{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T02:18:22Z","timestamp":1781576302863,"version":"3.54.5"},"reference-count":115,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,10,6]],"date-time":"2022-10-06T00:00:00Z","timestamp":1665014400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NSF","award":["PHY-1733907"],"award-info":[{"award-number":["PHY-1733907"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"We study bosonic quantum computations using the Segal-Bargmann representation of quantum states. We argue that this holomorphic representation is a natural one which not only gives a canonical description of bosonic quantum computing using basic elements of complex analysis but also provides a unifying picture which delineates the boundary between discrete- and continuous-variable quantum information theory. Using this representation, we show that the evolution of a single bosonic mode under a Gaussian Hamiltonian can be described as an integrable dynamical system of classical Calogero-Moser particles corresponding to the zeros of the holomorphic function, together with a conformal evolution of Gaussian parameters. We explain that the Calogero-Moser dynamics is due to unique features of bosonic Hilbert spaces such as squeezing. We then generalize the properties of this holomorphic representation to the multimode case, deriving a non-Gaussian hierarchy of quantum states and relating entanglement to factorization properties of holomorphic functions. Finally, we apply this formalism to discrete- and continuous- variable quantum measurements and obtain a classification of subuniversal models that are generalizations of Boson Sampling and Gaussian quantum computing.<\/jats:p>","DOI":"10.22331\/q-2022-10-06-831","type":"journal-article","created":{"date-parts":[[2022,10,6]],"date-time":"2022-10-06T15:06:12Z","timestamp":1665068772000},"page":"831","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":23,"title":["Holomorphic representation of quantum computations"],"prefix":"10.22331","volume":"6","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0135-9819","authenticated-orcid":false,"given":"Ulysse","family":"Chabaud","sequence":"first","affiliation":[{"name":"Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1323-360X","authenticated-orcid":false,"given":"Saeed","family":"Mehraban","sequence":"additional","affiliation":[{"name":"Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA"},{"name":"Computer Science, Tufts University, Medford, MA 02155, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2022,10,6]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"P. W. Shor, ``Algorithms for quantum computation: Discrete logarithms and factoring,'' in Proceedings 35th annual symposium on foundations of computer science, pp. 124\u2013134, IEEE. 1994.","DOI":"10.1109\/SFCS.1994.365700"},{"key":"1","doi-asserted-by":"publisher","unstructured":"A. W. Harrow and A. Montanaro, ``Quantum computational supremacy,'' Nature 549, 203\u2013209 (2017).","DOI":"10.1038\/nature23458"},{"key":"2","doi-asserted-by":"publisher","unstructured":"S. Wiesner, ``Conjugate coding,'' ACM Sigact News 15, 78\u201388 (1983).","DOI":"10.1145\/1008908.1008920"},{"key":"3","doi-asserted-by":"publisher","unstructured":"C. H. Bennett, G. Brassard, C. Cr\u00e9peau, R. Jozsa, A. Peres, and W. K. Wootters, ``Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,'' Physical Review Letters 70, 1895 (1993).","DOI":"10.1103\/PhysRevLett.70.1895"},{"key":"4","doi-asserted-by":"publisher","unstructured":"V. Giovannetti, S. Lloyd, and L. Maccone, ``Quantum-enhanced measurements: beating the standard quantum limit,'' Science 306, 1330\u20131336 (2004).","DOI":"10.1126\/science.1104149"},{"key":"5","doi-asserted-by":"publisher","unstructured":"M. A. Nielsen and I. L. Chuang, ``Quantum Computation and Quantum Information: 10th Anniversary Edition,''. Cambridge University Press, New York, NY, USA, 10th ed., 2011.","DOI":"10.1119\/1.1463744"},{"key":"6","doi-asserted-by":"publisher","unstructured":"S. L. Braunstein and P. van Loock, ``Quantum information with continuous variables,'' Rev. Mod. Phys. 77, 513\u2013577 (2005).","DOI":"10.1103\/RevModPhys.77.513"},{"key":"7","doi-asserted-by":"publisher","unstructured":"G. Adesso, S. Ragy, and A. R. Lee, ``Continuous variable quantum information: Gaussian states and beyond,'' Open Systems & Information Dynamics 21, 1440001 (2014).","DOI":"10.1142\/S1230161214400010"},{"key":"8","doi-asserted-by":"publisher","unstructured":"D. Gottesman, A. Kitaev, and J. Preskill, ``Encoding a qubit in an oscillator,'' Physical Review A 64, 012310 (2001).","DOI":"10.1103\/PhysRevA.64.012310"},{"key":"9","doi-asserted-by":"publisher","unstructured":"W. Cai, Y. Ma, W. Wang, C.-L. Zou, and L. Sun, ``Bosonic quantum error correction codes in superconducting quantum circuits,'' Fundamental Research 50\u201367 (2021).","DOI":"10.1016\/j.fmre.2020.12.006"},{"key":"10","doi-asserted-by":"publisher","unstructured":"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,'' Physical Review Letters 117, 110801 (2016).","DOI":"10.1103\/PhysRevLett.117.110801"},{"key":"11","doi-asserted-by":"publisher","unstructured":"S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J.-i. Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, ``Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,'' Nature Photonics 7, 982 (2013).","DOI":"10.1038\/nphoton.2013.287"},{"key":"12","doi-asserted-by":"publisher","unstructured":"S. Aaronson and A. Arkhipov, ``The computational Complexity of Linear Optics,'' Theory of Computing 9, 143 (2013).","DOI":"10.1145\/1993636.1993682"},{"key":"13","doi-asserted-by":"publisher","unstructured":"A. P. Lund, A. Laing, S. Rahimi-Keshari, T. Rudolph, J. L. O'Brien, and T. C. Ralph, ``Boson Sampling from a Gaussian State,'' Physical Review Letters 113, 100502 (2014).","DOI":"10.1103\/PhysRevLett.113.100502"},{"key":"14","doi-asserted-by":"publisher","unstructured":"C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, ``Gaussian boson sampling,'' Physical Review Letters 119, 170501 (2017).","DOI":"10.1103\/PhysRevLett.119.170501"},{"key":"15","doi-asserted-by":"publisher","unstructured":"T. Douce, D. Markham, E. Kashefi, E. Diamanti, T. Coudreau, P. Milman, P. van Loock, and G. Ferrini, ``Continuous-Variable Instantaneous Quantum Computing is hard to sample,'' Physical Review Letters 118, 070503 (2017).","DOI":"10.1103\/PhysRevLett.118.070503"},{"key":"16","doi-asserted-by":"publisher","unstructured":"A. P. Lund, S. Rahimi-Keshari, and T. C. Ralph, ``Exact Boson Sampling using Gaussian continuous variable measurements,'' Physical Review A 96, 022301 (2017).","DOI":"10.1103\/PhysRevA.96.022301"},{"key":"17","doi-asserted-by":"publisher","unstructured":"L. Chakhmakhchyan and N. J. Cerf, ``Boson sampling with Gaussian measurements,'' Physical Review A 96, 032326 (2017).","DOI":"10.1103\/PhysRevA.96.032326"},{"key":"18","doi-asserted-by":"publisher","unstructured":"U. Chabaud, T. Douce, D. Markham, P. Van Loock, E. Kashefi, and G. Ferrini, ``Continuous-variable sampling from photon-added or photon-subtracted squeezed states,'' Physical Review A 96, 062307 (2017).","DOI":"10.1103\/PhysRevA.96.062307"},{"key":"19","doi-asserted-by":"publisher","unstructured":"H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin, D. Wu, X. Ding, Y. Hu, et al., ``Quantum computational advantage using photons,'' Science 370, 1460\u20131463 (2020).","DOI":"10.1126\/science.abe8770"},{"key":"20","doi-asserted-by":"publisher","unstructured":"F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A. Buell, et al., ``Quantum supremacy using a programmable superconducting processor,'' Nature 574, 505\u2013510 (2019).","DOI":"10.1038\/s41586-019-1666-5"},{"key":"21","doi-asserted-by":"publisher","unstructured":"J. K. Horner and J. F. Symons, ``What Have Google\u2019s Random Quantum Circuit Simulation Experiments Demonstrated about Quantum Supremacy?,'' in Advances in Software Engineering, Education, and e-Learning, pp. 411\u2013419. Springer, 2021.","DOI":"10.1007\/978-3-030-70873-3_29"},{"key":"22","unstructured":"F. Pan and P. Zhang, ``Simulating the Sycamore quantum supremacy circuits,'' arXiv:2103.03074."},{"key":"23","unstructured":"B. Villalonga, M. Y. Niu, L. Li, H. Neven, J. C. Platt, V. N. Smelyanskiy, and S. Boixo, ``Efficient approximation of experimental Gaussian boson sampling,'' arXiv:2109.11525."},{"key":"24","doi-asserted-by":"publisher","unstructured":"Y. Tong, V. V. Albert, J. R. McClean, J. Preskill, and Y. Su, ``Provably accurate simulation of gauge theories and bosonic systems,'' arXiv:2110.06942.","DOI":"10.22331\/q-2022-09-22-816"},{"key":"25","doi-asserted-by":"publisher","unstructured":"S. Lloyd and S. L. Braunstein, ``Quantum computation over continuous variables,'' in Quantum Information with Continuous Variables, pp. 9\u201317. Springer, 1999.","DOI":"10.1007\/978-94-015-1258-9_2"},{"key":"26","doi-asserted-by":"publisher","unstructured":"S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, ``Efficient Classical Simulation of Continuous Variable Quantum Information Processes,'' Physical Review Letters 88, 097904 (2002).","DOI":"10.1103\/PhysRevLett.88.097904"},{"key":"27","doi-asserted-by":"publisher","unstructured":"R. Takagi and Q. Zhuang, ``Convex resource theory of non-Gaussianity,'' Physical Review A 97, 062337 (2018).","DOI":"10.1103\/PhysRevA.97.062337"},{"key":"28","doi-asserted-by":"publisher","unstructured":"Q. Zhuang, P. W. Shor, and J. H. Shapiro, ``Resource theory of non-Gaussian operations,'' Physical Review A 97, 052317 (2018).","DOI":"10.1103\/PhysRevA.97.052317"},{"key":"29","doi-asserted-by":"publisher","unstructured":"F. Albarelli, M. G. Genoni, M. G. Paris, and A. Ferraro, ``Resource theory of quantum non-Gaussianity and Wigner negativity,'' Physical Review A 98, 052350 (2018).","DOI":"10.1103\/PhysRevA.98.052350"},{"key":"30","doi-asserted-by":"publisher","unstructured":"J. Eisert, S. Scheel, and M. B. Plenio, ``Distilling Gaussian states with Gaussian operations is impossible,'' Physical Review Letters 89, 137903 (2002).","DOI":"10.1103\/PhysRevLett.89.137903"},{"key":"31","doi-asserted-by":"publisher","unstructured":"J. Fiur\u00e1\u0161ek, ``Gaussian transformations and distillation of entangled Gaussian states,'' Physical Review Letters 89, 137904 (2002).","DOI":"10.1103\/PhysRevLett.89.137904"},{"key":"32","doi-asserted-by":"publisher","unstructured":"J. Niset, J. Fiur\u00e1\u0161ek, and N. J. Cerf, ``No-go theorem for Gaussian quantum error correction,'' Physical Review Letters 102, 120501 (2009).","DOI":"10.1103\/PhysRevLett.102.120501"},{"key":"33","doi-asserted-by":"publisher","unstructured":"J. B. Conway, ``Functions of one complex variable II,'', vol. 159. Springer Science & Business Media, 2012.","DOI":"10.1007\/978-1-4612-0817-4"},{"key":"34","doi-asserted-by":"publisher","unstructured":"F. Calogero, ``Solution of the one-dimensional N-body problems with quadratic and\/or inversely quadratic pair potentials,'' Journal of Mathematical Physics 12, 419\u2013436 (1971).","DOI":"10.1063\/1.1665604"},{"key":"35","doi-asserted-by":"publisher","unstructured":"J. Radcliffe, ``Some properties of coherent spin states,'' Journal of Physics A: General Physics 4, 313 (1971).","DOI":"10.1088\/0305-4470\/4\/3\/009"},{"key":"36","doi-asserted-by":"publisher","unstructured":"F. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, ``Atomic coherent states in quantum optics,'' Physical Review A 6, 2211 (1972).","DOI":"10.1103\/PhysRevA.6.2211"},{"key":"37","doi-asserted-by":"publisher","unstructured":"D. Gross, ``Hudson\u2019s theorem for finite-dimensional quantum systems,'' Journal of mathematical physics 47, 122107 (2006).","DOI":"10.1063\/1.2393152"},{"key":"38","doi-asserted-by":"publisher","unstructured":"E. Majorana, ``Atomi orientati in campo magnetico variabile,'' Il Nuovo Cimento (1924-1942) 9, 43\u201350 (1932).","DOI":"10.1007\/BF02960953"},{"key":"39","unstructured":"A. W. Harrow, ``The church of the symmetric subspace,'' arXiv:1308.6595."},{"key":"40","doi-asserted-by":"publisher","unstructured":"U. Chabaud, D. Markham, and F. Grosshans, ``Stellar representation of non-Gaussian quantum states,'' Physical Review Letters 124, 063605 (2020).","DOI":"10.1103\/PhysRevLett.124.063605"},{"key":"41","unstructured":"I. E. Segal and G. W. Mackey, ``Mathematical problems of relativistic physics,'', vol. 2. American Mathematical Soc., 1963."},{"key":"42","doi-asserted-by":"publisher","unstructured":"V. Bargmann, ``On a Hilbert space of analytic functions and an associated integral transform part I,'' Communications on pure and applied mathematics 14, 187\u2013214 (1961).","DOI":"10.1002\/cpa.3160140303"},{"key":"43","doi-asserted-by":"publisher","unstructured":"K. Husimi, ``Some formal properties of the density matrix,'' Proceedings of the Physico-Mathematical Society of Japan. 3rd Series 22, 264\u2013314 (1940).","DOI":"10.11429\/ppmsj1919.22.4_264"},{"key":"44","doi-asserted-by":"publisher","unstructured":"F. Ricci, ``A contraction of SU (2) to the Heisenberg group,'' Monatshefte f\u00fcr Mathematik 101, 211\u2013225 (1986).","DOI":"10.1007\/BF01301660"},{"key":"45","doi-asserted-by":"publisher","unstructured":"A. Vourdas, ``Analytic representations in quantum mechanics,'' Journal of Physics A: Mathematical and General 39, R65 (2006).","DOI":"10.1088\/0305-4470\/39\/7\/R01"},{"key":"46","unstructured":"A. Ferraro, S. Olivares, and M. G. Paris, ``Gaussian states in continuous variable quantum information,'' quant-ph\/0503237."},{"key":"47","doi-asserted-by":"publisher","unstructured":"L. G. Valiant, ``The complexity of computing the permanent,'' Theoretical computer science 8, 189\u2013201 (1979).","DOI":"10.1016\/0304-3975(79)90044-6"},{"key":"48","doi-asserted-by":"publisher","unstructured":"E. Schr\u00f6dinger, ``An undulatory theory of the mechanics of atoms and molecules,'' Physical review 28, 1049 (1926).","DOI":"10.1103\/PhysRev.28.1049"},{"key":"49","doi-asserted-by":"publisher","unstructured":"P. Leboeuf, ``Phase space approach to quantum dynamics,'' Journal of Physics A: Mathematical and General 24, 4575 (1991).","DOI":"10.1088\/0305-4470\/24\/19\/021"},{"key":"50","doi-asserted-by":"publisher","unstructured":"F. Calogero, ``Exactly solvable two-dimensional many-body problems,'' Lettere al Nuovo Cimento (1971-1985) 16, 35\u201338 (1976).","DOI":"10.1007\/BF02746871"},{"key":"51","doi-asserted-by":"publisher","unstructured":"J. Moser, ``Three integrable Hamiltonian systems connected with isospectral deformations,'' in Surveys in applied mathematics, pp. 235\u2013258. Elsevier, 1976.","DOI":"10.1016\/B978-0-12-492150-4.50023-8"},{"key":"52","doi-asserted-by":"publisher","unstructured":"M. A. Olshanetsky and A. M. Perelomov, ``Geodesic flows on symmetric spaces of zero curvature and explicit solution of the generalized Calogero model for the classical case,'' Functional Analysis and Its Applications 10, 237\u2013239 (1976).","DOI":"10.1007\/BF01075536"},{"key":"53","doi-asserted-by":"publisher","unstructured":"M. A. Olshanetsky and A. M. Perelomov, ``Classical integrable finite-dimensional systems related to Lie algebras,'' Physics Reports 71, 313\u2013400 (1981).","DOI":"10.1016\/0370-1573(81)90023-5"},{"key":"54","doi-asserted-by":"crossref","unstructured":"F. Calogero. http:\/\/www.scholarpedia.org\/article\/Calogero-Moser_system, 2008.","DOI":"10.4249\/scholarpedia.7216"},{"key":"55","doi-asserted-by":"publisher","unstructured":"A. P. Polychronakos, ``The physics and mathematics of Calogero particles,'' Journal of Physics A: Mathematical and General 39, 12793 (2006).","DOI":"10.1088\/0305-4470\/39\/41\/S07"},{"key":"56","doi-asserted-by":"publisher","unstructured":"M. Walschaers, ``Non-Gaussian Quantum States and Where to Find Them,'' PRX Quantum 2, 030204 (2021).","DOI":"10.1103\/PRXQuantum.2.030204"},{"key":"57","doi-asserted-by":"publisher","unstructured":"M. Amy, D. Maslov, M. Mosca, and M. Roetteler, ``A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits,'' IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 32, 818\u2013830 (2013).","DOI":"10.1109\/TCAD.2013.2244643"},{"key":"58","doi-asserted-by":"publisher","unstructured":"M. Beverland, E. Campbell, M. Howard, and V. Kliuchnikov, ``Lower bounds on the non-Clifford resources for quantum computations,'' Quantum Science and Technology 5, 035009 (2020).","DOI":"10.1088\/2058-9565\/ab8963"},{"key":"59","unstructured":"A. Lvovsky, P. Grangier, A. Ourjoumtsev, V. Parigi, M. Sasaki, and R. Tualle-Brouri, ``Production and applications of non-Gaussian quantum states of light,'' arXiv:2006.16985."},{"key":"60","doi-asserted-by":"publisher","unstructured":"U. Chabaud, G. Roeland, M. Walschaers, F. Grosshans, V. Parigi, D. Markham, and N. Treps, ``Certification of non-Gaussian states with operational measurements,'' PRX Quantum 2, 020333 (2021).","DOI":"10.1103\/PRXQuantum.2.020333"},{"key":"61","doi-asserted-by":"publisher","unstructured":"J. Fiur\u00e1\u0161ek, ``Efficient construction of witnesses of the stellar rank of nonclassical states of light,'' Optics Express 30, 30630\u201330639 (2022).","DOI":"10.1364\/OE.466175"},{"key":"62","doi-asserted-by":"publisher","unstructured":"S. Bravyi and A. Kitaev, ``Universal quantum computation with ideal Clifford gates and noisy ancillas,'' Physical Review A 71, 022316 (2005).","DOI":"10.1103\/PhysRevA.71.022316"},{"key":"63","doi-asserted-by":"publisher","unstructured":"M. Yoganathan, R. Jozsa, and S. Strelchuk, ``Quantum advantage of unitary Clifford circuits with magic state inputs,'' Proceedings of the Royal Society A 475, 20180427 (2019).","DOI":"10.1098\/rspa.2018.0427"},{"key":"64","doi-asserted-by":"publisher","unstructured":"E. Knill, R. Laflamme, and G. J. Milburn, ``A scheme for efficient quantum computation with linear optics,'' Nature 409, 46\u201352 (2001).","DOI":"10.1038\/35051009"},{"key":"65","doi-asserted-by":"publisher","unstructured":"N. C. Menicucci, P. Van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, ``Universal quantum computation with continuous-variable cluster states,'' Physical Review Letters 97, 110501 (2006).","DOI":"10.1103\/PhysRevLett.97.110501"},{"key":"66","doi-asserted-by":"publisher","unstructured":"J. Zhang and S. L. Braunstein, ``Continuous-variable Gaussian analog of cluster states,'' Physical Review A 73, 032318 (2006).","DOI":"10.1103\/PhysRevA.73.032318"},{"key":"67","doi-asserted-by":"publisher","unstructured":"M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, ``Quantum computing with continuous-variable clusters,'' Physical Review A 79, 062318 (2009).","DOI":"10.1103\/PhysRevA.79.062318"},{"key":"68","doi-asserted-by":"publisher","unstructured":"B. Q. Baragiola, G. Pantaleoni, R. N. Alexander, A. Karanjai, and N. C. Menicucci, ``All-Gaussian universality and fault tolerance with the Gottesman-Kitaev-Preskill code,'' Physical Review Letters 123, 200502 (2019).","DOI":"10.1103\/PhysRevLett.123.200502"},{"key":"69","doi-asserted-by":"publisher","unstructured":"B. C. Hall, ``Holomorphic methods in analysis and mathematical physics. First Summer School in Analysis and Mathematical Physics,(Cuernavaca Morelos, 1998),'' Contemp. Math 260, 1\u201359.","DOI":"10.48550\/arXiv.quant-ph\/9912054"},{"key":"70","doi-asserted-by":"publisher","unstructured":"P. Leboeuf and A. Voros, ``Chaos-revealing multiplicative representation of quantum eigenstates,'' Journal of Physics A: Mathematical and General 23, 1765 (1990).","DOI":"10.1088\/0305-4470\/23\/10\/017"},{"key":"71","doi-asserted-by":"publisher","unstructured":"D. Ellinas and V. Kovanis, ``Motion of the wave-function zeros in spin-boson systems,'' Physical Review A 51, 4230 (1995).","DOI":"10.1103\/PhysRevA.51.4230"},{"key":"72","doi-asserted-by":"publisher","unstructured":"R. L. Hudson, ``When is the Wigner quasi-probability density non-negative?,'' Reports on Mathematical Physics 6, 249\u2013252 (1974).","DOI":"10.1016\/0034-4877(74)90007-X"},{"key":"73","doi-asserted-by":"publisher","unstructured":"F. Soto and P. Claverie, ``When is the Wigner function of multidimensional systems nonnegative?,'' Journal of Mathematical Physics 24, 97\u2013100 (1983).","DOI":"10.1063\/1.525607"},{"key":"74","doi-asserted-by":"publisher","unstructured":"N. L\u00fctkenhaus and S. M. Barnett, ``Nonclassical effects in phase space,'' Physical Review A 51, 3340 (1995).","DOI":"10.1103\/PhysRevA.51.3340"},{"key":"75","doi-asserted-by":"publisher","unstructured":"U. Chabaud, G. Ferrini, F. Grosshans, and D. Markham, ``Classical simulation of Gaussian quantum circuits with non-Gaussian input states,'' Physical Review Research 3, 033018 (2021).","DOI":"10.1103\/PhysRevResearch.3.033018"},{"key":"76","unstructured":"U. Chabaud and M. Walschaers, ``Resources for bosonic quantum computational advantage,'' arXiv:2207.11781."},{"key":"77","doi-asserted-by":"publisher","unstructured":"O. Bournez and M. L. Campagnolo, ``A survey on continuous time computations,'' in New computational paradigms, pp. 383\u2013423. Springer, 2008.","DOI":"10.1007\/978-0-387-68546-5_17"},{"key":"78","doi-asserted-by":"publisher","unstructured":"L. G. Valiant, ``Completeness classes in algebra,'' in Proceedings of the eleventh annual ACM symposium on Theory of computing, pp. 249\u2013261. 1979.","DOI":"10.1145\/800135.804419"},{"key":"79","doi-asserted-by":"publisher","unstructured":"L. Blum, M. Shub, and S. Smale, ``On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines,'' in The Collected Papers of Stephen Smale: Volume 3, pp. 1293\u20131338. World Scientific, 2000.","DOI":"10.1142\/9789812792839_0013"},{"key":"80","doi-asserted-by":"publisher","unstructured":"V. Bush, ``The differential analyzer. A new machine for solving differential equations,'' Journal of the Franklin Institute 212, 447\u2013488 (1931).","DOI":"10.1016\/S0016-0032(31)90616-9"},{"key":"81","doi-asserted-by":"publisher","unstructured":"S. Smale, ``Mathematical problems for the next century,'' The mathematical intelligencer 20, 7\u201315 (1998).","DOI":"10.1007\/BF03025291"},{"key":"82","doi-asserted-by":"publisher","unstructured":"L. D'Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, ``From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics,'' Advances in Physics 65, 239\u2013362 (2016).","DOI":"10.1080\/00018732.2016.1198134"},{"key":"83","doi-asserted-by":"publisher","unstructured":"S. Aaronson, A. Bouland, G. Kuperberg, and S. Mehraban, ``The computational complexity of ball permutations,'' in Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pp. 317\u2013327. 2017.","DOI":"10.1145\/3055399.3055453"},{"key":"84","doi-asserted-by":"publisher","unstructured":"K. Fukui and S. Takeda, ``Building a large-scale quantum computer with continuous-variable optical technologies,'' arXiv:2110.03247.","DOI":"10.1088\/1361-6455\/ac489c"},{"key":"85","doi-asserted-by":"publisher","unstructured":"D. Menzies and R. Filip, ``Gaussian-optimized preparation of non-Gaussian pure states,'' Physical Review A 79, 012313 (2009).","DOI":"10.1103\/PhysRevA.79.012313"},{"key":"86","doi-asserted-by":"publisher","unstructured":"R. Gomes, A. Salles, F. Toscano, P. S. Ribeiro, and S. Walborn, ``Quantum entanglement beyond Gaussian criteria,'' Proceedings of the National Academy of Sciences 106, 21517\u201321520 (2009).","DOI":"10.1073\/pnas.0908329106"},{"key":"87","doi-asserted-by":"publisher","unstructured":"A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, ``Preparation of non-local superpositions of quasi-classical light states,'' Nature Physics 5, 189\u2013192 (2009).","DOI":"10.1038\/nphys1199"},{"key":"88","unstructured":"C. M. Dawson and M. A. Nielsen, ``The Solovay\u2013Kitaev algorithm,'' quant-ph\/0505030."},{"key":"89","doi-asserted-by":"publisher","unstructured":"S. Becker, N. Datta, L. Lami, and C. Rouz\u00e9, ``Energy-Constrained Discrimination of Unitaries, Quantum Speed Limits, and a Gaussian Solovay-Kitaev Theorem,'' Physical Review Letters 126, 190504 (2021).","DOI":"10.1103\/PhysRevLett.126.190504"},{"key":"90","doi-asserted-by":"publisher","unstructured":"K. E. Cahill and R. J. Glauber, ``Density operators and quasiprobability distributions,'' Physical Review 177, 1882 (1969).","DOI":"10.1103\/PhysRev.177.1882"},{"key":"91","unstructured":"D. Petz, ``An Invitation to the algebra of canonical commutation relations,'' in Leuven notes in mathematical and theoretical physics, vol. 2. Leuven University Press, Leuven, Belgium, 1990."},{"key":"92","doi-asserted-by":"publisher","unstructured":"J. Derezi\u0144ski, ``Introduction to representations of the canonical commutation and anticommutation relations,'' in Large Coulomb Systems, pp. 63\u2013143. Springer, 2006.","DOI":"10.1007\/3-540-32579-4_3"},{"key":"93","doi-asserted-by":"publisher","unstructured":"A. Zavatta, S. Viciani, and M. Bellini, ``Quantum-to-classical transition with single-photon-added coherent states of light,'' science 306, 660\u2013662 (2004).","DOI":"10.1126\/science.1103190"},{"key":"94","doi-asserted-by":"publisher","unstructured":"A. Serafini, ``Quantum continuous variables: a primer of theoretical methods,''. CRC press, 2017.","DOI":"10.1201\/9781315118727"},{"key":"95","doi-asserted-by":"publisher","unstructured":"E. P. Wigner, ``On the quantum correction for thermodynamic equilibrium,'' in Part I: Physical Chemistry. Part II: Solid State Physics, pp. 110\u2013120. Springer, 1997.","DOI":"10.1007\/978-3-642-59033-7_9"},{"key":"96","doi-asserted-by":"publisher","unstructured":"C. Weedbrook, S. Pirandola, R. Garc\u00eda-Patr\u00f3n, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, ``Gaussian quantum information,'' Reviews of Modern Physics 84, 621 (2012).","DOI":"10.1103\/RevModPhys.84.621"},{"key":"97","doi-asserted-by":"publisher","unstructured":"G. S. Agarwal, ``Quantum optics,''. Cambridge University Press, 2012.","DOI":"10.1017\/CBO9781139035170"},{"key":"98","doi-asserted-by":"publisher","unstructured":"B. Demoen, P. Vanheuverzwijn, and A. Verbeure, ``Completely positive maps on ccr-algebra,'' Letters in mathematical physics 2, 161\u2013166 (1977).","DOI":"10.1007\/BF00398582"},{"key":"99","doi-asserted-by":"publisher","unstructured":"G. Giedke and J. I. Cirac, ``Characterization of Gaussian operations and distillation of Gaussian states,'' Physical Review A 66, 032316 (2002).","DOI":"10.1103\/PhysRevA.66.032316"},{"key":"100","doi-asserted-by":"publisher","unstructured":"M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, ``Experimental realization of any discrete unitary operator,'' Physical Review Letters 73, 58 (1994).","DOI":"10.1103\/PhysRevLett.73.58"},{"key":"101","doi-asserted-by":"publisher","unstructured":"D. J. Korteweg and G. De Vries, ``XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves,'' The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 39, 422\u2013443 (1895).","DOI":"10.1080\/14786449508620739"},{"key":"102","doi-asserted-by":"publisher","unstructured":"H. Airault, H. McKean, and J. Moser, ``Rational and elliptic solutions of the korteweg-de vries equation and a related many-body problem,'' Communications on Pure and Applied Mathematics 30, 95\u2013148 (1977).","DOI":"10.1002\/cpa.3160300106"},{"key":"103","unstructured":"B. B. Kadomtsev and V. I. Petviashvili, ``On the stability of solitary waves in weakly dispersing media,'' in Sov. Phys. Dokl, vol. 15, pp. 539\u2013541. 1970."},{"key":"104","doi-asserted-by":"publisher","unstructured":"I. M. Krichever, ``Rational solutions of the Kadomtsev\u2014Petviashvili equation and integrable systems of N particles on a line,'' Functional Analysis and Its Applications 12, 59\u201361 (1978).","DOI":"10.1007\/BF01077570"},{"key":"105","unstructured":"P. Etingof, ``Lectures on Calogero-Moser systems,'' math\/0606233."},{"key":"106","doi-asserted-by":"publisher","unstructured":"U. Shackerley-Bennett, A. Pitchford, M. G. Genoni, A. Serafini, and D. K. Burgarth, ``The reachable set of single-mode quadratic Hamiltonians,'' Journal of Physics A: Mathematical and Theoretical 50, 155203 (2017).","DOI":"10.1088\/1751-8121\/aa6243"},{"key":"107","unstructured":"U. J. Le Verrier, ``M\u00e9moire sur les variations s\u00e9culaires des \u00e9l\u00e9ments des orbites: pour les sept plan\u00e8tes principales, Mercure, V\u00e9nus, la Terre, Mars, Jupiter, Saturne et Uranus,''. Bachelier, 1845."},{"key":"108","unstructured":"D. K. Faddeev and I. S. Sominski\u012d, ``Problems in Higher algebra,''. Mir Publishers, 1972."},{"key":"109","doi-asserted-by":"publisher","unstructured":"M. Abramowitz and I. A. Stegun, ``Handbook of mathematical functions: with formulas, graphs, and mathematical tables,'', vol. 55. Courier Corporation, 1965.","DOI":"10.1119\/1.15378"},{"key":"110","doi-asserted-by":"publisher","unstructured":"A. Botero and B. Reznik, ``Modewise entanglement of Gaussian states,'' Physical Review A 67, 052311 (2003).","DOI":"10.1103\/PhysRevA.67.052311"},{"key":"111","unstructured":"G. Giedke, J. Eisert, J. I. Cirac, and M. B. Plenio, ``Entanglement transformations of pure Gaussian states,'' quant-ph\/0301038."},{"key":"112","doi-asserted-by":"publisher","unstructured":"M. Walschaers, C. Fabre, V. Parigi, and N. Treps, ``Statistical signatures of multimode single-photon-added and-subtracted states of light,'' Physical Review A 96, 053835 (2017).","DOI":"10.1103\/PhysRevA.96.053835"},{"key":"113","doi-asserted-by":"publisher","unstructured":"U. Leonhardt, ``Essential Quantum Optics,''. Cambridge University Press, Cambridge, UK, 1st ed., 2010.","DOI":"10.1017\/CBO9780511806117"},{"key":"114","doi-asserted-by":"publisher","unstructured":"A. W\u00fcnsche, ``Laguerre 2D-functions and their application in quantum optics,'' Journal of Physics A: Mathematical and General 31, 8267 (1998).","DOI":"10.1088\/0305-4470\/31\/40\/017"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2022-10-06-831\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,10,6]],"date-time":"2022-10-06T15:06:38Z","timestamp":1665068798000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2022-10-06-831\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,6]]},"references-count":115,"URL":"https:\/\/doi.org\/10.22331\/q-2022-10-06-831","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,6]]},"article-number":"831"}}