# Dynamical Behavior of Two Interacting Double Quantum Dots in 2D Materials for Feasibility of Controlled-NOT Operation

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}having 1T-phase embedded in 2H-phase with the aim to investigate the feasibility of controlled-NOT (CNOT) gate operation with the Coulomb interaction. The Hamiltonian of the system is constructed by two models, namely the 2D electronic potential model and the $4\times 4$ matrix model whose matrix elements are computed from the approximated two-level systems interaction. The dynamics of states are carried out by the Crank–Nicolson method in the potential model and by the fourth order Runge–Kutta method in the matrix model. Model parameters are analyzed to optimize the CNOT operation feasibility and fidelity, and investigate the behaviors of DQDs in different regimes. Results from both models are in excellent agreement, indicating that the constructed matrix model can be used to simulate dynamical behaviors of two interacting DQDs with lower computational resources. For CNOT operation, the two DQD systems with the Coulomb interaction are feasible, though optimization of engineering parameters is needed to achieve optimal fidelity.

## 1. Introduction

_{2}[21], which can determine essential parameters of DQD models. Recently, DFT was deployed to investigate the stability of 18 monolayer metal oxides [22], from which 9 monolayer structures were predicted for the first time, and at least, 2D InO has been synthesized experimentally [23]. More relevantly, DFT and phonon calculations has enabled the investigation of electrical properties and dynamical stability of 2D materials XBi and XBi

_{3}(X = B, Al, Ga, and In) [24], where the results suggested potentially compatible heterostructure systems. These systems exemplify heterostructures-based model systems with potential and viability for the purposes of the present work. Using the finite difference method, the effect of discontinuous effective mass was investigated in InAs/GaAs quantum dots [25], and, likewise, the energies of 2D-MoS

_{2}periodic QDs [21]. Using the tight-binding method, the dynamics of the one charge qubit in InAs/GaAs DQD under external field was investigated [26,27].

_{2}, have exhibited great potentials in a wide range of applications, such as in optoelectronics [28,29], solar cells [30], batteries [31], and recently in quantum computation and information [1,32], single photon sources [19,33,34], sensing [35,36,37,38], etc. In particular, the QD structure has been constructed in 2D materials, and yielded great benefits in these applications [13] as well. To combine the advantages of QDs, for example, the fabrication of the heterostructure QDs in MoS

_{2}have been demonstrated [21,39], where the 2H-phase MoS

_{2}is changed to the 1T-phase with triangular shape when irradiated by an electron beam. The energy gaps calculated by the finite difference method showed good agreement with those from the experiment [21]. This paves the way for exploring quantum sensing and information processing with 2D materials both experimentally and computationally. For instance, using tight-binding and configuration interaction methods, the two-qubit system is generated from valley isospins of two electrons localized in the double quantum dot created within a MoS

_{2}monolayer flake [32].

_{2}of 1T-phase and 2H-phase. The Hamiltonian of the two interacting DQDs are modeled by two models, namely the Coulomb electronic potential model and the two-level matrix model, whose matrix elements are obtained from the averaged interaction (partial trace) of the subsystem. The eigenenergies of DQD are investigated under various structural parameters: the base length b of QD, the inter-QD distance d between QDs, and the potential of the heterostructure. The dynamics of state are numerical simulation under situation of the CNOT operation and applied for the two qubit states. We remark that heterostructure of MoS

_{2}is only exemplary, and it can be modified to accommodate other materials by changing the potential parameter in the simulation.

## 2. Model and Methodology

#### 2.1. Structural Model

_{2}in the semi-conducting 2H-phase changes to the metallic 1T-phase when irradiated by an electron beam. The transformed 1T-phase has a triangular shape, whose size depends on the radius of the beam, embedded in the rectangular 2H-phase (see Figure 1). This structure is a guiding model for simulation in this work, in which we can also investigate other parameters (e.g., QD dimensions, band offset of the heterostructure) and resulted behaviors. In experiment by Ref. [21], this heterostructure was achieved at room temperature. The periodic monolayers of 1T-phase and 2H-phase of MoS

_{2}are calculated by DFT to evaluate the effective masses and potential parameters. The temperature was set to be 300 K in these DFT calculations. These effective masses and band offset values were used to construct the heterostructures, where the results of calculated energy gap (i.e., the energy difference of electron and hole) were compared favorably with those from experiments. We used these same values of parameters to model DQDs in our simulation. However, in our simulation, the dynamics of states are carried out by the time-dependent Schrödinger equation without decoherence and energy dissipation, which corresponds to zero temperature behavior. As reported in experiment by an annealing process [50], the 1T-MoS

_{2}thin film changes significantly to the 2H-MoS

_{2}phase at temperature higher than 498 K. Thus, at room temperature or lower, the considered heterostructure of 1T-MoS

_{2}and 2H-MoS

_{2}should be thermal stable.

_{2}is assumed to be a rectangle of size dimensions ${L}_{x}$ nm and ${L}_{y}$ nm. The DQD is constructed from QDs with base length b nm and height h nm in the triangular shape of the 1T-phase MoS

_{2}. The QDs are placed with the inter-QD distance of d nm symmetric about the center of the rectangular ${L}_{x}\times {L}_{y}$ supercell. This constructs one DQD. In this work, two identical DQDs are placed side by side along the x-axis with a width a nm. We define the occupancy of an electron in the left dot and the right dot of the left DQD as the states ${|0\rangle}_{l}$ and ${|1\rangle}_{l}$, respectively. Likewise, the states ${|0\rangle}_{r}$ and ${|1\rangle}_{r}$ define the occupancy in the right DQD, as identified by the bits 0 and 1 in Figure 1.

#### 2.2. Electronic Potential Model

_{2}shown by different colors in Figure 1 are represented by the electronic potential ${V}_{in}$ inside the QDs and ${V}_{out}$ outside the QDs, as described in Equation (1). For MoS

_{2}, the electron effective masses and potential parameters are taken from Ref. [21], in which these values were extracted from DFT calculations. The electron effective masses are ${m}_{e,2H}^{*}=0.54{m}_{e}$ for the 2H-phase (outside QD) and ${m}_{e,1T}^{*}=0.29{m}_{e}$ for the 1T-phase (inside QD), where ${m}_{e}$ is the mass of a free electron. The potentials of electron are ${V}_{in}=0$ inside the wells and ${V}_{out}=0.915\phantom{\rule{4pt}{0ex}}\mathrm{eV}$ outside the wells.

#### 2.3. Matrix Model

#### 2.4. Dynamics of States

#### 2.4.1. CNOT Operation

#### 2.4.2. Transition Probability

## 3. Results and Discussion

#### 3.1. Parameter Optimization for Energy Tuning of DQD

_{2}are obtained from Ref. [21]. The QD base length b is varied from 1.0 nm to 2.2 nm, and the height at $h=b/2$. The DQD energies converge when the supercell lengths (${L}_{x}$ and ${L}_{y}$) are sufficiently large, as shown in Section S2 of the Supplementary Materials. In this simulation, the supercell of lengths ${L}_{x}=9.0\phantom{\rule{4pt}{0ex}}\mathrm{nm}$ and ${L}_{y}=4.5\phantom{\rule{4pt}{0ex}}\mathrm{nm}$ are selected.

_{2}. Then, V is varied to cover the range of 0.60–2.00 eV, because some 2D materials have the energy band gap around 0–2 eV. Such variation can account for similar materials other than MoS

_{2}. Moreover, the external strain can adjust the band gap by a few tenths of eV around the original value [17,18,19,20]. Then, the energy gap is analyzed as a function of V and other engineering parameters. Below, we define a fitting function for the energy gap as a function of V, b, and d in the form:

#### 3.2. Dynamics of States on Bloch Sphere

#### 3.3. CNOT Gate Efficiency

_{2}in Ref. [21] are used, but can be changed to other artificial potential values V (see Section S6 in the Supplementary Materials). In Figure 11 and Figure 12, the inter-DQD distance a is varied in the x-axis; there the separate panels correspond to different values of the QD base length b, and the inter-QD distance d is varied with different symbols (colors).

_{2}can be constructed and optimized for CNOT gate operation.

## 4. Conclusions

_{2}consisting of the 1T-phase triangular shape embedded in the 2H-phase square supercell. The two interacting DQDs are investigated for the feasibility of CNOT gate operation.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DQD | Double quantum dots |

QD | Quantum dot |

CNOT | Controlled-NOT (gate) |

DFT | Density functional theory |

FD | Finite difference |

TB | Tight binding |

## References

- Georgescu, I.M.; Ashhab, S.; Nori, F. Quantum simulation. Rev. Mod. Phys.
**2014**, 86, 153–185. [Google Scholar] [CrossRef] [Green Version] - Nielsen, M.A.; Chuang, I.L. Quantum Computation and Quantum Information; University Press Cambridge: Cambridge, UK, 2000. [Google Scholar]
- Divincenzo, D.P. Topics in Quantum Computers. In Mesoscopic Electron Transport; Sohn, L.L., Kouwenhoven, L.P., Schön, G., Eds.; Springer: Dordrecht, The Netherlands, 1997; pp. 657–677. [Google Scholar] [CrossRef]
- Hayashi, T.; Fujisawa, T.; Cheong, H.D.; Jeong, Y.H.; Hirayama, Y. Coherent Manipulation of Electronic States in a Double Quantum Dot. Phys. Rev. Lett.
**2003**, 91, 226804. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Fujisawa, T.; Hayashi, T.; Cheong, H.; Jeong, Y.; Hirayama, Y. Rotation and phase-shift operations for a charge qubit in a double quantum dot. Phys. E
**2004**, 21, 1046–1052. [Google Scholar] [CrossRef] - Gorman, J.; Hasko, D.G.; Williams, D.A. Charge-Qubit Operation of an Isolated Double Quantum Dot. Phys. Rev. Lett.
**2005**, 95, 090502. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Fujisawa, T.; Hayashi, T.; Sasaki, S. Time-dependent single-electron transport through quantum dots. Rep. Prog. Phys.
**2006**, 69, 759–796. [Google Scholar] [CrossRef] - Shi, Z.; Simmons, C.B.; Ward, D.R.; Prance, J.R.; Mohr, R.T.; Koh, T.S.; Gamble, J.K.; Wu, X.; Savage, D.E.; Lagally, M.G.; et al. Coherent quantum oscillations and echo measurements of a Si charge qubit. Phys. Rev. B
**2013**, 88, 075416. [Google Scholar] [CrossRef] [Green Version] - Petersson, K.D.; Petta, J.R.; Lu, H.; Gossard, A.C. Quantum Coherence in a One-Electron Semiconductor Charge Qubit. Phys. Rev. Lett.
**2010**, 105, 246804. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Larsson, M.; Xu, H.Q. Charge state readout and hyperfine interaction in a few-electron InGaAs double quantum dot. Phys. Rev. B
**2011**, 83, 235302. [Google Scholar] [CrossRef] - Li, H.O.; Cao, G.; Yu, G.D.; Xiao, M.; Guo, G.C.; Jiang, H.W.; Guo, G.P. Conditional rotation of two strongly coupled semiconductor charge qubits. Nat. Commun.
**2015**, 6, 7681. [Google Scholar] [CrossRef] - MacQuarrie, E.R.; Neyens, S.F.; Dodson, J.P.; Corrigan, J.; Thorgrimsson, B.; Holman, N.; Palma, M.; Edge, L.F.; Friesen, M.; Coppersmith, S.N.; et al. Progress toward a capacitively mediated CNOT between two charge qubits in Si/SiGe. NPJ Quantum Inf.
**2020**, 6, 81. [Google Scholar] [CrossRef] - Wang, X.; Sun, G.; Li, N.; Chen, P. Quantum dots derived from two-dimensional materials and their applications for catalysis and energy. Chem. Soc. Rev.
**2016**, 45, 2239–2262. [Google Scholar] [CrossRef] [Green Version] - Senellart, P. Deterministic light–matter coupling with single quantum dots. In Quantum Dots: Optics, Electron Transport and Future Applications; Tartakovskii, A., Ed.; Cambridge University Press: Cambridge, UK, 2012; pp. 137–152. [Google Scholar] [CrossRef]
- Laferrière, P.; Yeung, E.; Miron, I.; Northeast, D.B.; Haffouz, S.; Lapointe, J.; Korkusinski, M.; Poole, P.J.; Williams, R.L.; Dalacu, D. Unity yield of deterministically positioned quantum dot single photon sources. Sci. Rep.
**2022**, 12, 6376. [Google Scholar] [CrossRef] [PubMed] - Kouwenhoven, L.P.; Marcus, C.M.; McEuen, P.L.; Tarucha, S.; Westervelt, R.M.; Wingreen, N.S. Electron Transport in Quantum Dots. In Mesoscopic Electron Transport; Sohn, L.L., Kouwenhoven, L.P., Schön, G., Eds.; Springer: Dordrecht, The Netherlands, 1997; pp. 105–214. [Google Scholar] [CrossRef]
- Kang, J.; Tongay, S.; Zhou, J.; Li, J.; Wu, J. Band offsets and heterostructures of two-dimensional semiconductors. Appl. Phys. Lett.
**2013**, 102, 012111. [Google Scholar] [CrossRef] [Green Version] - Patra, A.; Jana, S.; Samal, P.; Tran, F.; Kalantari, L.; Doumont, J.; Blaha, P. Efficient Band Structure Calculation of Two-Dimensional Materials from Semilocal Density Functionals. J. Phys. Chem. C
**2021**, 125, 11206–11215. [Google Scholar] [CrossRef] - Dev, P. Fingerprinting quantum emitters in hexagonal boron nitride using strain. Phys. Rev. Res.
**2020**, 2, 022050. [Google Scholar] [CrossRef] - Hunkao, R.; Kesorn, A.; Maezono, R.; Sinsarp, A.; Sukkabot, W.; Tivakornsasithorn, K.; Suwanna, S. Density Functional Theory Study of Strain-Induced Band Gap Tunability of Two-Dimensional Layered MoS
_{2}, MoO_{2}, WS_{2}and WO_{2}. In Proceedings of the 44th Congress on Science and Technology of Thailand (STT 44), Bangkok, Thailand, 29–31 October 2018; pp. 715–722. [Google Scholar] - Xei, X.; Kang, J.; Cao, W.; Chu, J.H.; Gong, Y.; Ajayan, P.M.; Banerjee, K. Designing artificial 2D crystals with site and size controlled quantum dots. Sci. Rep.
**2017**, 7, 9965. [Google Scholar] - Guo, Y.; Ma, L.; Mao, K.; Ju, M.; Bai, Y.; Zhao, J.; Zeng, X.C. Eighteen functional monolayer metal oxides: Wide bandgap semiconductors with superior oxidation resistance and ultrahigh carrier mobility. Nanoscale Horiz.
**2019**, 4, 592–600. [Google Scholar] [CrossRef] [Green Version] - Kakanakova-Georgieva, A.; Giannazzo, F.; Nicotra, G.; Cora, I.; Gueorguiev, G.K.; Persson, P.O.; Pécz, B. Material proposal for 2D indium oxide. Appl. Surf. Sci.
**2021**, 548, 149275. [Google Scholar] [CrossRef] - Freitas, R.R.Q.; de Brito Mota, F.; Rivelino, R.; de Castilho, C.M.C.; Kakanakova-Georgieva, A.; Gueorguiev, G.K. Spin-orbit-induced gap modification in buckled honeycomb XBi and XBi
_{3}(X = B, Al, Ga, and IN) sheets. J. Phys. Condens. Matter**2015**, 27, 485306. [Google Scholar] [CrossRef] [Green Version] - Woon, C.Y.; Gopir, G.; Othman, A.P. Discontinuity Mass of Finite Difference Calculation in InAs-GaAs Quantum Dots. Adv. Mater. Res.
**2014**, 895, 415–419. [Google Scholar] - Kesorn, A.; Sukkabot, W.; Suwanna, S. Dynamics and Quantum Leakage of InAs/GaAs Double Quantum Dots under Finite Time-Dependent Square-Pulsed Electric Field. Adv. Mater. Res.
**2016**, 1131, 97–105. [Google Scholar] - Kesorn, A.; Kalasuwan, P.; Sinsarp, A.; Sukkabot, W.; Suwanna, S. Effects of square electric field pulses with random fluctuation on state dynamics of InAs/GaAs double quantum dots. Integr. Ferroelectr.
**2016**, 175, 220–235. [Google Scholar] [CrossRef] - Brill, A.R.; Kafri, A.; Mohapatra, P.K.; Ismach, A.; de Ruiter, G.; Koren, E. Modulating the Optoelectronic Properties of MoS
_{2}by Highly Oriented Dipole-Generating Monolayers. ACS Appl. Mater. Interfaces**2021**, 13, 32590–32597. [Google Scholar] [CrossRef] [PubMed] - Islam, M.M.; Dev, D.; Krishnaprasad, A.; Tetard, L.; Roy, T. Optoelectronic synapse using monolayer MoS
_{2}field effect transistors. Sci. Rep.**2020**, 10, 21870. [Google Scholar] [CrossRef] - Singh, E.; Kim, K.S.; Yeom, G.Y.; Nalwa, H.S. Two-dimensional transition metal dichalcogenide-based counter electrodes for dye-sensitized solar cells. RSC Adv.
**2017**, 7, 28234–28290. [Google Scholar] [CrossRef] [Green Version] - Ladha, D.G. A review on density functional theory–based study on two-dimensional materials used in batteries. Mater. Today Chem.
**2019**, 11, 94–111. [Google Scholar] [CrossRef] - Pawłowski, J.; Bieniek, M.; Woźniak, T. Valley Two-Qubit System in a MoS
_{2}-Monolayer Gated Double Quantum dot. Phys. Rev. Appl.**2021**, 15, 054025. [Google Scholar] [CrossRef] - Sajid, A.; Thygesen, K.S. V
_{N}C_{B}defect as source of single photon emission from hexagonal boron nitride. 2D Mater.**2020**, 7, 031007. [Google Scholar] [CrossRef] - Thompson, J.J.P.; Brem, S.; Fang, H.; Frey, J.; Dash, S.P.; Wieczorek, W.; Malic, E. Criteria for deterministic single-photon emission in two-dimensional atomic crystals. Phys. Rev. Mater.
**2020**, 4, 084006. [Google Scholar] [CrossRef] - Vamivakas, A.N.; Zhao, Y.; Fält, S.; Badolato, A.; Taylor, J.M.; Atatüre, M. Nanoscale Optical Electrometer. Phys. Rev. Lett.
**2011**, 107, 166802. [Google Scholar] [CrossRef] - Cui, S.; Pu, H.; Wells, S.A.; Wen, Z.; Mao, S.; Chang, J.; Hersam, M.C.; Chen, J. Ultrahigh sensitivity and layer-dependent sensing performance of phosphorene-based gas sensors. Nat. Commun.
**2015**, 6, 8632. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Tyagi, D.; Wang, H.; Huang, W.; Hu, L.; Tang, Y.; Guo, Z.; Ouyang, Z.; Zhang, H. Recent advances in two-dimensional-material-based sensing technology toward health and environmental monitoring applications. Nanoscale
**2020**, 12, 3535–3559. [Google Scholar] [CrossRef] [PubMed] - Kajale, S.N.; Yadav, S.; Cai, Y.; Joy, B.; Sarkar, D. 2D material based field effect transistors and nanoelectromechanical systems for sensing applications. iScience
**2021**, 24, 103513. [Google Scholar] [CrossRef] [PubMed] - Lin, Y.C.; Dumcenco, D.O.; Huang, Y.S.; Suenaga, K. Atomic mechanism of the semiconducting-to- metallic phase transition in single-layered MoS
_{2}. Nat. Nanotechnol.**2014**, 9, 391–396. [Google Scholar] [CrossRef] [PubMed] - Barenco, A.; Bennett, C.H.; Cleve, R.; DiVincenzo, D.P.; Margolus, N.; Shor, P.; Sleator, T.; Smolin, J.A.; Weinfurter, H. Elementary gates for quantum computation. Phys. Rev. A
**1995**, 52, 3457–3467. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Plantenberg, J.H.; de Groot, P.C.; Harmans, C.J.P.M.; Mooij, J.E. Demonstration of controlled-NOT quantum gates on a pair of superconducting quantum bits. Nature
**2007**, 447, 836–839. [Google Scholar] [CrossRef] - Long, J.; Zhao, T.; Bal, M.; Zhao, R.; Barron, G.S.; Ku, H.s.; Howard, J.A.; Wu, X.; McRae, C.R.H.; Deng, X.H.; et al. A universal quantum gate set for transmon qubits with strong ZZ interactions. arXiv
**2021**, arXiv:2103.12305. [Google Scholar] [CrossRef] - Knill, E.; Laflamme, R.; Milburn, G.J. A scheme for efficient quantum computation with linear optics. Nature
**2001**, 409, 46–52. [Google Scholar] [CrossRef] - O’Brien, J.L.; Pryde, G.J.; White, A.G.; Ralph, T.C.; Branning, D. Demonstration of an all-optical quantum controlled-NOT gate. Nature
**2003**, 426, 264–267. [Google Scholar] [CrossRef] [Green Version] - Clements, W.R.; Humphreys, P.C.; Metcalf, B.J.; Kolthammer, W.S.; Walmsley, I.A. Optimal design for universal multiport interferometers. Optica
**2016**, 3, 1460–1465. [Google Scholar] [CrossRef] - Carolan, J.; Harrold, C.; Sparrow, C.; Martín-López, E.; Russell, N.J.; Silverstone, J.W.; Shadbolt, P.J.; Matsuda, N.; Oguma, M.; Itoh, M.; et al. Universal linear optics. Science
**2015**, 349, 711–716. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ewert, F.; van Loock, P. 3/4-Efficient Bell Measurement with Passive Linear Optics and Unentangled Ancillae. Phys. Rev. Lett.
**2014**, 113, 140403. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zeuner, J.; Sharma, A.N.; Tillmann, M.; Heilmann, R.; Gräfe, M.; Moqanaki, A.; Szameit, A.; Walther, P. Integrated-optics heralded controlled-NOT gate for polarization-encoded qubits. NPJ Quantum Inf.
**2018**, 4, 13. [Google Scholar] [CrossRef] [Green Version] - Lee, J.M.; Lee, W.J.; Kim, M.S.; Cho, S.; Ju, J.J.; Navickaite, G.; Fernandez, J. Controlled-NOT operation of SiN-photonic circuit using photon pairs from silicon-photonic circuit. Opt. Commun.
**2022**, 509, 127863. [Google Scholar] [CrossRef] - Li, L.; Chen, J.; Wu, K.; Cao, C.; Shi, S.; Cui, J. The Stability of Metallic MoS
_{2}Nanosheets and Their Property Change by Annealing. Nanomaterials**2019**, 9, 1366. [Google Scholar] [CrossRef] [Green Version] - Varga, K.; Driscoll, J.A. Computational Nanoscience: Applications for Molecules, Clusters, and Solids; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Grasselli, M.; Pelinovsky, D. Numerical Mathematics; Jones and Bartlett Publishers: Sudbury, MA, USA, 2008. [Google Scholar]
- Jelic, V.; Marsiglio, F. The double-well potential in quantum mechanics: A simple, numerically exact formulation. Eur. J. Phys.
**2012**, 33, 1651–1666. [Google Scholar] [CrossRef] [Green Version] - Griffiths, D.J. Introduction to Quantum Mechanics, 2nd ed.; Pearson Prentice Hall: Hoboken, NJ, USA, 2004. [Google Scholar]
- Santos, E.J.G.; Kaxiras, E. Electric-Field Dependence of the Effective Dielectric Constant in Graphene. Nano Lett.
**2013**, 13, 898–902. [Google Scholar] [CrossRef] - Keldysh, L.V. Coulomb interaction in thin semiconductor and semimetal films. JETP Lett.
**1979**, 29, 658. [Google Scholar] - Cudazzo, P.; Tokatly, I.V.; Rubio, A. Dielectric screening in two-dimensional insulators: Implications for excitonic and impurity states in graphane. Phys. Rev. B
**2011**, 84, 085406. [Google Scholar] [CrossRef] [Green Version] - Van Tuan, D.; Yang, M.; Dery, H. Coulomb interaction in monolayer transition-metal dichalcogenides. Phys. Rev. B
**2018**, 98, 125308. [Google Scholar] [CrossRef] [Green Version] - Pedersen, L.H.; Møller, N.M.; Mølmer, K. Fidelity of quantum operations. Phys. Lett. A
**2007**, 367, 47–51. [Google Scholar] [CrossRef]

**Figure 1.**Schematic diagram of 2D two DQDs which different colors representing the different electronic potentials. This is an illustration of MoS

_{2}in Ref. [21]. Yellow rectangles denote the 2H phase, and violet triangles denote the 1T phase.

**Figure 2.**The wavefunction of MoS

_{2}DQD with QD base length $b=2.0\phantom{\rule{4pt}{0ex}}\mathrm{nm}$ and inter-QD distance $d=3.0\phantom{\rule{4pt}{0ex}}\mathrm{nm}$. (

**top**) bonding state and (

**bottom**) anti-bonding state.

**Figure 3.**The DQD energies as a function of inter-QD distance d; the solid and dash lines represent the energies of the bonding and anti-bonding states, respectively. Different lengths of QD base b are shown in different colors and symbols.

**Figure 4.**The energy gap $\Delta $ between the bonding and anti-bonding states of electron as a function of the inter-QD distance d, the different lengths of QD base b are shown in different colors and symbols.

**Figure 5.**(

**a**) The contour of maximum energy gap ${\Delta}_{max}$ which defined at $d=b$. (

**b**) The contour of fitting exponential parameter $\alpha $ which modeled in Equation (12). The x-axes and y-axes are the potential V and QD base b, respectively.

**Figure 6.**The component $\alpha $ as a function of QD base length b with the different values of potential V, the solid lines are the linear fitting.

**Figure 7.**The slope ${m}_{1}$ as a function of the potential V with linear fitting in solid line as ${m}_{1}\left(V\right)=0.852V+0.295$.

**Figure 8.**The dynamics of states are simulated for CNOT operation with the electronic potential model. The probability as a function of time in two qubit states are shown for (

**a**) initial state $|01\rangle $ and (

**b**) initial state $|00\rangle $.

**Figure 9.**The dynamics of states $|L\rangle $ and $|R\rangle $ displayed in the Bloch sphere for the left and right qubits, corresponding to Figure 8. The solution of the composite system is written as a product state $|{\Psi}_{ps}\rangle =|L\rangle \otimes |R\rangle $. Red and blue colors are the trajectories of the initial states $|01\rangle $ and $|00\rangle $, respectively. The right qubit state $|R\rangle $ is fixed in either ${|0\rangle}_{r}$ or ${|1\rangle}_{r}$ as a control qubit.

**Figure 10.**The dynamics of states of two DQDs with the inter-DQD Coulomb interaction for the initial state $|01\rangle $, the QD base length $b=2.0\phantom{\rule{4pt}{0ex}}\mathrm{nm}$, the inter-QD distance $d=3.0\phantom{\rule{4pt}{0ex}}\mathrm{nm}$ and the two DQDs are separated with the inter-DQD distance $a=8.0\phantom{\rule{4pt}{0ex}}\mathrm{nm}$. The solid and dash lines are electronic potential and matrix models, respectively.

**Figure 11.**CNOT operation efficiency $\Delta P$ as a function of the inter-DQD distance a, the QD base length $b=1.6\phantom{\rule{4pt}{0ex}}\mathrm{nm}$ and $b=1.8\phantom{\rule{4pt}{0ex}}\mathrm{nm}$ for the top and bottom panels, respectively. The different inter-QD distance d is depicted by different symbols. The black dash lines and other color solid lines are from the matrix and electronic potential models, respectively.

**Figure 12.**CNOT operation efficiency $\Delta P$ as a function of the inter-DQD distance a, the QD base length $b=2.0\phantom{\rule{4pt}{0ex}}\mathrm{nm}$ and $b=2.2\phantom{\rule{4pt}{0ex}}\mathrm{nm}$ for the top and bottom panels, respectively. The different inter-QD distance d is depicted by different symbols. The black dash lines and other color solid lines are from the matrix and electronic potential models, respectively.

**Figure 13.**Maximum $\Delta P$ corresponding to the DQDs with the energy gap $\Delta $; values from the matrix model are in black and those from the electronic potential model are in other colors.

**Figure 14.**The inter-qubit coupling energy ${J}_{2}$ and the energy gap $\Delta $ which give the maximum $\Delta P$ of the DQDs are plotted in the log-log scale. Calculations from the matrix model are in black and those from the electronic potential model are in other colors.

**Figure 15.**CNOT gate efficiency with the average fidelity ${F}_{av}$ and $\Delta P$ as a function of the inter-DQD distance a. The DQD potential is MoS

_{2}, and the QD base length $b=2.2\phantom{\rule{4pt}{0ex}}\mathrm{nm}$. The inter-QD distance d is depicted by different symbols.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kesorn, A.; Hunkao, R.; Tivakornsasithorn, K.; Sinsarp, A.; Sukkabot, W.; Suwanna, S.
Dynamical Behavior of Two Interacting Double Quantum Dots in 2D Materials for Feasibility of Controlled-NOT Operation. *Nanomaterials* **2022**, *12*, 3599.
https://doi.org/10.3390/nano12203599

**AMA Style**

Kesorn A, Hunkao R, Tivakornsasithorn K, Sinsarp A, Sukkabot W, Suwanna S.
Dynamical Behavior of Two Interacting Double Quantum Dots in 2D Materials for Feasibility of Controlled-NOT Operation. *Nanomaterials*. 2022; 12(20):3599.
https://doi.org/10.3390/nano12203599

**Chicago/Turabian Style**

Kesorn, Aniwat, Rutchapon Hunkao, Kritsanu Tivakornsasithorn, Asawin Sinsarp, Worasak Sukkabot, and Sujin Suwanna.
2022. "Dynamical Behavior of Two Interacting Double Quantum Dots in 2D Materials for Feasibility of Controlled-NOT Operation" *Nanomaterials* 12, no. 20: 3599.
https://doi.org/10.3390/nano12203599