relationships of materials, (iii) application of excitations and loads, etc. [START_REF] Kerschen | Past, present and future of nonlinear system identification in structural dynamics[END_REF].

The literature presents several methods that can be used to identify systems subjected to nonlinear phenomena, suck as Hilbert transform, NARMAX Models, High-Order Frequency Response Functions (HOFRFs), Restoring Force Surface (RFS) method, etc. [START_REF] Kerschen | Nonlinear system identification in structural dynamics: current status and future directions[END_REF]. The use of Volterra series expanded in orthonormal bases -mainly the Kautz functions -is also frequently reported in the literature to describe nonlinear systems, with satisfactory results in oscillatory phenomena with polynomial nonlinearities [START_REF] Da Silva | Non-linear model updating of a three-dimensional portal frame based on Wiener series[END_REF][START_REF] Oliveira | An introduction to models based on Laguerre, Kautz and other related orthonormal functionspart II: non-linear models[END_REF]17]. The major difficulty for the use of Kautz functions based Volterra series is related to the definition of the basis parameters, which requires the solution of a nonlinear optimization problem [START_REF] Da Rosa | Choice of free parameters in expansions of discrete-time Volterra models using Kautz functions[END_REF][START_REF] Da Rosa | Exact search directions for optimization of linear and nonlinear models based on generalized orthonormal functions[END_REF], a task that is computationally expensive and very sensitive to the initial guess for the parameters. To make this task even more difficult, every real system is subject to uncertainties, which manifest themselves via (i) measurements noise in the system observations (experimental data); (ii) variabilities of the real system concerning its simple configuration (due to geometric imperfections, manufacturing irregularities, environmental conditions, etc.); or (iii) lack of knowledge about the model structure (ignorance about the system physics) [START_REF] Soize | Stochastic modeling of uncertainties in computational structural dynamics -recent theoretical advances[END_REF]. These uncertainties make the solution to the optimization problem mentioned above even more complex and unreliable. Nevertheless, to the best of the authors' knowledge, few papers in the literature consider the approach of the Volterra series (with Kautz bases) applied to the identification of systems taking into account the variabilities (uncertainties) present in the measured data [START_REF] Villani | Damage detection in an uncertain nonlinear beam based on stochastic Volterra series: An experimental application[END_REF][START_REF] Villani | On the detection of a nonlinear damage in an uncertain nonlinear beam using stochastic Volterra series[END_REF]. In this way, to conduct a careful identification process and, consequently, obtain an accurate response prediction, it is essential to consider the effect of these uncertainties. Some initiatives in this sense can be found in the literature, such as [START_REF] Oliveira | An introduction to models based on Laguerre, Kautz and other related orthonormal functions -part I: linear and uncertain models[END_REF], which presents the formulation of orthonormal bases in robust control, considering parametric uncertainties in linear systems. Additionally, some expansions to represent nonlinear Volterra kernels in an uncertain scenery are shown with generalized orthonormal basis functions (GOBF) in [START_REF] Bouzrara | Nonlinear predictive controller for uncertain process modelled by GOBF-Volterra models[END_REF], and Kautz functions in [START_REF] Da Rosa | An optimal expansion of Volterra models using independent Kautz bases for each kernel dimension[END_REF][START_REF] Da Rosa | Robust expansion of uncertain Volterra kernels into orthonormal series[END_REF]. In this latter work, Rosa et al. (2010) [START_REF] Da Rosa | Robust expansion of uncertain Volterra kernels into orthonormal series[END_REF] consider Volterra kernels as uncertain objects, and an optimization procedure is employed to determine the Kautz bases parameters to describe the kernel's limits and minimize the uncertainties influence. In [START_REF] Villani | Damage detection in an uncertain nonlinear beam based on stochastic Volterra series: An experimental application[END_REF][START_REF] Villani | On the detection of a nonlinear damage in an uncertain nonlinear beam using stochastic Volterra series[END_REF], the authors considered the stochastic version of the Volterra series, implemented using random Kautz functions. However, an optimization procedure to determine the pole values that define the Kautz functions was performed for each Monte Carlo run. This optimization problem is non-convex, which requires global search algorithms, e.g., genetic algorithms (very computationally costly) since the use of gradient-based optimizers becomes highly dependent on the initial guess. This procedure can be very costly when considering a high number of MC runs, making the development of new methodologies essential for the practical use of the stochastic model. This paper proposes to address the problem of nonlinear system identification in the presence of uncertainties -for instance, noise in experimental observationsusing a stochastic version of Kautz functions based on the Volterra series, without the use of an optmization procedure to determine exact values for the Kautz parameters. In this context, the Volterra series is defined in a parametric probabilistic framework. A stochastic model of uncertainties for the nonlinear system under analysis is identified, where model parameters are described by independent random variables (instead of de-terministic values), and the system response is treated as a random process (rather than a deterministic function of time). One advantage of this approach is that it allows the construction of confidence bands for the Volterra kernels, i. e., it provides more robust predictions for the model output once it considers potential variabilities of the system response. Furthermore, the main advantage of the proposed methodology is that it does not require the optimization of deterministic Kautz poles values, as they are considered to be random variables. A deterministic optimal value becomes nonrepresentative in situations such as described above because a single-pole value may not represent all the data variation in the measured signals, and an optimization considering data variation with many measured signals can be very costly.

The proposed identification methodology uses a parametric probabilistic approach to address model parameters uncertainties, where Kautz poles are treated as independent random variables. Their probability distributions are obtained conservatively utilizing the maximum entropy principle [5], taking into account only known information about these parameters. Statistics about the nonlinear system response are obtained using the Monte Carlo method [START_REF] Cunha | Uncertainty quantification through Monte Carlo method in a cloud computing setting[END_REF][START_REF] Rubinstein | Simulation and the Monte Carlo Method[END_REF]. The proposed methodology's effectiveness is tested by identifying a clampedfree beam undergoing large displacements. Measurements were performed on different days to induce variations in the data related to uncertainties in boundary conditions, sensor positions, screws tightening, etc. The results obtained with the new methodology's application have shown that the identified random Kautz poles allow the stochastic model to predict the variability of the system response with a certain level of confidence, ensuring some robustness to the model output concerning the underlying uncertainties. Such robustness can not be obtained with an optimum pole-based deterministic model.

Stochastic Identification via Volterra Series

System uncertainties

Uncertainties are classified in the literature as being of two types, aleatory or epistemic. The first type is intrinsic to scenarios with variabilities, such as those described in section 1, and can not be eliminated, only better characterized. The second type, epistemic uncertainties, is only due to the lack of information. By increasing knowledge about a particular system, these uncertainties can be mitigated [START_REF] Soize | Stochastic modeling of uncertainties in computational structural dynamics -recent theoretical advances[END_REF][START_REF] Cunha | Modeling and Quantification of Physical Systems Uncertainties in a Probabilistic Framework[END_REF].

The methodology adopted in this work only considers the aleatory uncertainties, also known as data uncer-tainties. Therefore, the uncertainties are materialized in variations in the model parameters. In this sense, the authors assume that the Volterra series can produce a reliable representation of nonlinearities present in the system's dynamics. The epistemic uncertainties must be considered in future works.

Additionally, the model parameters subjected to uncertainties are described as random variables or random processes, defined on the probability space (Θ, Σ, P), where Θ is a sample space, Σ is a σ-algebra over Θ, and P is a probability measure. It is assumed that any random variable θ ∈ Θ → V(θ) ∈ R in this probabilistic setting, with probability distribution P V (dv) on R, admits a probability density function (PDF) v → p V (v) with respect to dv [START_REF] Villani | Damage detection in an uncertain nonlinear beam based on stochastic Volterra series: An experimental application[END_REF].

Stochastic Volterra series

In this paper, a stochastic version of the Volterra series is employed to identify the nonlinear system of interest. In this sense, the discrete-time Volterra series describes the random system output as

y(θ, k) = ∞ η=1 N1-1 n1=0 . . . Nη-1 nη=0 H η (θ, n 1 , . . . , n η ) η i=1 u(k-n i ), (1)
where k ∈ Z → u(k) is a deterministic input signal, the random process (θ, k) ∈ Θ × Z → y(θ, k) is the system response and (θ, n 1 , .., n η ) ∈ Θ×Z η → H η (θ, n 1 , . . . , n η ) represents the random version of the η-order Volterra kernel. The representation in the form of higher-order convolutions allows one to directly split the system output into a sum of linear and nonlinear contributions (that are random processes in this case).

The main drawback is related to the difficulty of convergence of the series using a large number of terms. However, the problem is reduced expanding the Volterra kernels into an orthonormal basis (this work employs the use of Kautz functions [START_REF] Kautz | Transient synthesis in the time domain. Circuit Theory[END_REF][START_REF] Heuberger | Modelling and identification with rational orthogonal basis functions. Springer 5. Jaynes ET[END_REF]). Once the system response varies during the measuring process, it is natural to model the Kautz functions as random processes, since their definition are associated with the dynamics of the system response y(θ, k) and depend on the parameters of damping ratio and natural frequency, which are subjected to uncertainties. Considering the random Kautz orthonormal basis expansion, the random output is written as

y(θ, k) ≈ ∞ η=1 J1 i1=1 . . . Jη iη=1 B η (θ, i 1 , . . . , i η ) η j=1 l ij (θ, k) , (2)
where J 1 , . . . , J η are the number of samples in each orthonormal projections of the Volterra kernels, the random process (θ, i 1 , . . . , i η ) ∈ Θ × Z η → B η (θ, i 1 , . . . , i η ) represents the η-order random Volterra kernel, expanded in the orthonormal basis, and the random process (θ, k) ∈ Θ × Z → l ij (θ, k) is a simple filtering of the deterministic input signal u(k) by the random Kautz function

l ij (θ, k) = K-1 ni=0 ψ ij (θ, n i )u(k -n i ) , (3)
where

K = max{J 1 , . . . , J η } and (θ, n j ) ∈ Θ × Z → ψ ij (θ, n j )
represents the random version of the i j -th Kautz filter. More information on the identification approach based on deterministic Volterra series can be found in [START_REF] Da Silva | Non-linear model updating of a three-dimensional portal frame based on Wiener series[END_REF]. Details about deterministic Kautz functions are given in section 2.3. The reader is also encouraged to see [START_REF] Oliveira | An introduction to models based on Laguerre, Kautz and other related orthonormal functions -part I: linear and uncertain models[END_REF].

Finally, the coefficients of the kernels can be estimated considering Monte Carlo (MC) simulations and the least squares method

Φ = (Γ T Γ ) -1 Γ T Y , (4)
where, considering each stochastic realization θ and the series truncated on the third-order kernel, the matrix Γ can be completed with the input signal filtered l ij (θ, k)

Γ =      l 11 (θ, 1) . . . l J1 1 (θ, 1) l 12 (θ, 1)l 12 (θ, 1) l 12 (θ, 1)l 22 (θ, 1) . . . l J2 2 (θ, 1)l J2 2 (θ, 1) l 11 (θ, 2) . . . l J1 1 (θ,
) . . . l J3 3 (θ, n s )l J3 3 (θ, n s )l J3 3 (θ, n s )      (5)
where n s represents the number of points of the training data. The vector Y can be completed with the experimental output signal y(θ, k)

Y =          y(θ, 1) y(θ, 2) . . . y(θ, n s )          , (6)
and,

Φ =                                            B 1 (θ, 1) . . . B 1 (θ, J 1 ) B 2 (θ, 1, 1) B 2 (θ, 1, 2) . . . B 2 (θ, J 2 , J 2 ) B 3 (θ, 1, 1, 1) B 3 (θ, 1, 1, 2) . . . B 3 (θ, J 3 , J 3 , J 3 )                                            , ( 7
)
has the terms of the orthonormal kernels B η to each realization θ. The procedure is repeated until the convergence is achieved. Figure 1 shows a flowchart of the stochastic Volterra kernels identification.

Deterministic Kautz functions

To better understand the work it is helpful to know the deterministic form of Kautz functions. The generalized form of Kautz functions is written as [START_REF] Da Rosa | Choice of free parameters in expansions of discrete-time Volterra models using Kautz functions[END_REF]

ψ 2j-1 (z) = √ 1 -d 2 √ 1 -c 2 z 2 + d(c -1)z -c -cz 2 + d(c -1)z + 1 z 2 + d(c -1)z -c j-1 , (8)
ψ 2j (z) = ψ 2j-1 (z) z -d √ 1 -d 2 , (9)
being the values of c and d respectively defined by

c = -β 2η-1 β 2η . ( 10
) d = β 2η-1 + β 2η 1 + β 2η-1 β 2η , (11)
The functions β 2η-1 and β 2η are the Kautz poles

β 2η = -ξ 2η ω 2η -jω 2η 1 -ξ 2 2η , (12)
β 2η-1 = -ξ 2η-1 ω 2η-1 + jω 2η-1 1 -ξ 2 2η-1 , (13)
where the parameters ω η and ξ η are the natural frequency and damping ratio of the system, and η represents the number of the kernel. For a stable system one has β 2η-1 , β 2η < 1. In identification processes it is common use some optimization methodology to find ω η and ξ η [START_REF] Da Rosa | Choice of free parameters in expansions of discrete-time Volterra models using Kautz functions[END_REF].

Figure 1 Description of the random kernels identification methodology.

The stochastic Kautz parameters

In an uncertain framework, the definition of Kautz functions is difficult, and an optimization procedure is usually implemented to estimate the Kautz pole's parameters. However, the optimization problem can generate some errors in an uncertain scenario, even more, if we consider a deterministic estimation, besides being very costly. Therefore, to construct a robust method, it is necessary to provide a statistic certification (the reliability envelope for the model parameters). Such certification can be obtained using a stochastic model, where probability distributions are identified for the model parameters instead of deterministic scalar values. As mentioned before, the Kautz poles' parameters depend on the system dynamics. As we consider the system output a random process that varies in each experimental realization, it is natural to consider that the poles' parameters vary. Therefore, the parameters ω η and ξ η for each kernel are considered as random vari-

ables θ ∈ Θ → ξ η (θ) ∈ R, θ ∈ Θ → ω η (θ) ∈ R.
The random character of the poles will allow the construction of the random Kautz functions described in the last section. The critical step of the methodology is related to the definition of the distributions of ω η and ξ η . It is assumed that the only information known about each random variable is the support [a, b]. In this sense, the maximum entropy principle is employed to construct a consistent probabilistic model for the random Kautz parameters [5]. Thus, the probability density function that maximizes the entropy is a uniform distribution,

p(x) = 1 [a,b] (x) 1 b -a , ( 14
)
where p(x) is the probability density function of a uniformly distributed random variable, a and b are the inferior and superior limits to the distribution. Then, to each kernel, the Kautz parameters ω η and ξ η distributions are estimated between these limits. Finally, the Kautz filters random processes are computed through MC simulations [START_REF] Cunha | Uncertainty quantification through Monte Carlo method in a cloud computing setting[END_REF][START_REF] Rubinstein | Simulation and the Monte Carlo Method[END_REF], and then used to filter the system's deterministic input data the Volterra kernels identification process. In this paper, the Kautz parameter limits are defined based on the system response variation because the Kautz poles are associated with the system's dynamics. To be more clear, thought the estimation of natural frequencies and damping ratios of the system using the operation in the linear regime of motion.

3 Effectiveness test for the new methodology

System of interest

To test the efficacy of the new identification methodology, a nonlinear system composed of a magneto-elastic beam, subject to large displacement, is considered. The experimental setup that emulates this behavior is shown in Fig. 2. It is the same setup used in [17, [START_REF] Villani | Damage detection in an uncertain nonlinear beam based on stochastic Volterra series: An experimental application[END_REF][START_REF] Villani | Application of a stochastic version of the restoring force surface method to identify a duffing oscillator[END_REF], composed by an 300 × 18 × 3 [mm] aluminum beam with steel mass connected in the free end to cause a magnetic interaction between the beam and a magnet. A MODAL SHOP shaker (Model Number: K2004E01) is attached 50 mm from the clamped and used to excite the structure considering different levels of voltage amplitude 0.01 V (low level), 0.10 V (medium level), and 0.15 V (high level). The system presents nonlinear behavior for large displacement amplitudes when a high level of excitation amplitude is applied. A vibrometer laser Polytec (Model: OFV-525/-5000-S) and a Dytran load cell (Model: 1022V) are used to measure the velocity in the free end of the beam and the force excitation, respectively. The magnet positioned next to the free end of the beam interacts with the steel mass and generates a nonlinear hardening behavior [START_REF] Villani | Damage detection in an uncertain nonlinear beam based on stochastic Volterra series: An experimental application[END_REF].

Figure 2 Nonlinear system to be identified [START_REF] Villani | Application of a stochastic version of the restoring force surface method to identify a duffing oscillator[END_REF].

Definition of the Kautz functions

The uncertain system response is used to establish the limits of Kautz parameters without any optimization procedure. The natural frequency and damping ratio of the system are determined based on experimental data measured on different days and used to establish the limits of Kautz parameters distribution defined in section 2.4. To find the number of Kautz functions for each kernel, the accuracy of the model is observed. The Volterra series is truncated in the third-order component because the cubic effect can describe the polynomial nonlinearity related to the system in the study, the effect of cubic stiffness. Table 1 shows the parameters found to the Kautz functions for each kernel. The limits are determined based on the variation of the system's physical parameters, natural frequency, and damping ratio of the measured data. The PDFs of the parameters are obtained considering 250 samples, and the function showed in Eq. ( 14), then the Kautz functions are obtained via MC simulations. This number of samples is determined based on a MC convergence test applied in the identified kernels. Figure 3 shows the Kautz poles to the three kernels represented in the z plain, considering the dispersion of the parameters ω η and ξ η . It can be seen that the probabilistic distribution allows a sweep in a region of the plane, which assures the model the ability to describe the variation of the uncertain response of the system. With these parameters, it is possible to evaluate the stochastic Kautz functions (ψ η ), and use the estimated random functions in the process of Volterra kernels identification, which is described in the following section.

Volterra kernels identification

To identify the uncertain Volterra kernels, 195 samples of experimental data, measured on 5 different days, were used. The identification was performed in two steps [17]. First, a low-level chirp was applied in the structure (0.01 V), varying the excitation frequency between 10 and 50 Hz, in the first mode bandwidth to identify the linear kernel. After that, a high-level chirp was applied (0.15 V) with the same frequency range to identify the second and third kernels. The stochastic Kautz functions described in section 2.4 were used in the process of kernels estimation combining the functions with the experimental system output measured on different days. Their statistics were obtained via MC simulations within a total of 1250 samples. Figure 4 presents the mean value for the first kernel, represented in the time domain, with the respective envelope for a 99% confidence band. Figures 5 and6 show the mean value of the main diagonal of the second and third kernels, also represented in the time domain, with the respective 99% confidence bands. The dispersion for the second and third-order kernels is larger than for the first-order ones. The second-order kernel has a large variability because the system response is approximately symmetric, and its contribution to the response is low, making it difficult to estimate in an uncertain scenery. The large dispersion observed on the impulse response of the thirdorder kernel is associated with the number of random Kautz functions used, representing a higher propagation of uncertainties related to the Kautz poles. On the other hand, fewer functions do not allow the model to describe the system behavior adequately.

Model validation

Once the stochastic model is identified, it is important to validate the model and observe if it can describe the system behavior characteristics. First, the same chirp signal used in the identification process is applied, considering two amplitude levels. It is important to note that the signals used in this process are new samples of the experimental data (that were not used in the identification process), with similar dynamical characteristics. Figure 7 shows the model prediction with 99% of confidence bands, compared with experimental data measured on different days, considering linear and nonlinear regimes of motion, respectively. The results are satisfactory because the model can describe the system behavior in linear and nonlinear components and the data variation simultaneously (the experimental dispersion is lower than the confidence bands). The mean of the model output cannot describe the variability of the experimental data, which confirms the importance of constructing a stochastic model.

Finally, it is important to validate the model using an input signal with different characteristics than the ones used in the identification process. As the system is nonlinear, its behavior is strongly input-signal dependent. Therefore, a sine signal with a fixed frequency is used as input to validate the model. This signal has a high level of amplitude (0. (b) High level of input (nonlinear behavior). to the system's natural frequency (≈ 23 Hz), inducing the nonlinear behavior. Figure 8 shows the results in the frequency domain to help the visualization of the frequency components. The presence of multiple harmonics in the response shows that the system behavior is nonlinear in this condition, and the model can describe this behavior in all-important frequency components. The large dispersion in the model response is a consequence of the number of Kautz functions used. However, as previously commented, fewer functions in nonlinear kernels do not allow the model to describe the system's nonlinearities.

Conclusion

Identifying nonlinear systems subject to uncertainties is challenging, especially when a gray-box model that depends directly on the input/output signals, like the Volterra series, is used. In this scenario, the optimization of Kautz poles can be costly and considered irrelevant once a deterministic value of the pole cannot describe the fluctuations necessary in Kautz functions to approximate the variation in system response. This work presented a methodology for nonlinear system identification based on a stochastic version of the Volterra series (expanded using Kautz functions), where model parameters are described as independent random variables and the model output as a random process. In this methodology, the Kautz poles are determined based on the uncertain output signals of the system and the maximum entropy principle. The results show that this stochastic version of the Volterra series can describe the system response considering the experimental data variations obtained from measurements performed on different days. The stochastic model obtained is robust to uncertainties and can be adequately validated.

Figure 3

 3 Figure 3 Kautz poles represented in discret domain (z). •first, • -second, • -third kernel.

Figure 4

 4 Figure 4 First kernel represented in time domain with 99% confidence band. continuous line -mean value.

Figure 5

 5 Figure 5 Main diagonal of second Volterra kernel represented in time domain with 99% confidence band. continuous line -mean value.

Figure 6

 6 Figure 6 Main diagonal of third Volterra kernel represented in time domain with 99% confidence band. continuous linemean value.

Figure 7

 7 Figure 7 Volterra model response with 99% of confidence bands. blue line -mean model, red line -new experimental data.

Figure 8

 8 Figure 8 Volterra model response with 99% of confidence bands for a single frequency sine input. blue line -mean model, red line -new experimental data.

  J1 1 (θ, n s ) l 12 (θ, n s )l 12 (θ, n s ) l 12 (θ, n s )l 22 (θ, n s ) . . . l J2 2 (θ, n s )l J2 2 (θ, n s )

			2)	l 12 (θ, 2)l 12 (θ, 2)	l 12 (θ, 1)l 22 (θ, 2) . . .	l J2 2 (θ, 2)l J2 2 (θ, 2)
	. . .	. . .	. . .	. . .	. . .	. . .	. . .

l 11 (θ, n s ) . . . l l 13 (θ, 1)l 13 (θ, 1)l 13 (θ, 1) l 13 (θ, 1)l 13 (θ, 1)l 23 (θ, 1) . . . l J3 3 (θ, 1)l J3 3 (θ, 1)l J3 3 (θ, 1) l 13 (θ, 2)l 13 (θ, 2)l 13 (θ, 2) l 13 (θ, 2)l 13 (θ, 2)l 23 (θ, 2) . . .

l J3 3 (θ, 2)l J3 3 (θ, 2)l J3 3 (θ,

2) . . . . . . . . . . . . l 13 (θ, n s )l 13 (θ, n s )l 13 (θ, n s ) l 13 (θ, n s )l 13 (θ, n s )l 23 (θ, n s

Table 1

 1 Kautz functions parameters.

	Parameter	1 st kernel 2 sd kernel 3 th kernel
	Kautz Functions	2	2	4
	ω inf [rad/s]	145.5	145.5	145.5
	ω sup [rad/s]	147.5	147.5	147.5
	ξ inf [%]	1.5	1.5	1.5
	ξ sup [%]	2.0	2.0	2.0

Funding

The authors are thankful for the financial support provided by São Paulo Research Foundation (FAPESP), grant numbers 2012/09135-3 and and 2015/25676-2; Carlos Chagas Filho Research Foundation of Rio de Janeiro State (FAPERJ), grant numbers 211.304/2015, 210.021/2018, 210.167/2019, 211.037/2019, and 201.294/2021; Brazilian National Council for Scientific and Technological Development (CNPq), grant numbers 303403/2013-6 and 306526/2019-0; and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Brasil (CAPES), Finance Code 001.

Conflict of Interest

The authors declares that they have no conflict of interest.