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  • Open Access

Variational Processing of Multimode Squeezed Light

Aviv Karnieli1,2,*,†, Paul-Alexis Mor1,*,‡, Charles Roques-Carmes1,*,§, Eran Lustig1,2, Jamison Sloan1, Jelena Vučković1, David A.B. Miller1, and Shanhui Fan1

  • *These authors contributed equally to this work.
  • †Contact author: karnieli@technion.ac.il
  • ‡Contact author: paul-alexis.mor@mines.org
  • §Contact author: chrc@stanford.edu
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PRX Quantum 7, 020311 – Published 20 April, 2026

DOI: https://doi.org/10.1103/mcmq-qf4p

Abstract

Integrated multimode quantum optics is a promising platform for scalable continuous-variable quantum technologies leveraging multimode squeezing in both the spatial and spectral domains. However, on-chip measurement, routing and processing the relevant “supermodes” over which the squeezing resource is distributed still scales quadratically with the number of modes N, causing a rapid increase in photonic circuit size and number of required measurements. Here, we introduce a variational scheme, relying on self-configuring photonic networks (SCNs) that learn and extract the most-squeezed supermodes sequentially, reducing both the circuit size and the experimental overhead. Using homodyne measurement as a cost function, a sparse SCN discovers the l≪N most significant supermodes using O(lN) physical elements and optimization steps. We analyze and numerically simulate these architectures for both real-space and frequency-domain implementations, showing a fidelity close to unity between the learned circuit and the supermode decomposition, even in the presence of optical losses and detection noise. In the frequency domain, we show that circuit size can be further reduced by using inverse-designed surrogate networks, which emulate the layers learned thus far. Using two different frequency encoding schemes—uniformly and nonuniformly spaced frequency bins—we reduce an entire network (learning all N supermodes) to O(N) and even O(1) modulated cavities. Our results point toward chip-scale, resource-efficient quantum processing units and demultiplexers for continuous variable processing in multimode quantum optics, with applications ranging from quantum communication, metrology, and computation.

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Popular Summary

Squeezed light—light with reduced quantum noise—is a key resource for quantum technologies such as sensing, communication, and computation. Modern photonic platforms can generate squeezed light distributed across many spatial or spectral modes simultaneously. However, the useful quantum correlations are typically encoded in special superpositions of these modes, called supermodes, which are not directly accessible in experiments. Identifying and extracting these supermodes becomes increasingly difficult as the number of modes grows. In this work, we introduce a method for processing multimode squeezed light using variational photonic circuits. Our approach employs a self-configuring optical network that learns the structure of the quantum state directly from measurement feedback. By optimizing homodyne measurements, the network sequentially identifies the most strongly squeezed supermodes and routes them to specific outputs. This strategy significantly reduces the hardware complexity required to analyze multimode squeezed states and provides a scalable route toward integrated processors for high-dimensional quantum light with applications in sensing, communication, and continuous-variable quantum computing.

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References (94)

  1. C. Fabre and N. Treps, Modes and states in quantum optics, Rev. Mod. Phys. 92, 035005 (2020).
  2. O. Pfister, Continuous-variable quantum computing in the quantum optical frequency comb, J. Phys. B: At., Mol. Opt. Phys. 53, 012001 (2020).
  3. S. Armstrong, J.-F. Morizur, J. Janousek, B. Hage, N. Treps, P. K. Lam, and H.-A. Bachor, Programmable multimode quantum networks, Nat. Commun. 3, 1026 (2012).
  4. J. Roslund, R. M. de Araújo, S. Jiang, C. Fabre, and N. Treps, Wavelength-multiplexed quantum networks with ultrafast frequency combs, Nat. Photonics 8, 109 (2014).
  5. Y. Cai, J. Roslund, G. Ferrini, F. Arzani, X. Xu, C. Fabre, and N. Treps, Multimode entanglement in reconfigurable graph states using optical frequency combs, Nat. Commun. 8, 15645 (2017).
  6. S. Gerke, J. Sperling, W. Vogel, Y. Cai, J. Roslund, N. Treps, and C. Fabre, Full multipartite entanglement of frequency-comb Gaussian states, Phys. Rev. Lett. 114, 050501 (2015).
  7. C. S. Embrey, M. T. Turnbull, P. G. Petrov, and V. Boyer, Observation of localized multi-spatial-mode quadrature squeezing, Phys. Rev. X 5, 031004 (2015).
  8. P. R. Sharapova, O. V. Tikhonova, S. Lemieux, R. W. Boyd, and M. V. Chekhova, Bright squeezed vacuum in a nonlinear interferometer: Frequency and temporal schmidt-mode description, Phys. Rev. A 97, 053827 (2018).
  9. P. R. Sharapova, G. Frascella, M. Riabinin, A. M. Pérez, O. V. Tikhonova, S. Lemieux, R. W. Boyd, G. Leuchs, and M. V. Chekhova, Properties of bright squeezed vacuum at increasing brightness, Phys. Rev. Res. 2, 013371 (2020).
  10. T. Kouadou, F. Sansavini, M. Ansquer, J. Henaff, N. Treps, and V. Parigi, Spectrally shaped and pulse-by-pulse multiplexed multimode squeezed states of light, APL Photonics 8, 086113 (2023).
  11. B. Dioum, V. D’Auria, A. Zavatta, O. Pfister, and G. Patera, Universal quantum frequency comb measurements by spectral mode-matching, Opt. Quantum 2, 413 (2024).
  12. J. M. Arrazola et al., Quantum circuits with many photons on a programmable nanophotonic chip, Nature 591, 54 (2021).
  13. L. S. Madsen, F. Laudenbach, M. F. Askarani, F. Rortais, T. Vincent, J. F. F. Bulmer, F. M. Miatto, L. Neuhaus, L. G. Helt, M. J. Collins, A. E. Lita, T. Gerrits, S. W. Nam, V. D. Vaidya, M. Menotti, I. Dhand, Z. Vernon, N. Quesada, and J. Lavoie, Quantum computational advantage with a programmable photonic processor, Nature 606, 75 (2022).
  14. R. Li, B. An, N. Jiao, J. Liu, L. Chen, Y. Wang, and Y. Zheng, Bright squeezed light in the kilohertz frequency band, Light: Sci. Appl. 14, 310 (2025).
  15. Q. Zhuang, Z. Zhang, and J. H. Shapiro, Distributed quantum sensing using continuous-variable multipartite entanglement, Phys. Rev. A 97, 032329 (2018).
  16. F. Lenzini, J. Janousek, O. Thearle, M. Villa, B. Haylock, S. Kasture, L. Cui, H.-P. Phan, D. V. Dao, H. Yonezawa, P. K. Lam, E. H. Huntington, and M. Lobino, Integrated photonic platform for quantum information with continuous variables, Sci. Adv. 4, eaat9331 (2018).
  17. X. Guo, C. R. Breum, J. Borregaard, S. Izumi, M. V. Larsen, T. Gehring, M. Christandl, J. S. Neergaard-Nielsen, and U. L. Andersen, Distributed quantum sensing in a continuous-variable entangled network, Nat. Phys. 16, 281 (2020).
  18. G. Ferrini, J. P. Gazeau, T. Coudreau, C. Fabre, and N. Treps, Compact Gaussian quantum computation by multi-pixel homodyne detection, New J. Phys. 15, 093015 (2013).
  19. F. Centrone, F. Grosshans, and V. Parigi, Cost and routing of continuous-variable quantum networks, Phys. Rev. A 108, 042615 (2023).
  20. A. Gorlach, M. E. Tzur, M. Birk, M. Krüger, N. Rivera, O. Cohen, and I. Kaminer, High-harmonic generation driven by quantum light, Nat. Phys. 19, 1689 (2023).
  21. J. Heimerl, A. Mikhaylov, S. Meier, H. Höllerer, I. Kaminer, M. Chekhova, and P. Hommelhoff, Multiphoton electron emission with non-classical light, Nat. Phys. 20, 945 (2024).
  22. L. G. Wright, F. O. Wu, D. N. Christodoulides, and F. W. Wise, Physics of highly multimode nonlinear optical systems, Nat. Phys. 18, 1018 (2022).
  23. N. Rivera, S. Z. Uddin, J. Sloan, and M. Soljačić, Ultra-broadband and passive stabilization of ultrafast light sources by quantum light injection, Nanophotonics 14, 1857 (2025).
  24. S. Zia Uddin, N. Rivera, D. Seyler, J. Sloan, Y. Salamin, C. Roques-Carmes, S. Xu, M. Y. Sander, I. Kaminer, and M. Soljačić, Noise-immune quantum correlations of intense light, Nat. Photonics 19, 751 (2025).
  25. A. Dutt, K. Luke, S. Manipatruni, A. L. Gaeta, P. Nussenzveig, and M. Lipson, On-chip optical squeezing, Phys. Rev. Appl. 3, 044005 (2015).
  26. Y. Zhao, Y. Okawachi, J. K. Jang, X. Ji, M. Lipson, and A. L. Gaeta, Near-degenerate quadrature-squeezed vacuum generation on a silicon-nitride chip, Phys. Rev. Lett. 124, 193601 (2020).
  27. Z. Yang, M. Jahanbozorgi, D. Jeong, S. Sun, O. Pfister, H. Lee, and X. Yi, A squeezed quantum microcomb on a chip, Nat. Commun. 12, 4781 (2021).
  28. J. F. Tasker, J. Frazer, G. Ferranti, E. J. Allen, L. F. Brunel, S. Tanzilli, V. D’Auria, and J. C. F. Matthews, Silicon photonics interfaced with integrated electronics for 9 GHz measurement of squeezed light, Nat. Photonics 15, 11 (2021).
  29. Y. Zhang, M. Menotti, K. Tan, V. D. Vaidya, D. H. Mahler, L. G. Helt, L. Zatti, M. Liscidini, B. Morrison, and Z. Vernon, Squeezed light from a nanophotonic molecule, Nat. Commun. 12, 2233 (2021).
  30. R. Nehra, R. Sekine, L. Ledezma, Q. Guo, R. M. Gray, A. Roy, and A. Marandi, Few-cycle vacuum squeezing in nanophotonics, Science 377, 1333 (2022).
  31. M. A. Guidry, D. M. Lukin, K. Y. Yang, and J. Vučković, Multimode squeezing in soliton crystal microcombs, Optica 10, 694 (2023).
  32. E. Lustig, M. A. Guidry, D. M. Lukin, S. Fan, and J. Vuckovic, Quadrature-dependent lattice dynamics of dissipative microcombs, Nat. Photonics 19, 1247 (2025).
  33. Y. Shen, P.-Y. Hsieh, D. Srinivasan, A. Henry, G. Moille, S. K. Sridhar, A. Restelli, Y.-C. Chang, K. Srinivasan, T. A. Smith, and A. Dutt, Highly squeezed nanophotonic quantum microcombs with broadband frequency tunability, arXiv:2505.03734.
  34. M. Sennary, J. Rivera-Dean, M. ElKabbash, V. Pervak, M. Lewenstein, and M. T. Hassan, Attosecond quantum uncertainty dynamics and ultrafast squeezed light for quantum communication, Light: Sci. Appl. 14, 350 (2025).
  35. V. Cimini, M. M. Sohoni, F. Presutti, B. K. Malia, S.-Y. Ma, R. Yanagimoto, T. Wang, T. Onodera, L. G. Wright, and P. L. McMahon, Large-scale quantum reservoir computing using a Gaussian Boson Sampler, arXiv:2505.13695.
  36. Z. Wang, K. Li, Y. Wang, X. Zhou, Y. Cheng, B. Jing, F. Sun, J. Li, Z. Li, B. Wu, Q. Gong, Q. He, B.-B. Li, and Q.-F. Yang, Large-scale cluster quantum microcombs, Light: Sci. Appl. 14, 164 (2025).
  37. Y. Hu, M. Yu, D. Zhu, N. Sinclair, A. Shams-Ansari, L. Shao, J. Holzgrafe, E. Puma, M. Zhang, and M. Lončar, On-chip electro-optic frequency shifters and beam splitters, Nature 599, 587 (2021).
  38. See Supplemental Material at http://link.aps.org/supplemental/10.1103/mcmq-qf4p for background on the Bloch-Messiah and Williamson decomposition, sequential optimization algorithms for automatic Bloch-Messiah decomposition using homodyne detection and self-configuring optics, and model for optimization with non-uniform frequency bins, which also includes Refs. [91, 92, 93, 94].
  39. M. V. Larsen, X. Guo, C. R. Breum, J. S. Neergaard-Nielsen, and U. L. Andersen, Deterministic generation of a two-dimensional cluster state, Science 366, 369 (2019).
  40. C. Weedbrook, S. Pirandola, R. GarciaPatron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum information, Rev. Mod. Phys. 84, 621 (2012).
  41. Y. Zhou, J. Yu, Z. Yan, X. Jia, J. Zhang, C. Xie, and K. Peng, Quantum secret sharing among four players using multipartite bound entanglement of an optical field, Phys. Rev. Lett. 121, 150502 (2018).
  42. Y. Qin, J. Cheng, J. Ma, D. Zhao, Z. Yan, X. Jia, C. Xie, and K. Peng, Efficient and secure quantum secret sharing for eight users, Phys. Rev. Res. 6, 033036 (2024).
  43. L. L. Volpe, S. De, T. Kouadou, D. Horoshko, M. I. Kolobov, C. Fabre, V. Parigi, and N. Treps, Multimode single-pass spatio-temporal squeezing, Opt. Express 28, 12385 (2020).
  44. Y. Shaked, Y. Michael, R. Z. Vered, L. Bello, M. Rosenbluh, and A. Pe’er, Lifting the bandwidth limit of optical homodyne measurement with broadband parametric amplification, Nat. Commun. 9, 609 (2018).
  45. M. Amooei, G. Kulkarni, J. Upham, and R. W. Boyd, Efficient characterization of spatial Schmidt modes of multiphoton entangled states produced from high-gain parametric down-conversion, Phys. Rev. A 111, 023714 (2025).
  46. F. Presutti, L. G. Wright, S.-Y. Ma, T. Wang, B. K. Malia, T. Onodera, and P. L. McMahon, Highly multimode visible squeezed light with programmable spectral correlations through broadband up-conversion, arXiv:2401.06119.
  47. D. Barral, M. Walschaers, K. Bencheikh, V. Parigi, J. A. Levenson, N. Treps, and N. Belabas, Versatile photonic entanglement synthesizer in the spatial domain, Phys. Rev. Appl. 14, 044025 (2020).
  48. H. H. Zhu, H. Sen Chen, T. Chen, Y. Li, S. B. Luo, M. F. Karim, X. S. Luo, F. Gao, Q. Li, H. Cai, L. K. Chin, L. C. Kwek, B. Nordén, X. D. Zhang, and A. Q. Liu, Large-scale photonic network with squeezed vacuum states for molecular vibronic spectroscopy, Nat. Commun. 15, 6057 (2024).
  49. H. Aghaee Rad et al., Scaling and networking a modular photonic quantum computer, Nature 638, 912 (2025).
  50. N. C. Harris, J. Carolan, D. Bunandar, M. Prabhu, M. Hochberg, T. Baehr-Jones, M. L. Fanto, A. M. Smith, C. C. Tison, P. M. Alsing, and D. Englund, Linear programmable nanophotonic processors, Optica 5, 1623 (2018).
  51. J. Wang, F. Sciarrino, A. Laing, and M. G. Thompson, Integrated photonic quantum technologies, Nat. Photonics 14, 273 (2020).
  52. W. Bogaerts, D. Pérez, J. Capmany, D. A. B. Miller, J. Poon, D. Englund, F. Morichetti, and A. Melloni, Programmable photonic circuits, Nature 586, 207 (2020).
  53. J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, Multidimensional quantum entanglement with large-scale integrated optics, Science 360, 285 (2018).
  54. J. Wang, F. Sciarrino, A. Laing, and M. G. Thompson, Integrated photonic quantum technologies, Nat. Photonics 14, 273 (2020).
  55. H.-H. Lu, M. Liscidini, A. L. Gaeta, A. M. Weiner, and J. M. Lukens, Frequency-bin photonic quantum information, Optica 10, 1655 (2023).
  56. D. A. B. Miller, Self-configuring universal linear optical component [Invited], Photonics Res. 1, 1 (2013).
  57. D. A. B. Miller, Analyzing and generating multimode optical fields using self-configuring networks, Optica 7, 794 (2020).
  58. S. SeyedinNavadeh, M. Milanizadeh, F. Zanetto, G. Ferrari, M. Sampietro, M. Sorel, D. A. B. Miller, A. Melloni, and F. Morichetti, Determining the optimal communication channels of arbitrary optical systems using integrated photonic processors, Nat. Photonics 18, 149 (2024).
  59. C. Roques-Carmes, S. Fan, and D. A. B. Miller, Measuring, processing, and generating partially coherent light with self-configuring optics, Light: Sci. Appl. 13, 260 (2024).
  60. C. Roques-Carmes, A. Karnieli, D. A. B. Miller, and S. Fan, Automated modal analysis of entanglement with bipartite self-configuring optics, ACS Photonics 12, 3285 (2025).
  61. R. Hamerly, S. Bandyopadhyay, and D. Englund, Stability of self-configuring large multiport interferometers, Phys. Rev. Appl. 18, 024018 (2022).
  62. R. Hamerly, S. Bandyopadhyay, and D. Englund, Accurate self-configuration of rectangular multiport interferometers, Phys. Rev. Appl. 18, 024019 (2022).
  63. S. Bandyopadhyay, R. Hamerly, and D. Englund, Hardware error correction for programmable photonics, Optica 8, 1247 (2021).
  64. L. Yuan, Q. Lin, M. Xiao, and S. Fan, Synthetic dimension in photonics, Optica 5, 1396 (2018).
  65. S. L. Braunstein, Squeezing as an irreducible resource, Phys. Rev. A 71, 055801 (2005).
  66. Mor et al., Separating partially coherent light, arXiv:2603.15517
  67. E. Gouzien, S. Tanzilli, V. D’Auria, and G. Patera, Morphing supermodes: A full characterization for enabling multimode quantum optics, Phys. Rev. Lett. 125, 103601 (2020).
  68. A. I. Lvovsky and M. G. Raymer, Continuous-variable optical quantum-state tomography, Rev. Mod. Phys. 81, 299 (2009).
  69. D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, arXiv:1412.6980.
  70. W. M. J. Green, R. K. Lee, G. A. DeRose, A. Scherer, and A. Yariv, Hybrid InGaAsP-InP Mach-Zehnder racetrack resonator for thermooptic switching and coupling control, Opt. Express 13, 1651 (2005).
  71. R. Yang, L. Zhou, H. Zhu, and J. Chen, Low-voltage high-speed coupling modulation in silicon racetrack ring resonators, Opt. Express 23, 28993 (2015).
  72. Y.-X. Yin, X.-P. Zhang, X.-J. Yin, Y. Li, X.-R. Xu, J.-M. An, Y.-D. Wu, X.-P. Liu, and D.-M. Zhang, High-Q-factor tunable silica-based microring resonators, Photonics 8, 256 (2021).
  73. S. Buddhiraju, A. Dutt, M. Minkov, I. A. D. Williamson, and S. Fan, Arbitrary linear transformations for photons in the frequency synthetic dimension, Nat. Commun. 12, 2401 (2021).
  74. G. Breitenbach, S. Schiller, and J. Mlynek, Measurement of the quantum states of squeezed light, Nature 387, 471 (1997).
  75. M. Mehmet, S. Ast, T. Eberle, S. Steinlechner, H. Vahlbruch, and R. Schnabel, Squeezed light at 1550 nm with a quantum noise reduction of 12.3 dB, Opt. Express 19, 25763 (2011).
  76. H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, Detection of 15 dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency, Phys. Rev. Lett. 117, 110801 (2016).
  77. M. A. Nielsen, A simple formula for the average gate fidelity of a quantum dynamical operation, Phys. Lett. A 303, 249 (2002).
  78. C. Han, Z. Zheng, H. Shu, M. Jin, J. Qin, R. Chen, Y. Tao, B. Shen, B. Bai, F. Yang, Y. Wang, H. Wang, F. Wang, Z. Zhang, S. Yu, C. Peng, and X. Wang, Slow-light silicon modulator with 110-GHz bandwidth, Sci. Adv. 9, eadi5339 (2023).
  79. P. O. Weigel, J. Zhao, K. Fang, H. Al-Rubaye, D. Trotter, D. Hood, J. Mudrick, C. Dallo, A. T. Pomerene, A. L. Starbuck, C. T. DeRose, A. L. Lentine, G. Rebeiz, and S. Mookherjea, Bonded thin film lithium niobate modulator on a silicon photonics platform exceeding 100 GHz 3-dB electrical modulation bandwidth, Opt. Express 26, 23728 (2018).
  80. D. Goldring, U. Levy, and D. Mendlovic, Highly dispersive micro-ring resonator based on one dimensional photonic crystal waveguide design and analysis, Opt. Express 15, 3156 (2007).
  81. S. M. Lo, S. Hu, G. Gaur, Y. Kostoulas, S. M. Weiss, and P. M. Fauchet, Photonic crystal microring resonator for label-free biosensing, Opt. Express 25, 7046 (2017).
  82. K. Y. Yang, K. Beha, D. C. Cole, X. Yi, P. Del’Haye, H. Lee, J. Li, D. Y. Oh, S. A. Diddams, S. B. Papp, and K. J. Vahala, Broadband dispersion-engineered microresonator on a chip, Nat. Photonics 10, 316 (2016).
  83. W. D. Sacher, W. M. J. Green, S. Assefa, T. Barwicz, H. Pan, S. M. Shank, Y. A. Vlasov, and J. K. S. Poon, Coupling modulation of microrings at rates beyond the linewidth limit, Opt. Express 21, 9722 (2013).
  84. W. Jin, Q.-F. Yang, L. Chang, B. Shen, H. Wang, M. A. Leal, L. Wu, M. Gao, A. Feshali, M. Paniccia, K. J. Vahala, and J. E. Bowers, Hertz-linewidth semiconductor lasers using CMOS-ready ultra-high-Q microresonators, Nat. Photonics 15, 346 (2021).
  85. E. Abbasgholinejad, D. Malz, A. Asenjo-Garcia, and R. Trivedi, Theory of quantum-enhanced interferometry with general Markovian light sources, arXiv:2504.05111.
  86. G. Masada, K. Miyata, A. Politi, T. Hashimoto, J. L. O’Brien, and A. Furusawa, Continuous-variable entanglement on a chip, Nat. Photonics 9, 316 (2015).
  87. S. Shi, L. Tian, Y. Wang, Y. Zheng, C. Xie, and K. Peng, Demonstration of channel multiplexing quantum communication exploiting entangled sideband modes, Phys. Rev. Lett. 125, 070502 (2020).
  88. M. Erhard, M. Krenn, and A. Zeilinger, Advances in high-dimensional quantum entanglement, Nat. Rev. Phys. 2, 365 (2020).
  89. M. Kues, C. Reimer, J. M. Lukens, W. J. Munro, A. M. Weiner, D. J. Moss, and R. Morandotti, Quantum optical microcombs, Nat. Photonics 13, 170 (2019).
  90. S. W. Golomb, in Graph Theory and Computing, edited by R. C. Read (Academic Press, New York, 1972), p. 23.
  91. M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, Experimental realization of any discrete unitary operator, Phys. Rev. Lett. 73, 58 (1994).
  92. W. G. Cochran, The distribution of quadratic forms in a normal system, with applications to the analysis of covariance, Math. Proc. Cambridge Philos. Soc. 30, 178 (1934).
  93. C. Jin, R. Ge, P. Netrapalli, S. M. Kakade, and M. I. Jordan, How to escape saddle points efficiently, arXiv:1703.00887.
  94. S. Blanes, F. Casas, J. A. Oteo, and J. Ros, The Magnus expansion and some of its applications, Phys. Rep. 470, 151 (2009).

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PRX Quantum 7, 020311– Published 20 April, 2026
  • Received 20 September 2025
  • Revised 14 January 2026
  • Accepted 3 March 2026
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