Generalized Conditional Displacement
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AbstractConditional displacement with a qubit ancilla is a critical component in continuous-variable error correction protocols. We present the generalized conditional displacement operator, conditioned on a qudit ancilla, and explore potential implementations. We show how this operator can be used to enhance error correction with Gottesman-Kitaev-Preskill (GKP) codes.Popular summaryOur research focuses on hybrid continuous-variable discrete-variable quantum error correction and its applications for creating and stabilizing the Gottesman-Kitaev-Preskill (GKP) code. We generalize the conventional conditional displacement (CD) gate from qubits to multi-level qudits. By exploiting the extra levels of a qudit, the generalized CD can act on both quadratures of a quantum oscillator simultaneously, doubling stabilization speed and improving resilience to noise. We investigate the fundamental interactions of the GKP with various systems, including another harmonic oscillator, a quantum planar rotor, and an atom ensemble, that can create this gate, thereby connecting to leading experimental platforms.► BibTeX data@article{EvenHaim2025generalized, doi = {10.22331/q-2025-12-10-1934}, url = {https://doi.org/10.22331/q-2025-12-10-1934}, title = {Generalized {C}onditional {D}isplacement}, author = {Even-Haim, Shiran and Diringer, Asaf A. and Ruimy, Ron and Baranes, Gefen and Gorlach, Alexey and Hacohen-Gourgy, Shay and Kaminer, Ido}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1934}, month = dec, year = {2025} }► References [1] M. H. Michael et al., Phys. Rev. X, 6, 031006 (2016), 10.1103/PhysRevX.6.031006. https://doi.org/10.1103/PhysRevX.6.031006 [2] P. T. Cochrane, G. J. Milburn, and W. J. Munro, Phys. Rev. A, 59, 2631–2634 (1999), 10.1103/PhysRevA.59.2631. https://doi.org/10.1103/PhysRevA.59.2631 [3] D. Gottesman, A. Kitaev, and J. Preskill, Phys. Rev. A, 64, 012310 (2001), 10.1103/PhysRevA.64.012310. https://doi.org/10.1103/PhysRevA.64.012310 [4] Y. Liu et al., PRX Quantum (to appear) (2025), 10.1103/4rf7-9tfx. https://doi.org/10.1103/4rf7-9tfx [5] C. Flühmann et al., Nature, 566, 7745 (2019), 10.1038/s41586-019-0960-6. https://doi.org/10.1038/s41586-019-0960-6 [6] P. Campagne-Ibarcq et al., Nature, 584, 7821 (2020), 10.1038/s41586-020-2603-3. https://doi.org/10.1038/s41586-020-2603-3 [7] V. V. Sivak et al., Nature, 616, 7955 (2023), 10.1038/s41586-023-05782-6. https://doi.org/10.1038/s41586-023-05782-6 [8] A. Eickbusch et al., Nat. Phys., 18, 12 (2022), 10.1038/s41567-022-01776-9. https://doi.org/10.1038/s41567-022-01776-9 [9] B. M. Terhal, J. Conrad, and C. Vuillot, Quantum Sci. Technol., 5, 043001 (2020), 10.1088/2058-9565/ab98a5. https://doi.org/10.1088/2058-9565/ab98a5 [10] A. Asadian, P. Erker, M. Huber, and C. Klöckl, Phys. Rev. A, 94, 010301 (2016), 10.1103/PhysRevA.94.010301. https://doi.org/10.1103/PhysRevA.94.010301 [11] V. V. Albert, S. Pascazio, and M. H. Devoret, J. Phys. A: Math. Theor., 50, 504002 (2017), 10.1088/1751-8121/aa9314. https://doi.org/10.1088/1751-8121/aa9314 [12] K. R. Motes, B. Q. Baragiola, A. Gilchrist, and N. C. Menicucci, Phys. Rev. A, 95, 053819 (2017), 10.1103/PhysRevA.95.053819. https://doi.org/10.1103/PhysRevA.95.053819 [13] L. Li et al., Phys. Rev. Lett., 119, 030502 (2017), 10.1103/PhysRevLett.119.030502. https://doi.org/10.1103/PhysRevLett.119.030502 [14] A. L. Grimsmo, J. Combes, and B. Q. Baragiola, Phys. Rev. X, 10, 011058 (2020), 10.1103/PhysRevX.10.011058. https://doi.org/10.1103/PhysRevX.10.011058 [15] R. Dahan et al., Phys. Rev. X, 13, 031001 (2023), 10.1103/PhysRevX.13.031001. https://doi.org/10.1103/PhysRevX.13.031001 [16] G. Baranes et al., Phys. Rev. Research, 5, 043271 (2023), 10.1103/PhysRevResearch.5.043271. https://doi.org/10.1103/PhysRevResearch.5.043271 [17] J. Zak, Phys. Rev. Lett., 19, 1385–1387 (1967), 10.1103/PhysRevLett.19.1385. https://doi.org/10.1103/PhysRevLett.19.1385 [18] B. Royer, S. Singh, and S. M. Girvin, Phys. Rev. Lett., 125, 260509 (2020), 10.1103/PhysRevLett.125.260509. https://doi.org/10.1103/PhysRevLett.125.260509 [19] B. M. Terhal and D. Weigand, Phys. Rev. A, 93, 012315 (2016), 10.1103/PhysRevA.93.012315. https://doi.org/10.1103/PhysRevA.93.012315 [20] I. I. Rabi, Phys. Rev., 49, 324–328 (1936), 10.1103/PhysRev.49.324. https://doi.org/10.1103/PhysRev.49.324 [21] A. Frisk Kockum et al., Nat. Rev. Phys., 1, 1 (2019), 10.1038/s42254-018-0006-2. https://doi.org/10.1038/s42254-018-0006-2 [22] J. Hastrup et al., npj Quantum Inf., 7, 1 (2021), 10.1038/s41534-020-00353-3. https://doi.org/10.1038/s41534-020-00353-3 [23] B. W. Shore and J. H. Eberly, Opt. Commun., 24, 83–88 (1978), 10.1016/0030-4018(78)90272-9. https://doi.org/10.1016/0030-4018(78)90272-9 [24] V. V. Albert, J. P. Covey, and J. Preskill, Phys. Rev. X, 10, 031050 (2020), 10.1103/PhysRevX.10.031050. https://doi.org/10.1103/PhysRevX.10.031050 [25] R. H. Dicke, Phys. Rev., 93, 99–110 (1954), 10.1103/PhysRev.93.99. https://doi.org/10.1103/PhysRev.93.99 [26] M. Tavis and F. W. Cummings, Phys. Rev., 170, 379–384 (1968), 10.1103/PhysRev.170.379. https://doi.org/10.1103/PhysRev.170.379 [27] Y. Ouyang, Linear Algebra Appl., 532, 43–59 (2017), 10.1016/j.laa.2017.06.031. https://doi.org/10.1016/j.laa.2017.06.031 [28] O. Tziperman et al., ACS Nano (2025), 10.1021/acsnano.4c15257. https://doi.org/10.1021/acsnano.4c15257 [29] N. Gutman et al., Phys. Rev. Lett. 132, 153601 (2025), 10.1103/PhysRevLett.132.153601. https://doi.org/10.1103/PhysRevLett.132.153601 [30] D. M. Lukin et al., Phys. Rev. X, 13, 011005 (2023), 10.1103/PhysRevX.13.011005. https://doi.org/10.1103/PhysRevX.13.011005 [31] A. Tiranov et al., Science, 379, 389–393 (2023), 10.1126/science.ade9324. https://doi.org/10.1126/science.ade9324 [32] G. Ferioli et al., Phys. Rev. Lett. 132, 133601 (2025), 10.1103/PhysRevLett.132.133601. https://doi.org/10.1103/PhysRevLett.132.133601 [33] C. Liedl et al., Phys. Rev. X, 14, 011020 (2024), 10.1103/PhysRevX.14.011020. https://doi.org/10.1103/PhysRevX.14.011020 [34] S. M. Barnett and D. T. Pegg, J. Phys. A: Math. Gen., 19, 3849 (1986), 10.1088/0305-4470/19/18/030. https://doi.org/10.1088/0305-4470/19/18/030 [35] A. A. Diringer et al., Phys. Rev. X, 14, 011055 (2024), 10.1103/PhysRevX.14.011055. https://doi.org/10.1103/PhysRevX.14.011055 [36] M. J. Peterer et al., Phys. Rev. Lett., 114, 010501 (2015), 10.1103/PhysRevLett.114.010501. https://doi.org/10.1103/PhysRevLett.114.010501 [37] B. Barwick, D. J. Flannigan, and A. H. Zewail, Nature, 462, 1859–1863 (2009), 10.1038/nature08662. https://doi.org/10.1038/nature08662 [38] S. T. Park, M. Lin, and A. H. Zewail, New J. Phys., 12, 123028 (2010), 10.1088/1367-2630/12/12/123028. https://doi.org/10.1088/1367-2630/12/12/123028 [39] F. J. García de Abajo, A. Asenjo-Garcia, and M. Kociak, Nano Lett., 10, 1859–1863 (2010), 10.1021/nl100613s. https://doi.org/10.1021/nl100613s [40] O. Reinhardt et al., Ann. Phys., 533, 2000254 (2021), 10.1002/andp.202000254. https://doi.org/10.1002/andp.202000254 [41] R. Dahan et al., Nat. Phys., 16, 11 (2020), 10.1038/s41567-020-01042-w. https://doi.org/10.1038/s41567-020-01042-w [42] R. Dahan et al., Science, 373, eabj7128 (2021), 10.1126/science.abj7128. https://doi.org/10.1126/science.abj7128 [43] J.-W. Henke et al., Nature, 600, 7890 (2021), 10.1038/s41586-021-04197-5. https://doi.org/10.1038/s41586-021-04197-5 [44] A. Feist et al., Science, 377, 777–780 (2022), 10.1126/science.abo5037. https://doi.org/10.1126/science.abo5037 [45] Y. Adiv et al., Phys. Rev. X, 13, 011002 (2023), 10.1103/PhysRevX.13.011002. https://doi.org/10.1103/PhysRevX.13.011002Cited byCould not fetch Crossref cited-by data during last attempt 2025-12-10 11:31:17: Could not fetch cited-by data for 10.22331/q-2025-12-10-1934 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-12-10 11:31:17: 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. AbstractConditional displacement with a qubit ancilla is a critical component in continuous-variable error correction protocols. We present the generalized conditional displacement operator, conditioned on a qudit ancilla, and explore potential implementations. We show how this operator can be used to enhance error correction with Gottesman-Kitaev-Preskill (GKP) codes.Popular summaryOur research focuses on hybrid continuous-variable discrete-variable quantum error correction and its applications for creating and stabilizing the Gottesman-Kitaev-Preskill (GKP) code. We generalize the conventional conditional displacement (CD) gate from qubits to multi-level qudits. By exploiting the extra levels of a qudit, the generalized CD can act on both quadratures of a quantum oscillator simultaneously, doubling stabilization speed and improving resilience to noise. We investigate the fundamental interactions of the GKP with various systems, including another harmonic oscillator, a quantum planar rotor, and an atom ensemble, that can create this gate, thereby connecting to leading experimental platforms.► BibTeX data@article{EvenHaim2025generalized, doi = {10.22331/q-2025-12-10-1934}, url = {https://doi.org/10.22331/q-2025-12-10-1934}, title = {Generalized {C}onditional {D}isplacement}, author = {Even-Haim, Shiran and Diringer, Asaf A. and Ruimy, Ron and Baranes, Gefen and Gorlach, Alexey and Hacohen-Gourgy, Shay and Kaminer, Ido}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1934}, month = dec, year = {2025} }► References [1] M. H. Michael et al., Phys. Rev. X, 6, 031006 (2016), 10.1103/PhysRevX.6.031006. https://doi.org/10.1103/PhysRevX.6.031006 [2] P. T. Cochrane, G. J. Milburn, and W. J. Munro, Phys. Rev. A, 59, 2631–2634 (1999), 10.1103/PhysRevA.59.2631. https://doi.org/10.1103/PhysRevA.59.2631 [3] D. Gottesman, A. Kitaev, and J. Preskill, Phys. Rev. A, 64, 012310 (2001), 10.1103/PhysRevA.64.012310. https://doi.org/10.1103/PhysRevA.64.012310 [4] Y. Liu et al., PRX Quantum (to appear) (2025), 10.1103/4rf7-9tfx. https://doi.org/10.1103/4rf7-9tfx [5] C. Flühmann et al., Nature, 566, 7745 (2019), 10.1038/s41586-019-0960-6. https://doi.org/10.1038/s41586-019-0960-6 [6] P. Campagne-Ibarcq et al., Nature, 584, 7821 (2020), 10.1038/s41586-020-2603-3. https://doi.org/10.1038/s41586-020-2603-3 [7] V. V. Sivak et al., Nature, 616, 7955 (2023), 10.1038/s41586-023-05782-6. https://doi.org/10.1038/s41586-023-05782-6 [8] A. Eickbusch et al., Nat. Phys., 18, 12 (2022), 10.1038/s41567-022-01776-9. https://doi.org/10.1038/s41567-022-01776-9 [9] B. M. Terhal, J. Conrad, and C. Vuillot, Quantum Sci. Technol., 5, 043001 (2020), 10.1088/2058-9565/ab98a5. https://doi.org/10.1088/2058-9565/ab98a5 [10] A. Asadian, P. Erker, M. Huber, and C. Klöckl, Phys. Rev. A, 94, 010301 (2016), 10.1103/PhysRevA.94.010301. https://doi.org/10.1103/PhysRevA.94.010301 [11] V. V. Albert, S. Pascazio, and M. H. Devoret, J. Phys. A: Math. Theor., 50, 504002 (2017), 10.1088/1751-8121/aa9314. https://doi.org/10.1088/1751-8121/aa9314 [12] K. R. Motes, B. Q. Baragiola, A. Gilchrist, and N. C. Menicucci, Phys. Rev. A, 95, 053819 (2017), 10.1103/PhysRevA.95.053819. https://doi.org/10.1103/PhysRevA.95.053819 [13] L. Li et al., Phys. Rev. Lett., 119, 030502 (2017), 10.1103/PhysRevLett.119.030502. https://doi.org/10.1103/PhysRevLett.119.030502 [14] A. L. Grimsmo, J. Combes, and B. Q. Baragiola, Phys. Rev. X, 10, 011058 (2020), 10.1103/PhysRevX.10.011058. https://doi.org/10.1103/PhysRevX.10.011058 [15] R. Dahan et al., Phys. Rev. X, 13, 031001 (2023), 10.1103/PhysRevX.13.031001. https://doi.org/10.1103/PhysRevX.13.031001 [16] G. Baranes et al., Phys. Rev. Research, 5, 043271 (2023), 10.1103/PhysRevResearch.5.043271. https://doi.org/10.1103/PhysRevResearch.5.043271 [17] J. Zak, Phys. Rev. Lett., 19, 1385–1387 (1967), 10.1103/PhysRevLett.19.1385. https://doi.org/10.1103/PhysRevLett.19.1385 [18] B. Royer, S. Singh, and S. M. Girvin, Phys. Rev. Lett., 125, 260509 (2020), 10.1103/PhysRevLett.125.260509. https://doi.org/10.1103/PhysRevLett.125.260509 [19] B. M. Terhal and D. Weigand, Phys. Rev. A, 93, 012315 (2016), 10.1103/PhysRevA.93.012315. https://doi.org/10.1103/PhysRevA.93.012315 [20] I. I. Rabi, Phys. Rev., 49, 324–328 (1936), 10.1103/PhysRev.49.324. https://doi.org/10.1103/PhysRev.49.324 [21] A. Frisk Kockum et al., Nat. Rev. Phys., 1, 1 (2019), 10.1038/s42254-018-0006-2. https://doi.org/10.1038/s42254-018-0006-2 [22] J. Hastrup et al., npj Quantum Inf., 7, 1 (2021), 10.1038/s41534-020-00353-3. https://doi.org/10.1038/s41534-020-00353-3 [23] B. W. Shore and J. H. Eberly, Opt. Commun., 24, 83–88 (1978), 10.1016/0030-4018(78)90272-9. https://doi.org/10.1016/0030-4018(78)90272-9 [24] V. V. Albert, J. P. Covey, and J. Preskill, Phys. Rev. X, 10, 031050 (2020), 10.1103/PhysRevX.10.031050. https://doi.org/10.1103/PhysRevX.10.031050 [25] R. H. Dicke, Phys. Rev., 93, 99–110 (1954), 10.1103/PhysRev.93.99. https://doi.org/10.1103/PhysRev.93.99 [26] M. Tavis and F. W. Cummings, Phys. Rev., 170, 379–384 (1968), 10.1103/PhysRev.170.379. https://doi.org/10.1103/PhysRev.170.379 [27] Y. Ouyang, Linear Algebra Appl., 532, 43–59 (2017), 10.1016/j.laa.2017.06.031. https://doi.org/10.1016/j.laa.2017.06.031 [28] O. Tziperman et al., ACS Nano (2025), 10.1021/acsnano.4c15257. https://doi.org/10.1021/acsnano.4c15257 [29] N. Gutman et al., Phys. Rev. Lett. 132, 153601 (2025), 10.1103/PhysRevLett.132.153601. https://doi.org/10.1103/PhysRevLett.132.153601 [30] D. M. Lukin et al., Phys. Rev. X, 13, 011005 (2023), 10.1103/PhysRevX.13.011005. https://doi.org/10.1103/PhysRevX.13.011005 [31] A. Tiranov et al., Science, 379, 389–393 (2023), 10.1126/science.ade9324. https://doi.org/10.1126/science.ade9324 [32] G. Ferioli et al., Phys. Rev. Lett. 132, 133601 (2025), 10.1103/PhysRevLett.132.133601. https://doi.org/10.1103/PhysRevLett.132.133601 [33] C. Liedl et al., Phys. Rev. X, 14, 011020 (2024), 10.1103/PhysRevX.14.011020. https://doi.org/10.1103/PhysRevX.14.011020 [34] S. M. Barnett and D. T. Pegg, J. Phys. A: Math. Gen., 19, 3849 (1986), 10.1088/0305-4470/19/18/030. https://doi.org/10.1088/0305-4470/19/18/030 [35] A. A. Diringer et al., Phys. Rev. X, 14, 011055 (2024), 10.1103/PhysRevX.14.011055. https://doi.org/10.1103/PhysRevX.14.011055 [36] M. J. Peterer et al., Phys. Rev. Lett., 114, 010501 (2015), 10.1103/PhysRevLett.114.010501. https://doi.org/10.1103/PhysRevLett.114.010501 [37] B. Barwick, D. J. Flannigan, and A. H. Zewail, Nature, 462, 1859–1863 (2009), 10.1038/nature08662. https://doi.org/10.1038/nature08662 [38] S. T. Park, M. Lin, and A. H. Zewail, New J. Phys., 12, 123028 (2010), 10.1088/1367-2630/12/12/123028. https://doi.org/10.1088/1367-2630/12/12/123028 [39] F. J. García de Abajo, A. Asenjo-Garcia, and M. Kociak, Nano Lett., 10, 1859–1863 (2010), 10.1021/nl100613s. https://doi.org/10.1021/nl100613s [40] O. Reinhardt et al., Ann. Phys., 533, 2000254 (2021), 10.1002/andp.202000254. https://doi.org/10.1002/andp.202000254 [41] R. Dahan et al., Nat. Phys., 16, 11 (2020), 10.1038/s41567-020-01042-w. https://doi.org/10.1038/s41567-020-01042-w [42] R. Dahan et al., Science, 373, eabj7128 (2021), 10.1126/science.abj7128. https://doi.org/10.1126/science.abj7128 [43] J.-W. Henke et al., Nature, 600, 7890 (2021), 10.1038/s41586-021-04197-5. https://doi.org/10.1038/s41586-021-04197-5 [44] A. Feist et al., Science, 377, 777–780 (2022), 10.1126/science.abo5037. https://doi.org/10.1126/science.abo5037 [45] Y. Adiv et al., Phys. Rev. X, 13, 011002 (2023), 10.1103/PhysRevX.13.011002. https://doi.org/10.1103/PhysRevX.13.011002Cited byCould not fetch Crossref cited-by data during last attempt 2025-12-10 11:31:17: Could not fetch cited-by data for 10.22331/q-2025-12-10-1934 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-12-10 11:31:17: 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.