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Universal quantum computation by scattering in the Fermi-Hubbard model. (English) Zbl 1448.81222

Summary: The Hubbard model may be the simplest model of particles interacting on a lattice, but simulation of its dynamics remains beyond the reach of current numerical methods. In this article, we show that general quantum computations can be encoded into the physics of wave packets propagating through a planar graph, with scattering interactions governed by the fermionic Hubbard model. Therefore, simulating the model on planar graphs is as hard as simulating quantum computation. We give two different arguments, demonstrating that the simulation is difficult both for wave packets prepared as excitations of the fermionic vacuum, and for hole wave packets at filling fraction one-half in the limit of strong coupling. In the latter case, which is described by the \(t\)-\(J\) model, there is only reflection and no transmission in the scattering events, as would be the case for classical hard spheres. In that sense, the construction provides a quantum mechanical analog of the Fredkin-Toffoli billiard ball computer.

MSC:

81P68 Quantum computation
81V74 Fermionic systems in quantum theory
82C22 Interacting particle systems in time-dependent statistical mechanics
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[1] Gharibian S, Huang Y and Landau Z 2014 arXiv:1401.3916
[2] Trotter H F 1959 Proc. Am. Math. Soc.10 545 · Zbl 0099.10401 · doi:10.1090/S0002-9939-1959-0108732-6
[3] Kitaev A Y 1995 arXiv:quant-ph/9511026
[4] Cleve R, Ekert A, Macchiavello C and Mosca M 1998 Proc. R. Soc. A 454 339 · Zbl 0915.68050 · doi:10.1098/rspa.1998.0164
[5] Kitaev A Y, Shen A and Vyalyi M N 2002 Classical and Quantum Computation vol 47 (Providence, RI: American Mathematical Society) · Zbl 1022.68001
[6] Aharonov D, Gottesman D, Irani S and Kempe J 2009 Commun. Math. Phys.287 41 · Zbl 1180.68162 · doi:10.1007/s00220-008-0710-3
[7] Porras D and Cirac J 2004 Phys. Rev. Lett.92 207901 · doi:10.1103/PhysRevLett.92.207901
[8] Jordan S P, Lee K S and Preskill J 2012 Science336 1130 · doi:10.1126/science.1217069
[9] Marzuoli A and Rasetti M 2005 Ann. Phys., NY318 345 · Zbl 1072.81013 · doi:10.1016/j.aop.2005.01.005
[10] Terhal B M and DiVincenzo D P 2002 Phys. Rev. A 65 032325 · doi:10.1103/PhysRevA.65.032325
[11] Aaronson S and Arkhipov A 2011 Proc. 43rd Annual ACM Symp. on Theory of Computing pp 333-42 · Zbl 1288.68066
[12] Childs A M, Gosset D and Webb Z 2013 Science339 791 · Zbl 1355.68101 · doi:10.1126/science.1229957
[13] Emery V, Kivelson S and Lin H 1990 Phys. Rev. Lett.64 475 · doi:10.1103/PhysRevLett.64.475
[14] Essler F, Frahm H, Göhmann F, Klümper A and Korepin V 2005 The One-Dimensional Hubbard Model (Cambridge: Cambridge University Press) · Zbl 1107.82014 · doi:10.1017/CBO9780511534843
[15] Spałek J 2007 Acta Phys. Pol. A 111 409 · doi:10.12693/APhysPolA.111.409
[16] Yao A C-C 1993 Proc. 34th Annual Symp. on Foundations of Computer Science (IEEE) pp 352-61 · Zbl 0850.68166
[17] Briegel H, Browne D, Dür W, Raussendorf R and van den Nest M 2009 Nat. Phys.5 19 · doi:10.1038/nphys1157
[18] Farhi E, Goldstone J, Gutmann S and Sipser M 2000 arXiv:quant-ph/0001106
[19] Farhi E, Goldstone J, Gutmann S, Lapan J, Lundgren A and Preda D 2001 Science292 472 · Zbl 1226.81046 · doi:10.1126/science.1057726
[20] Freedman M, Kitaev A, Larsen M and Wang Z 2003 Bull. Am. Math. Soc.40 31 · Zbl 1019.81008 · doi:10.1090/S0273-0979-02-00964-3
[21] Kielpinski D, Monroe C and Wineland D J 2002 Nature417 709 · doi:10.1038/nature00784
[22] Monroe C 2002 Nature416 238 · doi:10.1038/416238a
[23] Loss D and DiVincenzo D P 1998 Phys. Rev. A 57 120 · doi:10.1103/PhysRevA.57.120
[24] Blais A, Huang R-S, Wallraff A, Girvin S and Schoelkopf R J 2004 Phys. Rev. A 69 062320 · doi:10.1103/PhysRevA.69.062320
[25] Ionicioiu R and Zanardi P 2002 Phys. Rev. A 66 050301 · doi:10.1103/PhysRevA.66.050301
[26] DiVincenzo D P, Bacon D, Kempe J, Burkard G and Whaley K B 2000 Nature408 339 · doi:10.1038/35042541
[27] Childs A M and Gosset D 2012 J. Math. Phys.53 102207 · Zbl 1278.81093 · doi:10.1063/1.4757665
[28] Childs A M, Gosset D, Nagaj D, Raha M and Webb Z 2015 Quant. Inf. Comput.15 601-21
[29] Świerczkowski S 1958 Fundam. Math.46 187 · Zbl 0085.27203 · doi:10.4064/fm-46-2-187-189
[30] Knuth D 1998 The Art of Computer Programming Sorting and Searching vol 3 (Reading, MA: Addison-Wesley) · Zbl 0895.65001
[31] Miller S J and Takloo-Bighash R 2006 An Invitation to Modern Number Theory vol 13 (Princeton, NJ: Princeton University Press) · Zbl 1155.11001
[32] Fredkin E and Toffoli T 1982 Int. J. Theor. Phys.21 219 · Zbl 0496.94015 · doi:10.1007/BF01857727
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