#
Parallel Implementation of CNOT^{N} and C_{2}NOT^{2} Gates via Homonuclear and Heteronuclear Förster Interactions of Rydberg Atoms

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## Abstract

**:**

## 1. Introduction

## 2. Scheme of Rydberg EIT CNOT${}^{\mathrm{N}}$ Gate

^{th}target atom, and will make the coupling radiation ${\Omega}_{c}$ off-resonant for the transition between intermediate excited and Rydberg state of the target atoms, as shown in Figure 2c. Thus, the conditions for EIT are not met anymore, and the Raman population transfer between the states $|A\rangle $ and $|B\rangle $ becomes possible. In the ideal limit of a blockade regime, the Rydberg states of the target atoms are never populated.

## 3. Homonuclear and Heteronuclear Interaction Energy

#### 3.1. Heteronuclear Architecture

#### 3.2. Homonuclear Architecture

## 4. Fidelity of Multiqubit Entangled States

#### 4.1. Homonuclear Architecture

#### 4.2. Heteronuclear Architecture

## 5. Scheme of Rydberg EIT C${}_{2}$NOT${}^{2}$

- We apply $\pi $-pulses to excite the control atoms from ground state $|1\rangle $ to highly excited Rydberg state $|r\rangle $ in sequence.
- Then we apply smooth Raman laser $\pi $ pulse to couple the ground states of the target atoms $|A\rangle $ and $|B\rangle $, simultaneously, to the intermediate dark state $|P\rangle $.
- Finally, we apply $\pi $ pulses to return the control atoms from highly excited Rydberg state $|r\rangle $ to ground state $|1\rangle $ in reversed sequence applied in step 1.

## 6. Gate Errors

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Model of Multiple Rydberg Interaction Channels

**Figure A1.**Hyperfine structure of the collective atomic energy levels with control ${}^{133}$Cs atom and target ${}^{87}$Rb atom and laser-induced transitions between ground and Rydberg states.

**Table A1.**The calculated dipole-dipole coefficient ${\mathcal{C}}_{3}^{\left(\alpha \right)}$ ($\mathrm{GHz}$$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$$\mathsf{\mu}{\mathrm{m}}^{3}$), the energy defect ${\delta}_{{E}_{\alpha}}$ ($\mathrm{GHz}$), the van der Waals ${\mathcal{C}}_{6}^{\left(\alpha \right)}$ ($\mathrm{GHz}$$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$$\mathsf{\mu}{\mathrm{m}}^{6}$) interaction coefficients for:

**Table (above)**, the heteronuclear asymmetric interaction channels $|{r}_{0}\rangle |{R}_{0}\rangle \to |{r}_{\alpha}\rangle |{R}_{\alpha}\rangle $ with $|{r}_{0}\rangle =|81{S}_{1/2},{m}_{j}=-1/2\rangle $ and $|{R}_{0}\rangle =|77{S}_{1/2},$${m}_{j}=1/2\rangle $ for Cs and Rb atoms, respectively.

**Table (below)**, the corresponding homonuclear symmetric interaction channels $|{R}_{0}\rangle |{R}_{0}\rangle \to |{R}_{\alpha}\rangle |{R}_{\alpha}\rangle $ for Rb atoms. ${\omega}_{j}$ ($j=1,2$) is the driving transition field that couples $|{r}_{0}\rangle \mapsto |{r}_{\alpha}\rangle $ ($|{R}_{0}\rangle \mapsto |{R}_{\alpha}\rangle $). ${\chi}_{\alpha}={\displaystyle \frac{{\mathcal{C}}_{3}^{\left(\alpha \right)}}{{\overline{R}}_{\mathrm{CT}}^{3}\phantom{\rule{0.166667em}{0ex}}{\delta}_{{F}_{\alpha}}}}$ is the dimensionless coupling strength factor. If ${\chi}_{\alpha}\ll 1$, then the leakage from the resonantly coupled state can be suppressed. We limited the results of heteronuclear interaction to be $\Delta n=2$ and ${\delta}_{{F}_{\alpha}}/2\pi \phantom{\rule{0.166667em}{0ex}}\in [-2,2]$ $\mathrm{GHz}$ and considered the corresponding interaction channels for the homonuclear interactions. ${\overline{R}}_{\mathrm{CT}}=5$ $\mathsf{\mu}\mathrm{m}$, ${\overline{R}}_{\mathrm{TT}}=\sqrt{2}\phantom{\rule{0.166667em}{0ex}}{\overline{R}}_{\mathrm{CT}}$. The values are taken from ARC using functions

`getDipoleMatrixElement`, and

`getEnergy`.

$\mathit{\alpha}$ | ${\mathit{\omega}}_{\mathbf{1}}$ | ${}^{\mathbf{87}}$Rb | ${\mathit{\omega}}_{\mathbf{2}}$ | ${}^{\mathbf{133}}$Cs | ${\mathcal{C}}_{\mathbf{3}}/\mathbf{2}\mathit{\pi}$ | ${\mathit{\delta}}_{\mathit{F}}/\mathbf{2}\mathit{\pi}$ | $\mathit{\chi}$ | ${\mathcal{C}}_{\mathbf{6}}/\mathbf{2}\mathit{\pi}$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{n}}_{\mathbf{1}}$ | ${\mathit{\ell}}_{\mathbf{1}}$ | ${\mathit{j}}_{\mathbf{1}}$ | ${\mathit{m}}_{\mathit{j}\mathbf{1}}$ | ${\mathit{n}}_{\mathbf{2}}$ | ${\ell}_{\mathbf{2}}$ | ${\mathit{j}}_{\mathbf{2}}$ | ${\mathit{m}}_{\mathit{j}\mathbf{2}}$ | |||||||

1 | ${\sigma}^{+}$ | 76 | 1 | $1.5$ | $1.5$ | ${\sigma}^{+}$ | 81 | 1 | $0.5$ | $0.5$ | $10.9$ | $1.87$ | $4.650\times {10}^{-2}$ | $5.812$ |

2 | $\pi $ | 77 | 1 | $0.5$ | $0.5$ | ${\sigma}^{+}$ | 80 | 1 | $0.5$ | $0.5$ | $5.88$ | $0.214$ | $2.201\times {10}^{-1}$ | $27.52$ |

3 | ${\sigma}^{+}$ | 77 | 1 | $1.5$ | $1.5$ | ${\sigma}^{+}$ | 80 | 1 | $0.5$ | $0.5$ | $10.0$ | $0.002$ | $3.656\times {10}^{+1}$ | 4570 |

4 | $\pi $ | 78 | 1 | $0.5$ | $0.5$ | ${\sigma}^{+}$ | 79 | 1 | $0.5$ | $0.5$ | $0.105$ | $-0.437$ | $1.927\times {10}^{-3}$ | $-0.241$ |

5 | ${\sigma}^{+}$ | 78 | 1 | $1.5$ | $1.5$ | ${\sigma}^{+}$ | 79 | 1 | $0.5$ | $0.5$ | $0.190$ | $-0.64$ | $2.374\times {10}^{-3}$ | $-0.297$ |

$\mathit{\alpha}$ | ${\mathit{\omega}}_{\mathbf{1}}$ | ${}^{\mathbf{87}}$Rb | ${\mathit{\omega}}_{\mathbf{2}}$ | ${}^{\mathbf{87}}$Rb | ${\mathcal{C}}_{\mathbf{3}}/\mathbf{2}\mathit{\pi}$ | ${\mathit{\delta}}_{\mathit{F}}/\mathbf{2}\mathit{\pi}$ | $\mathit{\chi}$ | ${\mathcal{C}}_{\mathbf{6}}/\mathbf{2}\mathit{\pi}$ | ||||||

${\mathit{n}}_{\mathbf{1}}$ | ${\ell}_{\mathbf{1}}$ | ${\mathit{j}}_{\mathbf{1}}$ | ${\mathit{m}}_{\mathit{j}\mathbf{1}}$ | ${\mathit{n}}_{\mathbf{2}}$ | ${\ell}_{\mathbf{2}}$ | ${\mathit{j}}_{\mathbf{2}}$ | ${\mathit{m}}_{\mathit{j}\mathbf{2}}$ | |||||||

1 | ${\sigma}^{+}$ | 76 | 1 | $1.5$ | $1.5$ | ${\sigma}^{+}$ | 76 | 1 | $1.5$ | $1.5$ | $11.2$ | $16.84$ | $6.657\times {10}^{-4}$ | $67.26$ |

2 | $\pi $ | 77 | 1 | $0.5$ | $0.5$ | $\pi $ | 77 | 1 | $0.5$ | $0.5$ | $4.29$ | $-15.40$ | $2.784\times {10}^{-4}$ | $-96.78$ |

3 | ${\sigma}^{+}$ | 77 | 1 | $1.5$ | $1.5$ | ${\sigma}^{+}$ | 77 | 1 | $1.5$ | $1.5$ | $12.5$ | $-15.82$ | $7.873\times {10}^{-4}$ | $-88.39$ |

4 | $\pi $ | 78 | 1 | $0.5$ | $0.5$ | $\pi $ | 78 | 1 | $0.5$ | $0.5$ | $0.060$ | $-46.79$ | $1.294\times {10}^{-6}$ | $-0.006$ |

5 | ${\sigma}^{+}$ | 78 | 1 | $1.5$ | $1.5$ | ${\sigma}^{+}$ | 78 | 1 | $1.5$ | $1.5$ | $0.197$ | $-47.20$ | $4.178\times {10}^{-6}$ | $-0.007$ |

**Figure A2.**(

**a**) The contour plots of fidelity for implementing CNOT${}^{N}$ gate using multi-Rydberg model as a function of ${R}_{\mathrm{CT}}$ and ${\Omega}_{c}/{\Omega}_{p}$ for (

**a**) $N=1$, (

**b**) $N=2$.

## Appendix B. Model of Multi EIT Gate via 6P_{3/2}

**Figure A3.**(

**a**) Scheme of energy levels of ${}^{87}$Rb target atom. (

**b**) Dependence of the fidelity of the transfer gate on two-photon Raman detuning. (

**c**) Dependence of the fidelity of the blocking gate on the two-photon Rydberg detuning.

## References

- Arute, F.; Arya, K.; Babbush, R.; Bacon, D.; Bardin, J.C.; Barends, R.; Biswas, R.; Boixo, S.; Brandao, F.G.; Buell, D.A.; et al. Quantum supremacy using a programmable superconducting processor. Nature
**2019**, 574, 505–510. [Google Scholar] [CrossRef] - Wu, Y.; Bao, W.S.; Cao, S.; Chen, F.; Chen, M.C.; Chen, X.; Chung, T.H.; Deng, H.; Du, Y.; Fan, D.; et al. Strong quantum computational advantage using a superconducting quantum processor. Phys. Rev. Lett.
**2021**, 127, 180501. [Google Scholar] [CrossRef] [PubMed] - Zhong, H.S.; Deng, Y.H.; Qin, J.; Wang, H.; Chen, M.C.; Peng, L.C.; Luo, Y.H.; Wu, D.; Gong, S.Q.; Su, H.; et al. Phase-programmable Gaussian boson sampling using stimulated squeezed light. Phys. Rev. Lett.
**2021**, 127, 180502. [Google Scholar] [CrossRef] - Morgado, M.; Whitlock, S. Quantum simulation and computing with Rydberg-interacting qubits. AVS Quantum Sci.
**2021**, 3, 023501. [Google Scholar] [CrossRef] - Graham, T.; Song, Y.; Scott, J.; Poole, C.; Phuttitarn, L.; Jooya, K.; Eichler, P.; Jiang, X.; Marra, A.; Grinkemeyer, B.; et al. Multi-qubit entanglement and algorithms on a neutral-atom quantum computer. Nature
**2022**, 604, 457. [Google Scholar] [CrossRef] [PubMed] - Zeng, Y.; Xu, P.; He, X.; Liu, Y.; Liu, M.; Wang, J.; Papoular, D.; Shlyapnikov, G.; Zhan, M. Entangling two individual atoms of different isotopes via Rydberg blockade. Phys. Rev. Lett.
**2017**, 119, 160502. [Google Scholar] [CrossRef] - Levine, H.; Keesling, A.; Semeghini, G.; Omran, A.; Wang, T.T.; Ebadi, S.; Bernien, H.; Greiner, M.; Vuletić, V.; Pichler, H.; et al. Parallel implementation of high-fidelity multiqubit gates with neutral atoms. Phys. Rev. Lett.
**2019**, 123, 170503. [Google Scholar] [CrossRef] [PubMed] - Graham, T.; Kwon, M.; Grinkemeyer, B.; Marra, Z.; Jiang, X.; Lichtman, M.; Sun, Y.; Ebert, M.; Saffman, M. Rydberg-mediated entanglement in a two-dimensional neutral atom qubit array. Phys. Rev. Lett.
**2019**, 123, 230501. [Google Scholar] [CrossRef] [PubMed] - Evered, S.J.; Bluvstein, D.; Kalinowski, M.; Ebadi, S.; Manovitz, T.; Zhou, H.; Li, S.H.; Geim, A.A.; Wang, T.T.; Maskara, N.; et al. High-fidelity parallel entangling gates on a neutral atom quantum computer. arXiv
**2023**, arXiv:2304.05420. [Google Scholar] [CrossRef] - Scholl, P.; Schuler, M.; Williams, H.J.; Eberharter, A.A.; Barredo, D.; Schymik, K.N.; Lienhard, V.; Henry, L.P.; Lang, T.C.; Lahaye, T.; et al. Quantum simulation of 2D antiferromagnets with hundreds of Rydberg atoms. Nature
**2021**, 595, 233–238. [Google Scholar] [CrossRef] - Ebadi, S.; Wang, T.T.; Levine, H.; Keesling, A.; Semeghini, G.; Omran, A.; Bluvstein, D.; Samajdar, R.; Pichler, H.; Ho, W.W.; et al. Quantum phases of matter on a 256-atom programmable quantum simulator. Nature
**2021**, 595, 227–232. [Google Scholar] [CrossRef] - Gallagher, T. Rydberg atoms. Rep. Prog. Phys.
**1988**, 51, 143. [Google Scholar] [CrossRef] - Singer, K.; Stanojevic, J.; Weidemüller, M.; Côté, R. Long-range interactions between alkali Rydberg atom pairs correlated to the ns–ns, np–np and nd–nd asymptotes. J. Phys. At. Mol. Opt. Phys.
**2005**, 38, S295. [Google Scholar] [CrossRef] - Jaksch, D.; Cirac, J.; Zoller, P.; Rolston, S.; Côté, R.; Lukin, M. Fast quantum gates for neutral atoms. Phys. Rev. Lett.
**2000**, 85, 2208. [Google Scholar] [CrossRef] - Lukin, M.D.; Fleischhauer, M.; Cote, R.; Duan, L.; Jaksch, D.; Cirac, J.I.; Zoller, P. Dipole blockade and quantum information processing in mesoscopic atomic ensembles. Phys. Rev. Lett.
**2001**, 87, 037901. [Google Scholar] [CrossRef] [PubMed] - Urban, E.; Johnson, T.A.; Henage, T.; Isenhower, L.; Yavuz, D.; Walker, T.; Saffman, M. Observation of Rydberg blockade between two atoms. Nat. Phys.
**2009**, 5, 110–114. [Google Scholar] [CrossRef] - Gaëtan, A.; Miroshnychenko, Y.; Wilk, T.; Chotia, A.; Viteau, M.; Comparat, D.; Pillet, P.; Browaeys, A.; Grangier, P. Observation of collective excitation of two individual atoms in the Rydberg blockade regime. Nat. Phys.
**2009**, 5, 115–118. [Google Scholar] [CrossRef] - Isenhower, L.; Urban, E.; Zhang, X.; Gill, A.; Henage, T.; Johnson, T.A.; Walker, T.; Saffman, M. Demonstration of a neutral atom controlled-NOT quantum gate. Phys. Rev. Lett.
**2010**, 104, 010503. [Google Scholar] [CrossRef] - Su, S.L.; Li, W. Dipole-dipole interaction driven antiblockade of two Rydberg atoms. Phys. Rev. A
**2021**, 104, 033716. [Google Scholar] [CrossRef] - Wu, J.L.; Wang, Y.; Han, J.X.; Feng, Y.K.; Su, S.L.; Xia, Y.; Jiang, Y.; Song, J. One-step implementation of Rydberg-antiblockade SWAP and controlled-SWAP gates with modified robustness. Photonics Res.
**2021**, 9, 814–821. [Google Scholar] [CrossRef] - Wu, J.L.; Wang, Y.; Han, J.X.; Su, S.L.; Xia, Y.; Jiang, Y.; Song, J. Unselective ground-state blockade of Rydberg atoms for implementing quantum gates. Front. Phys.
**2022**, 17, 22501. [Google Scholar] [CrossRef] - Khazali, M.; Mølmer, K. Fast multiqubit gates by adiabatic evolution in interacting excited-state manifolds of Rydberg atoms and superconducting circuits. Phys. Rev. X
**2020**, 10, 021054. [Google Scholar] [CrossRef] - Young, J.T.; Bienias, P.; Belyansky, R.; Kaufman, A.M.; Gorshkov, A.V. Asymmetric blockade and multiqubit gates via dipole-dipole interactions. Phys. Rev. Lett.
**2021**, 127, 120501. [Google Scholar] [CrossRef] - Li, M.; Li, J.Y.; Guo, F.Q.; Zhu, X.Y.; Liang, E.; Zhang, S.; Yan, L.L.; Feng, M.; Su, S.L. Multiple-Qubit CkUm Logic Gates of Rydberg Atoms via Optimized Geometric Quantum Operations. Ann. Der Phys.
**2022**, 534, 2100506. [Google Scholar] [CrossRef] - Sun, L.N.; Yan, L.L.; Su, S.L.; Jia, Y. One-Step Implementation of Time-Optimal-Control Three-Qubit Nonadiabatic Holonomic Controlled Gates in Rydberg Atoms. Phys. Rev. Appl.
**2021**, 16, 064040. [Google Scholar] [CrossRef] - Fu, Z.; Xu, P.; Sun, Y.; Liu, Y.Y.; He, X.D.; Li, X.; Liu, M.; Li, R.B.; Wang, J.; Liu, L.; et al. High-fidelity entanglement of neutral atoms via a Rydberg-mediated single-modulated-pulse controlled-phase gate. Phys. Rev. A
**2022**, 105, 042430. [Google Scholar] [CrossRef] - Auger, J.M.; Bergamini, S.; Browne, D.E. Blueprint for fault-tolerant quantum computation with Rydberg atoms. Phys. Rev. A
**2017**, 96, 052320. [Google Scholar] [CrossRef] - Bluvstein, D.; Levine, H.; Semeghini, G.; Wang, T.T.; Ebadi, S.; Kalinowski, M.; Keesling, A.; Maskara, N.; Pichler, H.; Greiner, M.; et al. A quantum processor based on coherent transport of entangled atom arrays. Nature
**2022**, 604, 451–456. [Google Scholar] [CrossRef] [PubMed] - Müller, M.; Lesanovsky, I.; Weimer, H.; Büchler, H.; Zoller, P. Mesoscopic Rydberg gate based on electromagnetically induced transparency. Phys. Rev. Lett.
**2009**, 102, 170502. [Google Scholar] [CrossRef] - Fleischhauer, M.; Imamoglu, A.; Marangos, J.P. Electromagnetically induced transparency: Optics in coherent media. Rev. Mod. Phys.
**2005**, 77, 633. [Google Scholar] [CrossRef] - Ji, Z.; Jiao, Y.; Xue, Y.; Hao, L.; Zhao, J.; Jia, S. Distinction of electromagnetically induced transparency and Autler-Towners splitting in a Rydberg-involved ladder-type cold atom system. Opt. Express
**2021**, 29, 11406–11415. [Google Scholar] [CrossRef] [PubMed] - Pritchard, J.D.; Maxwell, D.; Gauguet, A.; Weatherill, K.J.; Jones, M.; Adams, C.S. Cooperative atom-light interaction in a blockaded Rydberg ensemble. Phys. Rev. Lett.
**2010**, 105, 193603. [Google Scholar] [CrossRef] [PubMed] - Weimer, H.; Müller, M.; Lesanovsky, I.; Zoller, P.; Büchler, H.P. A Rydberg quantum simulator. Nat. Phys.
**2010**, 6, 382–388. [Google Scholar] [CrossRef] - Peyronel, T.; Firstenberg, O.; Liang, Q.Y.; Hofferberth, S.; Gorshkov, A.V.; Pohl, T.; Lukin, M.D.; Vuletić, V. Quantum nonlinear optics with single photons enabled by strongly interacting atoms. Nature
**2012**, 488, 57–60. [Google Scholar] [CrossRef] [PubMed] - Baur, S.; Tiarks, D.; Rempe, G.; Dürr, S. Single-photon switch based on Rydberg blockade. Phys. Rev. Lett.
**2014**, 112, 073901. [Google Scholar] [CrossRef] - Gorniaczyk, H.; Tresp, C.; Bienias, P.; Paris-Mandoki, A.; Li, W.; Mirgorodskiy, I.; Büchler, H.; Lesanovsky, I.; Hofferberth, S. Enhancement of Rydberg-mediated single-photon nonlinearities by electrically tuned Förster resonances. Nat. Commun.
**2016**, 7, 12480. [Google Scholar] [CrossRef] [PubMed] - Beterov, I.; Saffman, M. Rydberg blockade, Förster resonances, and quantum state measurements with different atomic species. Phys. Rev. A
**2015**, 92, 042710. [Google Scholar] [CrossRef] - Sheng, C.; Hou, J.; He, X.; Wang, K.; Guo, R.; Zhuang, J.; Mamat, B.; Xu, P.; Liu, M.; Wang, J.; et al. Defect-free arbitrary-geometry assembly of mixed-species atom arrays. Phys. Rev. Lett.
**2022**, 128, 083202. [Google Scholar] [CrossRef] - Singh, K.; Anand, S.; Pocklington, A.; Kemp, J.T.; Bernien, H. Dual-Element, Two-Dimensional Atom Array with Continuous-Mode Operation. Phys. Rev. X
**2022**, 12, 011040. [Google Scholar] [CrossRef] - Takekoshi, T.; Reichsöllner, L.; Schindewolf, A.; Hutson, J.M.; Le Sueur, C.R.; Dulieu, O.; Ferlaino, F.; Grimm, R.; Nägerl, H.C. Ultracold dense samples of dipolar RbCs molecules in the rovibrational and hyperfine ground state. Phys. Rev. Lett.
**2014**, 113, 205301. [Google Scholar] [CrossRef] - Molony, P.K.; Gregory, P.D.; Ji, Z.; Lu, B.; Köppinger, M.P.; Le Sueur, C.R.; Blackley, C.L.; Hutson, J.M.; Cornish, S.L. Creation of Ultracold
^{87}Rb^{133}Cs Molecules in the Rovibrational Ground State. Phys. Rev. Lett.**2014**, 113, 255301. [Google Scholar] [CrossRef] - Guttridge, A.; Ruttley, D.K.; Baldock, A.C.; González-Férez, R.; Sadeghpour, H.; Adams, C.; Cornish, S.L. Observation of Rydberg blockade due to the charge-dipole interaction between an atom and a polar molecule. Phys. Rev. Lett.
**2023**, 131, 013401. [Google Scholar] [CrossRef] - Tung, S.K.; Parker, C.; Johansen, J.; Chin, C.; Wang, Y.; Julienne, P.S. Ultracold mixtures of atomic 6 Li and 133 Cs with tunable interactions. Phys. Rev. A
**2013**, 87, 010702. [Google Scholar] [CrossRef] - Walker, T.G.; Saffman, M. Consequences of Zeeman degeneracy for the van der Waals blockade between Rydberg atoms. Phys. Rev. A
**2008**, 77, 032723. [Google Scholar] [CrossRef] - Beterov, I.; Hamzina, G.; Yakshina, E.; Tretyakov, D.; Entin, V.; Ryabtsev, I. Adiabatic passage of radio-frequency-assisted förster resonances in Rydberg atoms for two-qubit gates and the generation of bell states. Phys. Rev. A
**2018**, 97, 032701. [Google Scholar] [CrossRef] - McDonnell, K.; Keary, L.F.; Pritchard, J.D. Demonstration of a Quantum Gate Using Electromagnetically Induced Transparency. Phys. Rev. Lett.
**2022**, 129, 200501. [Google Scholar] [CrossRef] - Förster, T.; Sinanoglu, O. Modern Quantum Chemistry; Academic Press: New York, NY, USA, 1965; Volume 3, pp. 93–137. [Google Scholar]
- Saffman, M.; Walker, T. Analysis of a quantum logic device based on dipole-dipole interactions of optically trapped Rydberg atoms. Phys. Rev. A
**2005**, 72, 022347. [Google Scholar] [CrossRef] - Ryabtsev, I.; Tretyakov, D.; Beterov, I.; Entin, V. Observation of the Stark-tuned Förster resonance between two Rydberg atoms. Phys. Rev. Lett.
**2010**, 104, 073003. [Google Scholar] [CrossRef] - Tretyakov, D.; Beterov, I.; Yakshina, E.; Entin, V.; Ryabtsev, I.; Cheinet, P.; Pillet, P. Observation of the Borromean three-body Förster resonances for three interacting Rb Rydberg atoms. Phys. Rev. Lett.
**2017**, 119, 173402. [Google Scholar] [CrossRef] - Browaeys, A.; Lahaye, T. Many-body physics with individually controlled Rydberg atoms. Nat. Phys.
**2020**, 16, 132–142. [Google Scholar] [CrossRef] - Šibalić, N.; Pritchard, J.D.; Adams, C.S.; Weatherill, K.J. ARC: An open-source library for calculating properties of alkali Rydberg atoms. Comput. Phys. Commun.
**2017**, 220, 319–331. [Google Scholar] [CrossRef] - Weber, S.; Tresp, C.; Menke, H.; Urvoy, A.; Firstenberg, O.; Büchler, H.P.; Hofferberth, S. Tutorial: Calculation of Rydberg interaction potentials. J. Phys. B At. Mol. Opt. Phys.
**2017**, 50, 133001. [Google Scholar] [CrossRef] - Le Roy, R.J. Long-Range Potential Coefficients from RKR Turning Points: C
_{6}and C_{8}for B (${3\Pi}_{Ou}^{+}$)-State Cl_{2}, Br_{2}, and I_{2}. Can. J. Phys.**1974**, 52, 246–256. [Google Scholar] [CrossRef] - Yu, D.; Wang, H.; Liu, J.M.; Su, S.L.; Qian, J.; Zhang, W. Multiqubit Toffoli gates and optimal geometry with Rydberg atoms. Phys. Rev. Appl.
**2022**, 18, 034072. [Google Scholar] [CrossRef] - Nielsen, M.A.; Chuang, I.L. Quantum Computing and Cuantum Information; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Hillery, M.; Bužek, V.; Berthiaume, A. Quantum secret sharing. Phys. Rev. A
**1999**, 59, 1829. [Google Scholar] [CrossRef] - Cong, I.; Levine, H.; Keesling, A.; Bluvstein, D.; Wang, S.T.; Lukin, M.D. Hardware-efficient, fault-tolerant quantum computation with Rydberg atoms. Phys. Rev. X
**2022**, 12, 021049. [Google Scholar] [CrossRef] - Li, R.; Qian, J.; Zhang, W. Proposal for practical Rydberg quantum gates using a native two-photon excitation. Quantum Sci. Technol.
**2023**, 8, 035032. [Google Scholar] [CrossRef] - Farouk, A.M.; Beterov, I.; Xu, P.; Bergamini, S.; Ryabtsev, I. Data for: Parallel Implementation of CNOT
^{N}and C_{2}NOT^{2}Gates via Homonuclear and Heteronuclear Forster Interactions of Rydberg Atoms; Zenodo: Newton, NJ, USA, 2023. [Google Scholar] [CrossRef] - Saffman, M.; Walker, T.G.; Mølmer, K. Quantum information with Rydberg atoms. Rev. Mod. Phys.
**2010**, 82, 2313. [Google Scholar] [CrossRef] - Ponciano-Ojeda, F.; Hernández-Gómez, S.; López-Hernández, O.; Mojica-Casique, C.; Colín-Rodríguez, R.; Ramírez-Martínez, F.; Flores-Mijangos, J.; Sahagún, D.; Jáuregui, R.; Jiménez-Mier, J. Observation of the 5p
_{3/2}→6p_{3/2}electric-dipole-forbidden transition in atomic rubidium using optical-optical double-resonance spectroscopy. Phys. Rev. A**2015**, 92, 042511. [Google Scholar] [CrossRef] - Ramírez-Martínez, F.; Ponciano-Ojeda, F.S.; Hernández-Gómez, S.; Del Angel, A.; Mojica-Casique, C.; Hoyos-Campo, L.M.; Flores-Mijangos, J.; Sahagún, D.; Jáuregui, R.; Jiménez-Mier, J.I. Use of an electric-dipole forbidden transition to optically probe the Autler Townes effect. arXiv
**2019**, arXiv:1909.01293. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Scheme of generation of multi-atom entangled GHZ-state using a sequence of CNOT gates applied to different target atoms. Firstly, we apply Hadamard gate H on the control atom and do measurement after performing the CNOT gates. (

**b**,

**c**) Scheme of spatial configurations of Rb and Cs atoms for implementation of the CNOT${}^{\mathrm{N}}$ gate in the case of (

**b**) symmetric homonuclear interaction of Rb atoms and (

**c**) asymmetric heteronuclear interaction between Cs control atom and four Rb target atoms.

**Figure 2.**(

**a**) Sequence of laser pulses for Rydberg EIT gate. ${\Omega}_{c}$ couples intermediate excited state $|P\rangle $ and Rydberg state $|r\rangle $ of the target atom. Two-photon smooth Raman $\pi $ pulse ${\Omega}_{p}\left(t\right)$ couples logical states of the target atoms $|{A}^{N}\rangle $ and $|{B}^{N}\rangle $ of the target atom. Laser $\pi $ pulses ${\Omega}_{r}$ excite and de-excite Rydberg states of control atom; (

**b**) Scheme of CNOT gate operation in the regime of blocked population transfer (the control atom in ground state $|0\rangle $). No population transfer between states $|A\rangle $ and $|B\rangle $ is allowed; (

**c**) Scheme of CNOT gate operation in the transfer regime. The ground state of control atom $|1\rangle $ is coupled to the Rydberg state $|r\rangle $ by a $\pi $-pulse ${\Omega}_{r}$. The ground states of the target atom $|A\rangle $ and $|B\rangle $ are coupled to the intermediate state $|P\rangle $ by a smooth Raman $\pi $-pulse ${\Omega}_{p}\left(t\right)$, and the intermediate state is coupled to the Rydberg state $|R\rangle $ by resonant laser radiation with Rabi frequency ${\Omega}_{c}$. The dipole-dipole interaction between control and target atoms results from coupling of Rydberg states $|r\rangle $ and $|R\rangle $ to $|{r}^{\prime}\rangle $ and $|{R}^{\prime}\rangle $, respectively.

**Figure 3.**Spatial configurations of control and target atoms. The control atom is at the origin. Here $\widehat{z}$ is the quantization axis and R is the interatomic distance between the control and the target atoms for linear configurations (

**a**) with single target atom, (

**b**) with two target atoms; (

**c**) Triangular configuration with three target atoms equally displaced from the control atom. The coordinates of the target atoms are $(-R,0,0)$, $(0,R,0)$, and $(R/\sqrt{2},-R/\sqrt{2},0)$; (

**d**) Rectangular configuration with four target atoms. The coordinates of the target atoms are $(R,0,0)$, $(-R,0,0)$, $(0,R,0)$, and $(0,-R,0)$.

**Figure 4.**(

**a**) The dependence of the probability of blocking the population transfer $|0\rangle |A\rangle \to |0\rangle |A\rangle $ on the ratio between ${\Omega}_{c}$ and ${\Omega}_{p}$. The inset shows the behavior of the population of initial state in the region when it is close to 1. (

**b**–

**d**) Time dependence of the population of the collective states ${P}_{|0\rangle |A\rangle}$ (solid-blue curve) and ${P}_{|0\rangle |B\rangle}$ (solid-red curve) for the case of blocked population transfer during CNOT gate for (

**b**) ${\Omega}_{c}=2\pi \times 7.5$ $\mathrm{MHz}$; (

**c**) ${\Omega}_{c}=2\pi \times 100$ $\mathrm{MHz}$ and (

**d**) ${\Omega}_{c}=2\pi \times 400$ $\mathrm{MHz}$. The solid-orange curve illustrates the population transfer to the Rydberg state of the target atom in the case of blocked population transfer between logical states.

**Figure 5.**(

**a**) The dipole-dipole interaction coefficient ${C}_{3}$ ($\mathrm{MHz}$.$\mathsf{\mu}{\mathrm{m}}^{3}$) as a function of the angle $\theta $ between the interatomic axis and the quantization axis for the heteronuclear interaction between Rb $|77{S}_{1/2},1/2\rangle $ and Cs $|81{S}_{1/2},{m}_{j}\rangle $. Red curve (Violet curve) represent the projection of the total angular momentum on z-axis ${m}_{j}=-\frac{1}{2}$ (${m}_{j}=\frac{1}{2}$) for the most dominant interaction channel. (

**b**,

**c**) The evolution of blockade radius (blue curve) and Le Roy radius (red curve) as a function of the principal quantum number n for homonuclear interaction of two Rb atoms and heteronuclear interaction between Rb and Cs atoms, respectively. Local minima of blockade radius of the heteronuclear interaction of pair states $|(n-4){S}_{1/2},n{S}_{1/2}\rangle $ can be attributed to principal quantum numbers where the dipole-dipole couplings to different pair states compensate each other.

**Figure 6.**Contour plot of the fidelity of entangled states ${F}_{\mathrm{R}\mathrm{b}\text{-}\mathrm{R}\mathrm{b}}$ as a function of interatomic distance ${R}_{\mathrm{CT}}$ ($\mathsf{\mu}\mathrm{m}$) and coupling Rabi frequency ${\Omega}_{c}/{\Omega}_{p}$, for homonuclear symmetric interaction of $|77{S}_{1/2},{m}_{j}=1/2\rangle $ Rb atoms for different spatial configurations of $N=1\u20134$ target atoms see [Figure 3].

**Figure 7.**Contour plot of fidelity ${F}_{\mathrm{C}\mathrm{s}\text{-}\mathrm{R}\mathrm{b}}$ as a function of interatomic distance ${R}_{\mathrm{CT}}$ ($\mathsf{\mu}\mathrm{m}$) and coupling Rabi frequency ${\Omega}_{c}/{\Omega}_{p}$, for the case of heteronuclear asymmetric interaction of Cs control atom in state $|r\rangle =|81{S}_{1/2},{m}_{j}=-1/2\rangle $ and Rb target atoms in state ${|R\rangle}_{i}=|77{S}_{1/2},{m}_{j}=1/2\rangle $ for different spatial configurations of $N=1\u20134$ target atoms [see Figure 3].

**Figure 8.**(

**a**) Scheme of spatial configurations of homonuclear/heteronuclear interactions for the implementation of C${}_{2}$NOT${}^{2}$ gate. The atoms are located on the vertices of a rhombus with perpendicular diagonals and ${R}_{\mathrm{TT}}=2\phantom{\rule{0.166667em}{0ex}}{R}_{\mathrm{CC}}$. ${R}_{\mathrm{CC}}$ (${R}_{\mathrm{TT}}$) is the interatomic distance between control (target) atoms. (

**b**) The sequence of laser pulses to perform C${}_{2}$NOT${}^{2}$ gate.

**Figure 9.**Contour plot of fidelity ${F}_{\mathrm{C}\mathrm{s}\text{-}\mathrm{R}\mathrm{b}}^{{C}_{2}NO{T}^{2}}$ for the case of heteronuclear interaction of Cs control atom in state ${|r\rangle}_{i}=|81{S}_{1/2},{m}_{j}=-1/2\rangle $ and Rb target atoms in state ${|R\rangle}_{j}=|77{S}_{1/2},{m}_{j}=1/2\rangle $ as a function of the interatomic distance ${R}_{\mathrm{CT}}$ and the ratio ${\Omega}_{c}/{\Omega}_{p}$. The system is initially prepared in state $\frac{1}{\sqrt{2}}\left(|00\rangle |AA\rangle +|11\rangle |AA\rangle \right)$. ${\Omega}_{p}=2\pi \times 50$ $\mathrm{MHz}$, and gate duration $\tau =1.53$ $\mathsf{\mu}\mathrm{s}$. (

**a**) ${V}_{\mathrm{CC}}\ne 0$; (

**b**) ${V}_{\mathrm{CC}}=0$; (

**c**) The evolution of interaction energies as a function of the interatomic distances according to the considered spatial arrangement in Figure 8a.

**Figure 10.**Investigating the source of error resulting from exciting the target atom from ground state through the first of the second resonance level of intermediate state for different configurations: (

**a**) CNOT${}^{2}$, (

**b**) CNOT${}^{3}$, and (

**c**) C${}_{2}$NOT${}^{2}$ gates. We plot the evolution of fidelity as a function of the interatomic distance ${R}_{\mathrm{CT}}$ taking into account the finite lifetimes of excited states. Solid (dashed) curves represent the case of intermediate state of the Rb target atoms to be $|P\rangle =|6{P}_{3/2},{m}_{j}=3/2\rangle $ ($|P\rangle =|5{P}_{3/2},{m}_{j}=3/2\rangle $) [see main text]. ${\Omega}_{c}=2.5\phantom{\rule{3.33333pt}{0ex}}{\Omega}_{p}=2\pi \times 125$ $\mathrm{MHz}$, ${T}_{\mathrm{C}}=1$ $\mathsf{\mu}\mathrm{s}$.

**Figure 11.**The fidelity for different spatial arrangement of qubits in CNOT${}^{N}$ (C${}_{2}$NOT${}^{2}$) gate for ${\Omega}_{c}\simeq 2.5\phantom{\rule{3.33333pt}{0ex}}{\Omega}_{p}$ and interatomic distance ${R}_{\mathrm{CT}}=5$ $\mathsf{\mu}\mathrm{m}$ in (

**a**), ${R}_{\mathrm{CT}}=7$ $\mathsf{\mu}\mathrm{m}$ in (

**b**).

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## Share and Cite

**MDPI and ACS Style**

M. Farouk, A.; Beterov, I.I.; Xu , P.; Bergamini , S.; Ryabtsev , I.I.
Parallel Implementation of CNOT^{N} and C_{2}NOT^{2} Gates via Homonuclear and Heteronuclear Förster Interactions of Rydberg Atoms. *Photonics* **2023**, *10*, 1280.
https://doi.org/10.3390/photonics10111280

**AMA Style**

M. Farouk A, Beterov II, Xu P, Bergamini S, Ryabtsev II.
Parallel Implementation of CNOT^{N} and C_{2}NOT^{2} Gates via Homonuclear and Heteronuclear Förster Interactions of Rydberg Atoms. *Photonics*. 2023; 10(11):1280.
https://doi.org/10.3390/photonics10111280

**Chicago/Turabian Style**

M. Farouk, Ahmed, Ilya I. Beterov, Peng Xu , Silvia Bergamini , and Igor I. Ryabtsev .
2023. "Parallel Implementation of CNOT^{N} and C_{2}NOT^{2} Gates via Homonuclear and Heteronuclear Förster Interactions of Rydberg Atoms" *Photonics* 10, no. 11: 1280.
https://doi.org/10.3390/photonics10111280