Page curves and typical entanglement in linear optics

Joseph T. Iosue1,2, Adam Ehrenberg1,2, Dominik Hangleiter2,1, Abhinav Deshpande3, and Alexey V. Gorshkov1,2

1Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
2Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
3Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA

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Abstract

Bosonic Gaussian states are a special class of quantum states in an infinite dimensional Hilbert space that are relevant to universal continuous-variable quantum computation as well as to near-term quantum sampling tasks such as Gaussian Boson Sampling. In this work, we study entanglement within a set of squeezed modes that have been evolved by a random linear optical unitary. We first derive formulas that are asymptotically exact in the number of modes for the Rényi-2 Page curve (the average Rényi-2 entropy of a subsystem of a pure bosonic Gaussian state) and the corresponding Page correction (the average information of the subsystem) in certain squeezing regimes. We then prove various results on the typicality of entanglement as measured by the Rényi-2 entropy by studying its variance. Using the aforementioned results for the Rényi-2 entropy, we upper and lower bound the von Neumann entropy Page curve and prove certain regimes of entanglement typicality as measured by the von Neumann entropy. Our main proofs make use of a symmetry property obeyed by the average and the variance of the entropy that dramatically simplifies the averaging over unitaries. In this light, we propose future research directions where this symmetry might also be exploited. We conclude by discussing potential applications of our results and their generalizations to Gaussian Boson Sampling and to illuminating the relationship between entanglement and computational complexity.

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What gives quantum computers an advantage over their classical counterparts? It is known that entanglement is necessary for quantum advantage, but a quantitative link between entanglement and complexity is missing. The first step toward building such a link is understanding the entanglement of quantum states that are hard to classically simulate. Such a study has not been done even for the first sampling scheme shown to have a quantum advantage, namely output states of linear optical circuits. In this work, we address this by characterizing the typical entanglement of such states.

Specifically, we study bipartite entanglement within quantum states generated by random linear optical circuits acting on specially prepared inputs. We derive an exact formula for the average entanglement and prove that, in certain regimes, the probability that a random state’s entanglement deviates from the average vanishes asymptotically in the system size. Our results are obtained through a combination of methods coming from quantum optics and quantum information as well as from a novel technique that we develop based on a powerful symmetry present in the entanglement structure. We further propose how this new technique may be useful to study bipartite entanglement in different settings.

These results provide a stepping stone to a better understanding of the typical behavior of random linear optical circuits and the certification of quantum advantage in a linear optics sampling experiment.

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► References

[1] Don N. Page. ``Average entropy of a subsystem''. Physical Review Letters 71, 1291–1294 (1993).
https:/​/​doi.org/​10.1103/​PhysRevLett.71.1291

[2] S. K. Foong and S. Kanno. ``Proof of Page’s conjecture on the average entropy of a subsystem''. Physical Review Letters 72, 1148–1151 (1994).
https:/​/​doi.org/​10.1103/​PhysRevLett.72.1148

[3] Jorge Sánchez-Ruiz. ``Simple proof of Page’s conjecture on the average entropy of a subsystem''. Physical Review E 52, 5653–5655 (1995).
https:/​/​doi.org/​10.1103/​PhysRevE.52.5653

[4] Siddhartha Sen. ``Average Entropy of a Quantum Subsystem''. Physical Review Letters 77, 1–3 (1996).
https:/​/​doi.org/​10.1103/​PhysRevLett.77.1

[5] Don N. Page. ``Information in black hole radiation''. Physical Review Letters 71, 3743–3746 (1993).
https:/​/​doi.org/​10.1103/​PhysRevLett.71.3743

[6] Patrick Hayden and John Preskill. ``Black holes as mirrors: quantum information in random subsystems''. Journal of High Energy Physics 2007, 120–120 (2007).
https:/​/​doi.org/​10.1088/​1126-6708/​2007/​09/​120

[7] Eugenio Bianchi, Tommaso De Lorenzo, and Matteo Smerlak. ``Entanglement entropy production in gravitational collapse: covariant regularization and solvable models''. Journal of High Energy Physics 2015, 180 (2015).
https:/​/​doi.org/​10.1007/​JHEP06(2015)180

[8] Patrick Hayden, Debbie W. Leung, and Andreas Winter. ``Aspects of generic entanglement''. Communications in Mathematical Physics 265, 95–117 (2006).
https:/​/​doi.org/​10.1007/​s00220-006-1535-6

[9] Pavan Hosur, Xiao-Liang Qi, Daniel A. Roberts, and Beni Yoshida. ``Chaos in quantum channels''. Journal of High Energy Physics 2016, 4 (2016).
https:/​/​doi.org/​10.1007/​JHEP02(2016)004

[10] Hiroyuki Fujita, Yuya O. Nakagawa, Sho Sugiura, and Masataka Watanabe. ``Page curves for general interacting systems''. Journal of High Energy Physics 2018, 112 (2018).
https:/​/​doi.org/​10.1007/​JHEP12(2018)112

[11] Tsung-Cheng Lu and Tarun Grover. ``Renyi entropy of chaotic eigenstates''. Physical Review E 99, 032111 (2019).
https:/​/​doi.org/​10.1103/​PhysRevE.99.032111

[12] Yuya O. Nakagawa, Masataka Watanabe, Hiroyuki Fujita, and Sho Sugiura. ``Universality in volume-law entanglement of scrambled pure quantum states''. Nature Communications 9, 1635 (2018).
https:/​/​doi.org/​10.1038/​s41467-018-03883-9

[13] Lev Vidmar and Marcos Rigol. ``Entanglement entropy of eigenstates of quantum chaotic hamiltonians''. Physical Review Letters 119, 220603 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.220603

[14] Lev Vidmar, Lucas Hackl, Eugenio Bianchi, and Marcos Rigol. ``Entanglement entropy of eigenstates of quadratic fermionic hamiltonians''. Physical Review Letters 119, 020601 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.020601

[15] Lucas Hackl, Lev Vidmar, Marcos Rigol, and Eugenio Bianchi. ``Average eigenstate entanglement entropy of the XY chain in a transverse field and its universality for translationally invariant quadratic fermionic models''. Physical Review B 99, 075123 (2019).
https:/​/​doi.org/​10.1103/​PhysRevB.99.075123

[16] Sheldon Goldstein, Joel L. Lebowitz, Roderich Tumulka, and Nino Zanghi. ``Canonical typicality''. Physical Review Letters 96, 050403 (2006).
https:/​/​doi.org/​10.1103/​PhysRevLett.96.050403

[17] Luca D'Alessio, Yariv Kafri, Anatoli Polkovnikov, and Marcos Rigol. ``From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics''. Advances in Physics 65, 239–362 (2016).
https:/​/​doi.org/​10.1080/​00018732.2016.1198134

[18] Patrick Hayden, Debbie Leung, Peter W. Shor, and Andreas Winter. ``Randomizing Quantum States: Constructions and Applications''. Communications in Mathematical Physics 250, 371–391 (2004).
https:/​/​doi.org/​10.1007/​s00220-004-1087-6

[19] Benoit Collins, Carlos E. Gonzalez-Guillen, and David Pérez-Garcia. ``Matrix Product States, Random Matrix Theory and the Principle of Maximum Entropy''. Communications in Mathematical Physics 320, 663–677 (2013).
https:/​/​doi.org/​10.1007/​s00220-013-1718-x

[20] M. B. Hastings. ``Random MERA States and the Tightness of the Brandao-Horodecki Entropy Bound''. arXiv.1505.06468 (2015).
https:/​/​doi.org/​10.48550/​arXiv.1505.06468

[21] Silvano Garnerone, Thiago R. de Oliveira, and Paolo Zanardi. ``Typicality in random matrix product states''. Physical Review A 81, 032336 (2010).
https:/​/​doi.org/​10.1103/​PhysRevA.81.032336

[22] Sandu Popescu, Anthony J. Short, and Andreas Winter. ``Entanglement and the foundations of statistical mechanics''. Nature Physics 2, 754–758 (2006).
https:/​/​doi.org/​10.1038/​nphys444

[23] D. Gross, S. T. Flammia, and J. Eisert. ``Most Quantum States Are Too Entangled To Be Useful As Computational Resources''. Physical Review Letters 102, 190501 (2009).
https:/​/​doi.org/​10.1103/​PhysRevLett.102.190501

[24] Michael J. Bremner, Caterina Mora, and Andreas Winter. ``Are Random Pure States Useful for Quantum Computation?''. Physical Review Letters 102, 190502 (2009).
https:/​/​doi.org/​10.1103/​PhysRevLett.102.190502

[25] Eugenio Bianchi and Pietro Donà. ``Typical entanglement entropy in the presence of a center: Page curve and its variance''. Physical Review D 100, 105010 (2019).
https:/​/​doi.org/​10.1103/​PhysRevD.100.105010

[26] Eugenio Bianchi, Lucas Hackl, and Mario Kieburg. ``Page curve for fermionic Gaussian states''. Physical Review B 103, L241118 (2021).
https:/​/​doi.org/​10.1103/​PhysRevB.103.L241118

[27] Oscar CO Dahlsten, Cosmo Lupo, Stefano Mancini, and Alessio Serafini. ``Entanglement typicality''. Journal of Physics A: Mathematical and Theoretical 47, 363001 (2014).
https:/​/​doi.org/​10.1088/​1751-8113/​47/​36/​363001

[28] Michael A. Nielsen and Isaac L. Chuang. ``Quantum computation and quantum information''. Cambridge University Press (2010). 10th anniversary edition.

[29] Ingemar Bengtsson and Karol Życzkowski. ``Geometry of quantum states: An introduction to quantum entanglement''. Cambridge University Press (2008).

[30] Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral. ``Entanglement in many-body systems''. Reviews of Modern Physics 80, 517–576 (2008).
https:/​/​doi.org/​10.1103/​RevModPhys.80.517

[31] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. ``Quantum entanglement''. Reviews of Modern Physics 81, 865–942 (2009).
https:/​/​doi.org/​10.1103/​RevModPhys.81.865

[32] Mark Wilde. ``Quantum information theory''. Cambridge University Press (2017). Second edition.

[33] Richard Jozsa and Noah Linden. ``On the role of entanglement in quantum-computational speed-up''. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 459, 2011–2032 (2003).
https:/​/​doi.org/​10.1098/​rspa.2002.1097

[34] Guifré Vidal. ``Efficient Classical Simulation of Slightly Entangled Quantum Computations''. Physical Review Letters 91, 147902 (2003).
https:/​/​doi.org/​10.1103/​PhysRevLett.91.147902

[35] F. Verstraete, J. J. Garcia-Ripoll, and J. I. Cirac. ``Matrix Product Density Operators: Simulation of Finite-Temperature and Dissipative Systems''. Physical Review Letters 93, 207204 (2004).
https:/​/​doi.org/​10.1103/​PhysRevLett.93.207204

[36] 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).
https:/​/​doi.org/​10.1103/​PhysRevLett.113.100502

[37] Craig S. Hamilton, Regina Kruse, Linda Sansoni, Sonja Barkhofen, Christine Silberhorn, and Igor Jex. ``Gaussian Boson Sampling''. Physical Review Letters 119, 170501 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.170501

[38] Regina Kruse, Craig S. Hamilton, Linda Sansoni, Sonja Barkhofen, Christine Silberhorn, and Igor Jex. ``Detailed study of Gaussian boson sampling''. Physical Review A 100, 032326 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.100.032326

[39] Abhinav Deshpande, Arthur Mehta, Trevor Vincent, Nicolás Quesada, Marcel Hinsche, Marios Ioannou, Lars Madsen, Jonathan Lavoie, Haoyu Qi, Jens Eisert, Dominik Hangleiter, Bill Fefferman, and Ish Dhand. ``Quantum computational advantage via high-dimensional Gaussian boson sampling''. Science Advances 8, eabi7894 (2022).
https:/​/​doi.org/​10.1126/​sciadv.abi7894

[40] Daniel Grier, Daniel J. Brod, Juan Miguel Arrazola, Marcos Benicio De Andrade Alonso, and Nicolás Quesada. ``The Complexity of Bipartite Gaussian Boson Sampling''. Quantum 6, 863 (2022).
https:/​/​doi.org/​10.22331/​q-2022-11-28-863

[41] Ulysse Chabaud and Mattia Walschaers. ``Resources for Bosonic Quantum Computational Advantage''. Physical Review Letters 130, 090602 (2023).
https:/​/​doi.org/​10.1103/​PhysRevLett.130.090602

[42] Quntao Zhuang, Zheshen Zhang, and Jeffrey H. Shapiro. ``Distributed quantum sensing using continuous-variable multipartite entanglement''. Physical Review A 97, 032329 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.032329

[43] Emanuele Polino, Mauro Valeri, Nicolò Spagnolo, and Fabio Sciarrino. ``Photonic quantum metrology''. AVS Quantum Science 2, 024703 (2020).
https:/​/​doi.org/​10.1116/​5.0007577

[44] Changhun Oh, Changhyoup Lee, Seok Hyung Lie, and Hyunseok Jeong. ``Optimal distributed quantum sensing using Gaussian states''. Physical Review Research 2, 023030 (2020).
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.023030

[45] Marco Malitesta, Augusto Smerzi, and Luca Pezzè. ``Distributed Quantum Sensing with Squeezed-Vacuum Light in a Configurable Network of Mach-Zehnder Interferometers''. arXiv:2109.09178 (2021).
https:/​/​doi.org/​10.48550/​ARXIV.2109.09178

[46] Marco Barbieri. ``Optical Quantum Metrology''. PRX Quantum 3, 010202 (2022).
https:/​/​doi.org/​10.1103/​PRXQuantum.3.010202

[47] Gerardo Adesso. ``Entanglement of Gaussian states''. arXiv:0702069 [quant-ph] (2007).
https:/​/​doi.org/​10.48550/​arXiv.quant-ph/​0702069
arXiv:quant-ph/0702069

[48] Han-Sen Zhong, Hui Wang, Yu-Hao Deng, Ming-Cheng Chen, Li-Chao Peng, Yi-Han Luo, Jian Qin, Dian Wu, Xing Ding, Yi Hu, Peng Hu, Xiao-Yan Yang, Wei-Jun Zhang, Hao Li, Yuxuan Li, Xiao Jiang, Lin Gan, Guangwen Yang, Lixing You, Zhen Wang, Li Li, Nai-Le Liu, Chao-Yang Lu, and Jian-Wei Pan. ``Quantum computational advantage using photons''. Science 370, 1460–1463 (2020).
https:/​/​doi.org/​10.1126/​science.abe8770

[49] Han-Sen Zhong, Yu-Hao Deng, Jian Qin, Hui Wang, Ming-Cheng Chen, Li-Chao Peng, Yi-Han Luo, Dian Wu, Si-Qiu Gong, Hao Su, Yi Hu, Peng Hu, Xiao-Yan Yang, Wei-Jun Zhang, Hao Li, Yuxuan Li, Xiao Jiang, Lin Gan, Guangwen Yang, Lixing You, Zhen Wang, Li Li, Nai-Le Liu, Jelmer J. Renema, Chao-Yang Lu, and Jian-Wei Pan. ``Phase-Programmable Gaussian Boson Sampling Using Stimulated Squeezed Light''. Physical Review Letters 127, 180502 (2021).
https:/​/​doi.org/​10.1103/​PhysRevLett.127.180502

[50] Lars S. Madsen, Fabian Laudenbach, Mohsen Falamarzi. Askarani, Fabien Rortais, Trevor Vincent, Jacob F. F. Bulmer, Filippo M. Miatto, Leonhard Neuhaus, Lukas G. Helt, Matthew J. Collins, Adriana E. Lita, Thomas Gerrits, Sae Woo Nam, Varun D. Vaidya, Matteo Menotti, Ish Dhand, Zachary Vernon, Nicolás Quesada, and Jonathan Lavoie. ``Quantum computational advantage with a programmable photonic processor''. Nature 606, 75–81 (2022).
https:/​/​doi.org/​10.1038/​s41586-022-04725-x

[51] Budhaditya Bhattacharjee, Pratik Nandy, and Tanay Pathak. ``Eigenstate capacity and Page curve in fermionic Gaussian states''. Physical Review B 104, 214306 (2021).
https:/​/​doi.org/​10.1103/​PhysRevB.104.214306

[52] A Serafini, OCO Dahlsten, D Gross, and MB Plenio. ``Canonical and micro-canonical typical entanglement of continuous variable systems''. Journal of Physics A: Mathematical and Theoretical 40, 9551 (2007).
https:/​/​doi.org/​10.1088/​1751-8113/​40/​31/​027

[53] Alessio Serafini, Oscar C. O. Dahlsten, and Martin B. Plenio. ``Teleportation Fidelities of Squeezed States from Thermodynamical State Space Measures''. Physical Review Letters 98, 170501 (2007).
https:/​/​doi.org/​10.1103/​PhysRevLett.98.170501

[54] Motohisa Fukuda and Robert Koenig. ``Typical entanglement for Gaussian states''. Journal of Mathematical Physics 60, 112203 (2019).
https:/​/​doi.org/​10.1063/​1.5119950

[55] Gerardo Adesso, Davide Girolami, and Alessio Serafini. ``Measuring Gaussian Quantum Information and Correlations Using the Rényi Entropy of Order 2''. Physical Review Letters 109, 190502 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.109.190502

[56] Giancarlo Camilo, Gabriel T. Landi, and Sebas Eliëns. ``Strong subadditivity of the Rényi entropies for bosonic and fermionic Gaussian states''. Physical Review B 99, 045155 (2019).
https:/​/​doi.org/​10.1103/​PhysRevB.99.045155

[57] V. Bužek, C. H. Keitel, and P. L. Knight. ``Sampling entropies and operational phase-space measurement. I. General formalism''. Physical Review A 51, 2575–2593 (1995).
https:/​/​doi.org/​10.1103/​PhysRevA.51.2575

[58] Gerardo Adesso and R Simon. ``Strong subadditivity for log-determinant of covariance matrices and its applications''. Journal of Physics A: Mathematical and Theoretical 49, 34LT02 (2016).
https:/​/​doi.org/​10.1088/​1751-8113/​49/​34/​34LT02

[59] Ludovico Lami, Christoph Hirche, Gerardo Adesso, and Andreas Winter. ``Schur Complement Inequalities for Covariance Matrices and Monogamy of Quantum Correlations''. Physical Review Letters 117, 220502 (2016).
https:/​/​doi.org/​10.1103/​PhysRevLett.117.220502

[60] Alessio Serafini. ``Quantum continuous variables: a primer of theoretical methods''. CRC Press (2017).

[61] F. C. Khanna, J. M. C. Malbouisson, A. E. Santana, and E. S. Santos. ``Maximum entanglement in squeezed boson and fermion states''. Physical Review A 76, 022109 (2007).
https:/​/​doi.org/​10.1103/​PhysRevA.76.022109

[62] Stasja Stanisic, Noah Linden, Ashley Montanaro, and Peter S. Turner. ``Generating entanglement with linear optics''. Physical Review A 96, 043861 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.96.043861

[63] Marko Petkovšek, Herbert S. Wilf, and Doron Zeilberger. ``A=B''. A K Peters (1996).

[64] W.N. Bailey. ``Generalized Hypergeometric Series, By W.N. Bailey''. Cambridge Tracts in Mathematics and Mathematical Physics, No. 32. Camrbridge University Press (1964). url: books.google.com/​books?id=TVyswgEACAAJ.
https:/​/​books.google.com/​books?id=TVyswgEACAAJ

[65] Wadim Zudilin. ``Hypergeometric heritage of W. N. Bailey''. Notices of the International Congress of Chinese Mathematicians 7, 32–46 (2019).
https:/​/​doi.org/​10.4310/​ICCM.2019.v7.n2.a4

[66] Lucy Joan Slater. ``Generalized hypergeometric functions''. Cambridge Univ. PressCambridge (1966).

[67] ``Hypergeometric2F1''. WolframResearch (2001). https:/​/​functions.wolfram.com/​HypergeometricFunctions/​Hypergeometric2F1/​03/​02/​0002/​.
https:/​/​functions.wolfram.com/​HypergeometricFunctions/​Hypergeometric2F1/​03/​02/​0002/​

[68] Joseph T. Iosue. ``GLO''. GitHub (2022). https:/​/​github.com/​jtiosue/​GLO.
https:/​/​github.com/​jtiosue/​GLO

[69] Don Weingarten. ``Asymptotic behavior of group integrals in the limit of infinite rank''. Journal of Mathematical Physics 19, 999–1001 (1978).
https:/​/​doi.org/​10.1063/​1.523807

[70] Benoit Collins. ``Moments and Cumulants of Polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability''. arXiv.math-ph/​0205010 (2002).
https:/​/​doi.org/​10.48550/​arXiv.math-ph/​0205010

[71] N. Alexeev, A. Pologova, and M. A. Alekseyev. ``Generalized Hultman Numbers and Cycle Structures of Breakpoint Graphs''. Journal of Computational Biology 24, 93–105 (2017).
https:/​/​doi.org/​10.1089/​cmb.2016.0190

[72] Max Alekseyev, Adam Ehrenberg, Joseph T. Iosue, and Alexey V. Gorshkov. ``Calculating entanglement in linear optics via breakpoint graphs''. In preparation.

[73] Ilki Kim. ``Rényi-$\alpha$ entropies of quantum states in closed form: Gaussian states and a class of non-Gaussian states''. Physical Review E 97, 062141 (2018).
https:/​/​doi.org/​10.1103/​PhysRevE.97.062141

[74] Mark Wilde and Kunal Sharma. ``PHYS 7895: Gaussian Quantum Information, Lecture 10'' (2019). https:/​/​markwilde.com/​teaching/​2019-spring-gqi/​scribe-notes/​lecture-10-scribed.pdf.
https:/​/​markwilde.com/​teaching/​2019-spring-gqi/​scribe-notes/​lecture-10-scribed.pdf

[75] Lucas Hackl and Eugenio Bianchi. ``Bosonic and fermionic Gaussian states from Kähler structures''. SciPost Physics Core 4, 025 (2021).
https:/​/​doi.org/​10.21468/​SciPostPhysCore.4.3.025

[76] Quntao Zhuang, Thomas Schuster, Beni Yoshida, and Norman Y Yao. ``Scrambling and complexity in phase space''. Physical Review A 99, 062334 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.99.062334

[77] Tianci Zhou and Xiao Chen. ``Non-unitary Entanglement Dynamics in Continuous Variable Systems''. Physical Review B 104, L180301 (2021). arXiv:2103.06507.
https:/​/​doi.org/​10.1103/​PhysRevB.104.L180301
arXiv:2103.06507

[78] Bingzhi Zhang and Quntao Zhuang. ``Entanglement formation in continuous-variable random quantum networks''. npj Quantum Information 7, 1–12 (2021).
https:/​/​doi.org/​10.1038/​s41534-021-00370-w

[79] Abhinav Deshpande, Bill Fefferman, Minh C. Tran, Michael Foss-Feig, and Alexey V. Gorshkov. ``Dynamical Phase Transitions in Sampling Complexity''. Physical Review Letters 121, 030501 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.121.030501

[80] Gopikrishnan Muraleedharan, Akimasa Miyake, and Ivan H. Deutsch. ``Quantum computational supremacy in the sampling of bosonic random walkers on a one-dimensional lattice''. New Journal of Physics 21, 055003 (2019).
https:/​/​doi.org/​10.1088/​1367-2630/​ab0610

[81] Changhun Oh, Youngrong Lim, Bill Fefferman, and Liang Jiang. ``Classical simulation of bosonic linear-optical random circuits beyond linear light cone''. arXiv:2102.10083 (2021).
https:/​/​doi.org/​10.48550/​ARXIV.2102.10083

[82] Nishad Maskara, Abhinav Deshpande, Adam Ehrenberg, Minh C. Tran, Bill Fefferman, and Alexey V. Gorshkov. ``Complexity Phase Diagram for Interacting and Long-Range Bosonic Hamiltonians''. Physical Review Letters 129, 150604 (2022).
https:/​/​doi.org/​10.1103/​PhysRevLett.129.150604

[83] Changhun Oh, Youngrong Lim, Bill Fefferman, and Liang Jiang. ``Classical Simulation of Boson Sampling Based on Graph Structure''. Physical Review Letters 128, 190501 (2022).
https:/​/​doi.org/​10.1103/​PhysRevLett.128.190501

[84] OEIS Foundation Inc. ``The On-Line Encyclopedia of Integer Sequences'' (2022). Published electronically at http:/​/​oeis.org.
http:/​/​oeis.org

[85] Ewan Delanoy. ``Determinant of matrix defined by binomial coefficient''. Mathematics Stack Exchange (2017). https:/​/​math.stackexchange.com/​q/​2277633.
https:/​/​math.stackexchange.com/​q/​2277633

[86] Milton Abramowitz and Irene A. Stegun, editors. ``Handbook of mathematical functions: with formulas, graphs, and mathematical tables''. Dover books on mathematics. Dover Publ (2013).

[87] Darij Grinberg. ``A hyperfactorial divisibility''. Darij Grinberg homepage.
http:/​/​www.cip.ifi.lmu.de/​~grinberg/​hyperfactorialBRIEF.pdf

[88] Motohisa Fukuda, Robert König, and Ion Nechita. ``RTNI—A symbolic integrator for Haar-random tensor networks''. Journal of Physics A: Mathematical and Theoretical 52, 425303 (2019).
https:/​/​doi.org/​10.1088/​1751-8121/​ab434b

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