On the time-independent Hamiltonian in real-time and imaginary-time quantum annealing

doi:10.1088/1674-1056/aba2db

Abstract

We present the analog analogue of Grover’s problem as an example of the time-independent Hamiltonian for applying the speed limit of the imaginary-time Schrödinger equation derived by Okuyama and Ohzeki and the new class of energy-time uncertainty relation proposed by Kieu. It is found that the computational time of the imaginary-time quantum annealing of this Grover search can be exponentially small, while the counterpart of the quantum evolution driven by the real-time Schrödinger equation could only provide square root speedup, compared with classic search. The present results are consistent with the cases of the time-dependent quantum evolution of the natural Grover problem in previous works. We once again emphasize that the logarithm and square root algorithmic performances are generic in imaginary-time quantum annealing and quantum evolution driven by real-time Schrödinger equation, respectively. Also, we provide evidences to search deep reasons why the imaginary-time quantum annealing can lead to exponential speedup and the real-time quantum annealing can make square root speedup.

Keywords: time-independent Hamiltonian imaginary-time quantum annealing quantum speed limit the energy-time uncertainty relation continuous Grover’s search

Received: 04 March 2020
Revised: 26 May 2020
Accepted manuscript online: 06 July 2020

PACS:	03.67.Ac	(Quantum algorithms, protocols, and simulations)
	03.67.-a	(Quantum information)

Corresponding Authors: ^†E-mail: sunjie_hust@sina.com ^‡Corresponding author. E-mail: lusongfeng@hotmail.com

About author:

†Corresponding author. E-mail: sunjie_hust@sina.com

‡Corresponding author. E-mail: lusongfeng@hotmail.com

* Project supported by the China Postdoctoral Science Foundation (Grant No. 2017M620322), the Priority Fund for the Postdoctoral Scientific and Technological Program of Hubei Province in 2017, the Seed Foundation of Huazhong University of Science and Technology (Grant No. 2017KFYXJJ070), and the Science and Technology Program of Shenzhen of China (Grant No. JCYJ 20180306124612893).

Cite this article:

Jie Sun(孙杰)† and Songfeng Lu(路松峰)‡ On the time-independent Hamiltonian in real-time and imaginary-time quantum annealing 2020 Chin. Phys. B 29 100303

[1]	Bengtsson I, Zyczkowski K 2008 Geometry of Quantum States: An Introduction to Quantum Entanglement Cambridge Cambridge University Press 419
[2]	Levitin L B, Toffoli T 2009 Phys. Rev. Lett. 103 160502 DOI: 10.1103/PhysRevLett.103.160502
[3]	Mandelstam L, Tamm I 1991 Selected Papers Berlin Springer 115 123
[4]	Mulga G, Mayato R S, Egusquiza I 2008 Time in Quantum Mechanics 2 Berlin Springer 73 105
[5]	Aharonov Y, Bohm D 1961 Phys. Rev. 122 1649 DOI: 10.1103/PhysRev.122.1649
[6]	Anandan J, Aharonov Y 1990 Phys. Rev. Lett. 65 1697 DOI: 10.1103/PhysRevLett.65.1697
[7]	Vaidman L 1991 Am. J. Phys. 60 182 DOI: 10.1119/1.16940
[8]	Uffink J 1993 Am. J. Phys. 61 935 DOI: 10.1119/1.17368
[9]	Brody D C 2003 J. Phys. A: Math. Gen. 36 5587 DOI: 10.1088/0305-4470/36/20/314
[10]	Giovannetti V, Lloyd S, Maccone L 2003 Phys. Rev. A 67 052109 DOI: 10.1103/PhysRevA.67.052109
[11]	Giovannetti V, Lloyd S, Maccone L 2004 J. Opt. B: Quantum Semiclass. Opt. 6 S807 DOI: 10.1088/1464-4266/6/8/028
[12]	Bures D J C 1969 Trans. Am. Math. Soc. 135 199 DOI: 10.1090/S0002-9947-1969-0236719-2
[13]	Kieu T D 2019 Proc. R. Soc. Lond A 475 20190148 DOI: 10.1098/rspa.2019.0148
[14]	Farhi E, Goldstone J, Gutmann S, Sipser M 2000 arXiv:quant-ph/0001106v1 [quant-ph]
[15]	Farhi E, Goldstone J, Gutmann S, Lapan J, Lundgren A, Preda D 2001 Science 292 472 DOI: 10.1126/science.1057726
[16]	Grover L K 1997 Phys. Rev. Lett. 79 325 DOI: 10.1103/PhysRevLett.79.325
[17]	Okuyama M, Ohzeki M 2018 arXiv:1806.09040 [quant-ph]
[18]	Okada S, Ohzeki M, Tanaka K 2019 J. Phys. Soc. Jpn. 88 024803 DOI: 10.7566/JPSJ.88.024803
[19]	Farhi E, Gutmann S 1998 Phys. Rev. A 57 2403 DOI: 10.1103/PhysRevA.57.2403
[20]	Roland J, Cerf N J 2002 Phys. Rev. A 65 042308 DOI: 10.1103/PhysRevA.65.042308
[21]	Bender C M, Brody D C, Jones H F, Meister B K 2007 Phys. Rev. Lett. 98 040403 DOI: 10.1103/PhysRevLett.98.040403
[22]	Anandan J, Aharonov Y 1990 Phys. Rev. Lett. 65 1697 DOI: 10.1103/PhysRevLett.65.1697
[23]	Andrecut M, Ali M K 2004 Int. J. Theor. Phys. 43 969 DOI: 10.1023/B:IJTP.0000048594.07696.e5
[24]	Saurya D, Randy K, Gabor K 2003 J. Phys. A: Math. Gen. 36 2839 DOI: 10.1088/0305-4470/36/11/313
[25]	Galve F, Lutz E 2009 Phys. Rev. A 79 055804 DOI: 10.1103/PhysRevA.79.055804
[26]	Sebastian D, Eric L 2008 Phys. Rev. E 77 021128 DOI: 10.1103/PhysRevE.77.021128
[27]	Dominik S W, Sarang G, Michael K, Norman Y Y, Mikhail D L 2016 Phys. Rev. Lett. 117 150501 DOI: 10.1103/PhysRevLett.117.150501
[28]	Kaneko K, Nishimori H 2015 J. Phys. Soc. Jpn. 84 094001 DOI: 10.7566/JPSJ.84.094001
[29]	Hu F, Wang B N, Wang N, Wang C 2019 Quantum Engineering 1 e12 DOI: 10.1002/que2.12
[30]	Zhang Y, Ni Q 2020 Quantum Engineering 2 e34 DOI: 10.1002/que2.34
[31]	Li F G, Bao W, Zhang S, Wang X, Huang H L, Li T, Ma B W 2018 Chin. Phys. B 27 010308 DOI: 10.1088/1674-1056/27/1/010308
[32]	Zhang S, Duan Q H, Li T, Fu X Q, Huang H L, Wang X, Bao W S 2020 Chin. Phys. B 29 010308 DOI: 10.1088/1674-1056/ab5f02
[33]	Sun J, Lu S 2018 Chin. Phys. B 27 110306 DOI: 10.1088/1674-1056/27/11/110306
[34]	Carlini A, Hosoya A, Koike T, Okudaira Y 2006 Phys. Rev. Lett. 96 060503 DOI: 10.1103/PhysRevLett.96.060503
[35]	Gyongyosi L 2020 Quantum Engineering 2 e30 DOI: 10.1002/que2.30
[36]	Adler R J, Santiago D I 1999 Mod. Phys. Lett. A 14 1371 DOI: 10.1142/S0217732399001462
[37]	Scardigli F 1999 Phys. Lett. B 452 39 DOI: 10.1016/S0370-2693(99)00167-7

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