Suppressing crosstalk for Rydberg quantum gates
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AbstractWe present a method to suppress crosstalk from implementing controlled-Z gates via local addressing in neutral atom quantum computers. In these systems, a fraction of the laser light that is applied locally to implement gates typically leaks to other atoms. We analyze the resulting crosstalk in a setup of two gate atoms and one neighboring third atom. We then perturbatively derive a spin-echo-inspired gate protocol that suppresses the leading order of the amplitude error, which dominates the crosstalk. Numerical simulations demonstrate that our gate protocol improves the fidelity by two orders of magnitude across a broad range of experimentally relevant parameters. To further reduce the infidelity, we develop a circuit to cancel remaining phase errors. Our results pave the way for using local addressing for high-fidelity quantum gates on Rydberg-based quantum computers.Featured image: Setup for studying the crosstalk of a two-qubit gate affecting a third neighboring atom. A local laser (depicted in yellow) drives the gate on atoms 1 and 2. We assume that atom 3 is affected by a small fraction of the laser light.Popular summaryThe neutral-atom platform is a promising candidate for realizing universal quantum computers. Two-qubit gates are implemented by temporarily exciting atoms to strongly interacting Rydberg states. There are two complementary methods for executing gates on individual atomic qubits. In one method, gate atoms are shifted close to each other while having a large distance to other atoms. Then, a global excitation pulse has a non-trivial effect on the gate atoms only. Alternatively, local addressing can be used, where a laser is tightly focused onto the gate atoms. The latter avoids time-consuming shifts and enables denser atomic arrays. However, a fraction of the laser light typically leaks to other atoms. This crosstalk introduces gate errors. In our work, we analyze the crosstalk and develop a gate protocol to suppress it. We consider two gate atoms on which a controlled-Z gate is performed, and a third atom which is subject to leaking laser light. Using perturbation theory, we design a spin-echo-inspired protocol that suppresses gate errors in leading order. Numerical simulations demonstrate a suppression of the gate infidelity by two orders of magnitude across a broad range of experimentally relevant parameters. Our results pave the way for using local addressing for high-fidelity quantum gates on neutral-atom platforms.► BibTeX data@article{Warttmann2026suppressing, doi = {10.22331/q-2026-03-24-2045}, url = {https://doi.org/10.22331/q-2026-03-24-2045}, title = {Suppressing crosstalk for {R}ydberg quantum gates}, author = {Warttmann, Gina and Meinert, Florian and B{\"{u}}chler, Hans Peter and Weber, Sebastian}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2045}, month = mar, year = {2026} }► References [1] L. Henriet, L. Beguin, A. Signoles, T. Lahaye, A. Browaeys, G.-O. Reymond, and C. Jurczak, Quantum computing with neutral atoms, Quantum 4, 327 (2020). https://doi.org/10.22331/q-2020-09-21-327 [2] M. Saffman, T. G. Walker, and K. Mølmer, Quantum information with rydberg atoms, Rev. Mod. Phys. 82, 2313 (2010). https://doi.org/10.1103/RevModPhys.82.2313 [3] X. Wu, X. Liang, Y. Tian, F. Yang, C. Chen, Y.-C. Liu, M. K. Tey, and L. You, A concise review of rydberg atom based quantum computation and quantum simulation, Chinese Phys. B 30, 020305 (2021). https://doi.org/10.1088/1674-1056/abd76f [4] P. Huft, Y. Song, T. M. Graham, K. Jooya, S. Deshpande, C. Fang, M. Kats, and M. Saffman, Simple, passive design for large optical trap arrays for single atoms, Phys. Rev. A 105, 063111 (2022). https://doi.org/10.1103/PhysRevA.105.063111 [5] L. Pause, L. Sturm, M. Mittenbühler, S. Amann, T. Preuschoff, D. Schäffner, M. Schlosser, and G. Birkl, Supercharged two-dimensional tweezer array with more than 1000 atomic qubits, Optica 11, 222 (2024). https://doi.org/10.1364/OPTICA.513551 [6] H. J. Manetsch, G. Nomura, E. Bataille, X. Lv, K. H. Leung, and M. Endres, A tweezer array with 6,100 highly coherent atomic qubits, Nature 647, 60 (2025). https://doi.org/10.1038/s41586-025-09641-4 [7] G. Pichard, D. Lim, É. Bloch, J. Vaneecloo, L. Bourachot, G.-J. Both, G. Mériaux, et al., Rearrangement of individual atoms in a 2000-site optical-tweezer array at cryogenic temperatures, Phys. Rev. Appl. 22, 024073 (2024). https://doi.org/10.1103/PhysRevApplied.22.024073 [8] M. A. Norcia, H. Kim, W. B. Cairncross, M. Stone, A. Ryou, M. Jaffe, M. O. Brown, et al., Iterative assembly of ${}^{171}Yb$ atom arrays with cavity-enhanced optical lattices, PRX Quantum 5, 030316 (2024). https://doi.org/10.1103/PRXQuantum.5.030316 [9] D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, Fast quantum gates for neutral atoms, Phys. Rev. Lett. 85, 2208 (2000). https://doi.org/10.1103/PhysRevLett.85.2208 [10] T. M. Graham, Y. Song, J. Scott, C. Poole, L. Phuttitarn, K. Jooya, P. Eichler, et al., Multi-qubit entanglement and algorithms on a neutral-atom quantum computer, Nature 604, 457–462 (2022). https://doi.org/10.1038/s41586-022-04603-6 [11] M. Saffman, Quantum computing with atomic qubits and rydberg interactions: progress and challenges, J. Phys. B: At. Mol. Opt. Phys. 49, 202001 (2016). https://doi.org/10.1088/0953-4075/49/20/202001 [12] H. Levine, A. Keesling, G. Semeghini, A. Omran, T. T. Wang, S. Ebadi, H. Bernien, et al., Parallel implementation of high-fidelity multiqubit gates with neutral atoms, Phys. Rev. Lett. 123, 170503 (2019). https://doi.org/10.1103/PhysRevLett.123.170503 [13] A. Pagano, S. Weber, D. Jaschke, T. Pfau, F. Meinert, S. Montangero, and H. P. Büchler, Error budgeting for a controlled-phase gate with strontium-88 rydberg atoms, Phys. Rev. Res. 4, 033019 (2022). https://doi.org/10.1103/PhysRevResearch.4.033019 [14] S. Jandura and G. Pupillo, Time-Optimal Two- and Three-Qubit Gates for Rydberg Atoms, Quantum 6, 712 (2022). https://doi.org/10.22331/q-2022-05-13-712 [15] J. A. Muniz, M. Stone, D. T. Stack, M. Jaffe, J. M. Kindem, L. Wadleigh, E. Zalys-Geller, et al., High-fidelity universal gates in the ${}^{171}\mathrm{Yb}$ ground-state nuclear-spin qubit, PRX Quantum 6, 020334 (2025a). https://doi.org/10.1103/PRXQuantum.6.020334 [16] R. B.-S. Tsai, X. Sun, A. L. Shaw, R. Finkelstein, and M. Endres, Benchmarking and fidelity response theory of high-fidelity rydberg entangling gates, PRX Quantum 6, 010331 (2025). https://doi.org/10.1103/PRXQuantum.6.010331 [17] A. Radnaev, W. Chung, D. Cole, D. Mason, T. Ballance, M. Bedalov, D. Belknap, et al., Universal neutral-atom quantum computer with individual optical addressing and nondestructive readout, PRX Quantum 6, 030334 (2025). https://doi.org/10.1103/66s8-jj18 [18] A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, et al., Elementary gates for quantum computation, Phys. Rev. A 52, 3457 (1995). https://doi.org/10.1103/PhysRevA.52.3457 [19] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, 2010). https://doi.org/10.1017/CBO9780511976667 [20] D. Bluvstein, H. Levine, G. Semeghini, T. T. Wang, S. Ebadi, M. Kalinowski, A. Keesling, et al., A quantum processor based on coherent transport of entangled atom arrays, Nature 604, 451–456 (2022). https://doi.org/10.1038/s41586-022-04592-6 [21] K. Barnes, P. Battaglino, B. J. Bloom, K. Cassella, R. Coxe, N. Crisosto, J. P. King, et al., Assembly and coherent control of a register of nuclear spin qubits, Nature Commun. 13, 2779 (2022). https://doi.org/10.1038/s41467-022-29977-z [22] D. B. Tan, W.-H. Lin, and J. Cong, Compilation for dynamically field-programmable qubit arrays with efficient and provably near-optimal scheduling, in Proceedings of the 30th Asia and South Pacific Design Automation Conference (Association for Computing Machinery, New York, NY, USA, 2025) p. 921–929. https://doi.org/10.1145/3658617.3697778 [23] H. Wang, D. B. Tan, P. Liu, Y. Liu, J. Gu, J. Cong, and S. Han, in Proceedings of the 61st ACM/IEEE Design Automation Conference, DAC '24 (Association for Computing Machinery, New York, NY, USA, 2024) p. 1–6. https://doi.org/10.1145/3649329.3658470 [24] D. B. Tan, W.-H. Lin, and J. Cong, Compilation for dynamically field-programmable qubit arrays with efficient and provably near-optimal scheduling, in Proceedings of the 30th Asia and South Pacific Design Automation Conference (Association for Computing Machinery, New York, NY, USA, 2025) p. 921–929. https://doi.org/10.1145/3658617.3697778 [25] H. Wang, P. Liu, D. B. Tan, Y. Liu, J. Gu, D. Z. Pan, J. Cong, et al., in Proceedings of the 51st Annual International Symposium on Computer Architecture, ISCA '24 (IEEE Press, 2025) p. 293–309. https://doi.org/10.1109/ISCA59077.2024.00030 [26] B. W. Reichardt, A. Paetznick, D. Aasen, I. Basov, J. M. Bello-Rivas, P. Bonderson, R. Chao, et al., Fault-tolerant quantum compupillotation with a neutral atom processor, arXiv:2411.11822 (2024). arXiv:2411.11822 [27] J. A. Muniz, D. Crow, H. Kim, J. M. Kindem, W. B. Cairncross, A. Ryou, T. C. Bohdanowicz, et al., Repeated ancilla reuse for logical computation on a neutral atom quantum computer, Phys. Rev. X 15, 041040 (2025b). https://doi.org/10.1103/v7ny-fg31 [28] A. P. Burgers, S. Ma, S. Saskin, J. Wilson, M. A. Alarcón, C. H. Greene, and J. D. Thompson, Controlling rydberg excitations using ion-core transitions in alkaline-earth atom-tweezer arrays, PRX Quantum 3, 020326 (2022). https://doi.org/10.1103/PRXQuantum.3.020326 [29] L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, Demonstration of a neutral atom controlled-not quantum gate, Phys. Rev. Lett. 104, 010503 (2010). https://doi.org/10.1103/PhysRevLett.104.010503 [30] T. M. Graham, M. Kwon, B. Grinkemeyer, Z. Marra, X. Jiang, M. T. Lichtman, Y. Sun, et al., Rydberg-mediated entanglement in a two-dimensional neutral atom qubit array, Phys. Rev. Lett. 123, 230501 (2019). https://doi.org/10.1103/PhysRevLett.123.230501 [31] A. S. Sotirova, B. Sun, J. D. Leppard, A. Wang, M. Wang, A. Vazquez-Brennan, D. P. Nadlinger, et al., Low cross-talk optical addressing of trapped-ion qubits using a novel integrated photonic chip, Light. Sci. & Appl 13, 199 (2024). https://doi.org/10.1038/s41377-024-01542-x [32] K. Tanaka, N. Saga, and K. Hauchi, Focusing of a gaussian beam through a finite aperture lens, Appl. Opt. 24, 1098 (1985). https://doi.org/10.1364/AO.24.001098Cited byCould not fetch Crossref cited-by data during last attempt 2026-03-24 16:05:06: Could not fetch cited-by data for 10.22331/q-2026-03-24-2045 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-03-24 16:05:06: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractWe present a method to suppress crosstalk from implementing controlled-Z gates via local addressing in neutral atom quantum computers. In these systems, a fraction of the laser light that is applied locally to implement gates typically leaks to other atoms. We analyze the resulting crosstalk in a setup of two gate atoms and one neighboring third atom. We then perturbatively derive a spin-echo-inspired gate protocol that suppresses the leading order of the amplitude error, which dominates the crosstalk. Numerical simulations demonstrate that our gate protocol improves the fidelity by two orders of magnitude across a broad range of experimentally relevant parameters. To further reduce the infidelity, we develop a circuit to cancel remaining phase errors. Our results pave the way for using local addressing for high-fidelity quantum gates on Rydberg-based quantum computers.Featured image: Setup for studying the crosstalk of a two-qubit gate affecting a third neighboring atom. A local laser (depicted in yellow) drives the gate on atoms 1 and 2. We assume that atom 3 is affected by a small fraction of the laser light.Popular summaryThe neutral-atom platform is a promising candidate for realizing universal quantum computers. Two-qubit gates are implemented by temporarily exciting atoms to strongly interacting Rydberg states. There are two complementary methods for executing gates on individual atomic qubits. In one method, gate atoms are shifted close to each other while having a large distance to other atoms. Then, a global excitation pulse has a non-trivial effect on the gate atoms only. Alternatively, local addressing can be used, where a laser is tightly focused onto the gate atoms. The latter avoids time-consuming shifts and enables denser atomic arrays. However, a fraction of the laser light typically leaks to other atoms. This crosstalk introduces gate errors. In our work, we analyze the crosstalk and develop a gate protocol to suppress it. We consider two gate atoms on which a controlled-Z gate is performed, and a third atom which is subject to leaking laser light. Using perturbation theory, we design a spin-echo-inspired protocol that suppresses gate errors in leading order. Numerical simulations demonstrate a suppression of the gate infidelity by two orders of magnitude across a broad range of experimentally relevant parameters. Our results pave the way for using local addressing for high-fidelity quantum gates on neutral-atom platforms.► BibTeX data@article{Warttmann2026suppressing, doi = {10.22331/q-2026-03-24-2045}, url = {https://doi.org/10.22331/q-2026-03-24-2045}, title = {Suppressing crosstalk for {R}ydberg quantum gates}, author = {Warttmann, Gina and Meinert, Florian and B{\"{u}}chler, Hans Peter and Weber, Sebastian}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2045}, month = mar, year = {2026} }► References [1] L. Henriet, L. Beguin, A. Signoles, T. Lahaye, A. Browaeys, G.-O. Reymond, and C. Jurczak, Quantum computing with neutral atoms, Quantum 4, 327 (2020). https://doi.org/10.22331/q-2020-09-21-327 [2] M. Saffman, T. G. Walker, and K. Mølmer, Quantum information with rydberg atoms, Rev. Mod. Phys. 82, 2313 (2010). https://doi.org/10.1103/RevModPhys.82.2313 [3] X. Wu, X. Liang, Y. Tian, F. Yang, C. Chen, Y.-C. Liu, M. K. Tey, and L. You, A concise review of rydberg atom based quantum computation and quantum simulation, Chinese Phys. B 30, 020305 (2021). https://doi.org/10.1088/1674-1056/abd76f [4] P. Huft, Y. Song, T. M. Graham, K. Jooya, S. Deshpande, C. Fang, M. Kats, and M. Saffman, Simple, passive design for large optical trap arrays for single atoms, Phys. Rev. A 105, 063111 (2022). https://doi.org/10.1103/PhysRevA.105.063111 [5] L. Pause, L. Sturm, M. Mittenbühler, S. Amann, T. Preuschoff, D. Schäffner, M. Schlosser, and G. Birkl, Supercharged two-dimensional tweezer array with more than 1000 atomic qubits, Optica 11, 222 (2024). https://doi.org/10.1364/OPTICA.513551 [6] H. J. Manetsch, G. Nomura, E. Bataille, X. Lv, K. H. Leung, and M. Endres, A tweezer array with 6,100 highly coherent atomic qubits, Nature 647, 60 (2025). https://doi.org/10.1038/s41586-025-09641-4 [7] G. Pichard, D. Lim, É. Bloch, J. Vaneecloo, L. Bourachot, G.-J. Both, G. Mériaux, et al., Rearrangement of individual atoms in a 2000-site optical-tweezer array at cryogenic temperatures, Phys. Rev. Appl. 22, 024073 (2024). https://doi.org/10.1103/PhysRevApplied.22.024073 [8] M. A. Norcia, H. Kim, W. B. Cairncross, M. Stone, A. Ryou, M. Jaffe, M. O. Brown, et al., Iterative assembly of ${}^{171}Yb$ atom arrays with cavity-enhanced optical lattices, PRX Quantum 5, 030316 (2024). https://doi.org/10.1103/PRXQuantum.5.030316 [9] D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, Fast quantum gates for neutral atoms, Phys. Rev. Lett. 85, 2208 (2000). https://doi.org/10.1103/PhysRevLett.85.2208 [10] T. M. Graham, Y. Song, J. Scott, C. Poole, L. Phuttitarn, K. Jooya, P. Eichler, et al., Multi-qubit entanglement and algorithms on a neutral-atom quantum computer, Nature 604, 457–462 (2022). https://doi.org/10.1038/s41586-022-04603-6 [11] M. Saffman, Quantum computing with atomic qubits and rydberg interactions: progress and challenges, J. Phys. B: At. Mol. Opt. Phys. 49, 202001 (2016). https://doi.org/10.1088/0953-4075/49/20/202001 [12] H. Levine, A. Keesling, G. Semeghini, A. Omran, T. T. Wang, S. Ebadi, H. Bernien, et al., Parallel implementation of high-fidelity multiqubit gates with neutral atoms, Phys. Rev. Lett. 123, 170503 (2019). https://doi.org/10.1103/PhysRevLett.123.170503 [13] A. Pagano, S. Weber, D. Jaschke, T. Pfau, F. Meinert, S. Montangero, and H. P. Büchler, Error budgeting for a controlled-phase gate with strontium-88 rydberg atoms, Phys. Rev. Res. 4, 033019 (2022). https://doi.org/10.1103/PhysRevResearch.4.033019 [14] S. Jandura and G. Pupillo, Time-Optimal Two- and Three-Qubit Gates for Rydberg Atoms, Quantum 6, 712 (2022). https://doi.org/10.22331/q-2022-05-13-712 [15] J. A. Muniz, M. Stone, D. T. Stack, M. Jaffe, J. M. Kindem, L. Wadleigh, E. Zalys-Geller, et al., High-fidelity universal gates in the ${}^{171}\mathrm{Yb}$ ground-state nuclear-spin qubit, PRX Quantum 6, 020334 (2025a). https://doi.org/10.1103/PRXQuantum.6.020334 [16] R. B.-S. Tsai, X. Sun, A. L. Shaw, R. Finkelstein, and M. Endres, Benchmarking and fidelity response theory of high-fidelity rydberg entangling gates, PRX Quantum 6, 010331 (2025). https://doi.org/10.1103/PRXQuantum.6.010331 [17] A. Radnaev, W. Chung, D. Cole, D. Mason, T. Ballance, M. Bedalov, D. Belknap, et al., Universal neutral-atom quantum computer with individual optical addressing and nondestructive readout, PRX Quantum 6, 030334 (2025). https://doi.org/10.1103/66s8-jj18 [18] A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, et al., Elementary gates for quantum computation, Phys. Rev. A 52, 3457 (1995). https://doi.org/10.1103/PhysRevA.52.3457 [19] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, 2010). https://doi.org/10.1017/CBO9780511976667 [20] D. Bluvstein, H. Levine, G. Semeghini, T. T. Wang, S. Ebadi, M. Kalinowski, A. Keesling, et al., A quantum processor based on coherent transport of entangled atom arrays, Nature 604, 451–456 (2022). https://doi.org/10.1038/s41586-022-04592-6 [21] K. Barnes, P. Battaglino, B. J. Bloom, K. Cassella, R. Coxe, N. Crisosto, J. P. King, et al., Assembly and coherent control of a register of nuclear spin qubits, Nature Commun. 13, 2779 (2022). https://doi.org/10.1038/s41467-022-29977-z [22] D. B. Tan, W.-H. Lin, and J. Cong, Compilation for dynamically field-programmable qubit arrays with efficient and provably near-optimal scheduling, in Proceedings of the 30th Asia and South Pacific Design Automation Conference (Association for Computing Machinery, New York, NY, USA, 2025) p. 921–929. https://doi.org/10.1145/3658617.3697778 [23] H. Wang, D. B. Tan, P. Liu, Y. Liu, J. Gu, J. Cong, and S. Han, in Proceedings of the 61st ACM/IEEE Design Automation Conference, DAC '24 (Association for Computing Machinery, New York, NY, USA, 2024) p. 1–6. https://doi.org/10.1145/3649329.3658470 [24] D. B. Tan, W.-H. Lin, and J. Cong, Compilation for dynamically field-programmable qubit arrays with efficient and provably near-optimal scheduling, in Proceedings of the 30th Asia and South Pacific Design Automation Conference (Association for Computing Machinery, New York, NY, USA, 2025) p. 921–929. https://doi.org/10.1145/3658617.3697778 [25] H. Wang, P. Liu, D. B. Tan, Y. Liu, J. Gu, D. Z. Pan, J. Cong, et al., in Proceedings of the 51st Annual International Symposium on Computer Architecture, ISCA '24 (IEEE Press, 2025) p. 293–309. https://doi.org/10.1109/ISCA59077.2024.00030 [26] B. W. Reichardt, A. Paetznick, D. Aasen, I. Basov, J. M. Bello-Rivas, P. Bonderson, R. Chao, et al., Fault-tolerant quantum compupillotation with a neutral atom processor, arXiv:2411.11822 (2024). arXiv:2411.11822 [27] J. A. Muniz, D. Crow, H. Kim, J. M. Kindem, W. B. Cairncross, A. Ryou, T. C. Bohdanowicz, et al., Repeated ancilla reuse for logical computation on a neutral atom quantum computer, Phys. Rev. X 15, 041040 (2025b). https://doi.org/10.1103/v7ny-fg31 [28] A. P. Burgers, S. Ma, S. Saskin, J. Wilson, M. A. Alarcón, C. H. Greene, and J. D. Thompson, Controlling rydberg excitations using ion-core transitions in alkaline-earth atom-tweezer arrays, PRX Quantum 3, 020326 (2022). https://doi.org/10.1103/PRXQuantum.3.020326 [29] L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, Demonstration of a neutral atom controlled-not quantum gate, Phys. Rev. Lett. 104, 010503 (2010). https://doi.org/10.1103/PhysRevLett.104.010503 [30] T. M. Graham, M. Kwon, B. Grinkemeyer, Z. Marra, X. Jiang, M. T. Lichtman, Y. Sun, et al., Rydberg-mediated entanglement in a two-dimensional neutral atom qubit array, Phys. Rev. Lett. 123, 230501 (2019). https://doi.org/10.1103/PhysRevLett.123.230501 [31] A. S. Sotirova, B. Sun, J. D. Leppard, A. Wang, M. Wang, A. Vazquez-Brennan, D. P. Nadlinger, et al., Low cross-talk optical addressing of trapped-ion qubits using a novel integrated photonic chip, Light. Sci. & Appl 13, 199 (2024). https://doi.org/10.1038/s41377-024-01542-x [32] K. Tanaka, N. Saga, and K. Hauchi, Focusing of a gaussian beam through a finite aperture lens, Appl. Opt. 24, 1098 (1985). https://doi.org/10.1364/AO.24.001098Cited byCould not fetch Crossref cited-by data during last attempt 2026-03-24 16:05:06: Could not fetch cited-by data for 10.22331/q-2026-03-24-2045 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-03-24 16:05:06: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.