Quantum control of nuclear magnetic resonance spin systems

Li Jun; Cui Jiang-Yu; Yang Xiao-Dong; Luo Zhi-Huang; Pan Jian; Yu Qi; Li Zhao-Kai; Peng Xin-Hua; Du Jiang-Feng

doi:10.7498/aps.64.167601

Abstract

With the development of quantum information science, the active manipulation of quantum systems is becoming an important research frontier. To build realistic quantum information processors, one of the challenges is to implement arbitrary desired operations with high precision on quantum systems. A large number of quantum control methods and relevant numerical techniques have been put forward in recent years, such as quantum optimal control and quantum feedback control. Nuclear magnetic resonance (NMR) spin systems offer an excellent testbed to develop benchmark tools and techniques for controlling quantum systems. In this review paper, we briefly introduce some of the basic control ideas developed for NMR systems in recent years. We first explain, for the liquid spin systems, the physics of various couplings and the causes of relaxation effects. These mechanisms govern the system dynamics, and thus are crucial for constructing rigorous and efficient control models. We also identify three types of available control means: 1) raido-frequency fields as coherent controls; 2) phase cycling, gradient fields and relaxation effects as non-unitary controls; 3) radiation damping effect as feedback control mechanism. Then, we elucidate some important control tasks, which may arise from the conventional NMR spectroscopy (e.g., pulse design and polarization transfer) or from quantum information science (e.g., algorithmic cooling and pseudo-pure state preparation). In the last part, we review some of the most important control methods that are applicable to NMR control tasks. For systems with a relatively small number of spins, it is possible to use analytic optimal control theory to realize the target unitary operations. However, for larger systems, numerical methods are necessary. The gradient ascent pulse engineering algorithm and pulse compiler techniques are the most successful techniques for implementing complicated quantum networks currently. There are some interesting topics of utilizing radiation damping and relaxation effects to achieve more powerful controls. Finally, we give an outline of the possible future work.

Keywords:

Authors and contacts

1.
Hefei National Laboratory for Physical Sciences at Microscale, Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China;
2.
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China
Funds: Project supported by the National Basic Research Program of China (Grant Nos. 2013CB921800, 2014CB848700), the National Natural Science Fund for Distinguished Young Scholars of China (Grant No. 11425523), the National Natural Science Foundation of China (Grant Nos. 11375167, 11227901, 91021005), the Strategic Priority Research Program (B) of Chinese Academy of Sciences (Grant No. XDB01030400), and the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20113402110044).

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Cited By

[1]	Wiseman H M, Milburn G J 2010 Quantum Measurement and Control (Cambridge: Cambridge University Press)
[2]	Dong D, Petersen I R 2010 IET Control Theory Appl. 4 2651
[3]	Alessandro D D 2007 Introduction to Quantum Control and Dynamics (London: Chapman & Hall)
[4]	Brif C, Chakrabarti R, Rabitz H 2010 New J. Phys. 12 075008
[5]	Feynman R P 1982 Int. J. Theor. Phys. 21 467
[6]	Nielsen M A, Chuang I L 2010 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press)
[7]	Vandersypen L M K 2001 Ph. D. Dissertation (Stanford: Stanford University)
[8]	Vandersypen L M K, Chuang I L
[9]	Ryan C A, Negrevergne C, Laforest M, Knill E, Laflamme R 2008 Phys. Rev. A 78 012328
[10]	Chuang I, Vandersypen L, Zhou X, Leung D, Lloyd S 1998 Nature 393 143
[11]	Wu Z, Li J, Zheng W Q, Luo J, Feng M, Peng X H 2011 Phys. Rev. A 84 042312
[12]	Jones J, Mosca M, Hansen R 1998 Nature 393 344
[13]	Vandersypen L M K, Steffen M, Breyta G, Yannoni C S, Sherwood M H, Chuang I L 2001 Nature 414 883
[14]	Lu D W, Zhu J, Zou P, Peng X H, Yu Y H, Zhang S M, Chen Q, Du J F 2010 Phys. Rev. A 81 022308
[15]	Du J F, Xu N Y, Peng X H, Wang P F, Wu S F, Lu D W 2010 Phys. Rev. Lett. 104 030502
[16]	Peng X H, Wu S F, Li J, Suter D, Du J F 2010 Phys. Rev. Lett. 105 240405
[17]	Feng G R, Long G L, Laflamme R 2013 Phys. Rev. A 88 022305
[18]	Laflamme R, Knill E, Zurek W, Catasti P, Mariappan S 1998 Philos. Trans. R. Soc. London Ser. A 356 1941
[19]	Cory D G, Price M D, Maas W, Knill E, Laflamme R, Zurek W H, Havel T F, Somaroo S S 1998 Phys. Rev. Lett. 81 2152
[20]	Negrevergne C, Mahesh T S, Ryan C A, Ditty M, Cyr-Racine F, Power W, Boulant N, Havel T, Cory D G, Laflamme R 2006 Phys. Rev. Lett. 96 170501
[21]	Jones J A 2011 Prog. Nucl. Mag. Res. Sp. 59 91
[22]	Lu D W, Li H, Trottier D A, Li J, Brodutch A, Krismanich A P, Ghavami A, Dmitrienko G I, Long G L, Baugh J, Laflamme R 2015 Phys. Rev. Lett. 114 140505
[23]	Dirac P A M 1958 The Principles of Quantum Mechanics (Oxford: Oxford University Press)
[24]	Rivas Á, Huelga S F 2012 Open Quantum Systems: An Introduction (New York: Springer-Verlag)
[25]	Heinosaari T, Ziman M 2012 The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement (Cambridge: Cambridge University Press)
[26]	Hayashi M 2006 Quantum Information: An Introduction (New York: Springer-Verlag)
[27]	Barenco A, Bennett C H, Cleve R, DiVincenzo D P, Margolus N, Shor P, Sleator T, Smolin J A, Weinfurter H 1995 Phy. Rev. A 52 3457
[28]	Breuer H P, Petruccione F 2002 The Theory of Open Quantum Systems (Oxford: Oxford University Press)
[29]	Schrödinger E 1926 Phys. Rev. 28 1049
[30]	Kraus K 1983 States, Effects and Operations: Fundamental Notions of Quantum Theory (New York: Springer-Verlag)
[31]	Lindblad G 1976 Commun. Math. Phys. 48 119
[32]	Weiss U 1999 Quantum Dissipative Systems (Singapore: World Scientific)
[33]	Laforest M 2008 Ph. D. Dissertation (Waterloo: University of Waterloo)
[34]	Andersson E, Cresser J D, Hall M J W 2007 J. Mod. Opt. 54 1695
[35]	Werschnik J, Gross E K U J
[36]	Elliott D L 2009 Bilinear Control Systems: Matrices in Action (New York: Springer-Verlag)
[37]	Fouquieres P, Schirmer S G 2010 arXiv:1004.3492v1 [quant-ph]
[38]	Stengel R F 1994 Optimal Control and Estimation (New York: Dover Publications)
[39]	Ramakrishna V, Rabitz H 1996 Phys. Rev. A 54 1715
[40]	Albertini F, D'Alessandro D 2003 IEEE Trans. Autom. Control 48 1399
[41]	Ibort A, Pérez-Pardo J M 2009 Phys. A: Math. Theor. 42 205301
[42]	Levitt M H 2008 Spin Dynamics: Basics of Nuclear Magnetic Resonance (England: John Wiley & Sons Ltd)
[43]	Smith S A, Palke W E, Gerig J T 1992 Concepts in Magnetic Resonance 4 107
[44]	Smith S A, Palke W E, Gerig J T 1992 Concepts in Magnetic Resonance 4 181
[45]	Smith S A, Palke W E, Gerig J T
[46]	Smith S A, Palke W E, Gerig J T
[47]	Kowalewski J Mäler L 2006 Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications (Boca Raton: Taylor & Francis Group)
[48]	Freeman R 1998 Prog. Nucl. Mag. Res. Sp. 32 59
[49]	Bauer C, Freeman R, Frenkiel T, Keeler J, Shaka A J
[50]	Temps A J, Brewer C F
[51]	Warren W S 1984 J. Chem. Phys. 81 5437
[52]	Peng X H, Zhu X W, Fang X M, Feng M, Liu M L, Gao K L 2004 J. Chem. Phys. 120 3579
[53]	Silver M S, Joseph R I, Hoult D I 1984 J. Magn. Reson. 59 347
[54]	Geen H, Freeman R 1991 J. Magn. Reson. 93 93
[55]	Steffen M, Vandersypen L, Chuang I 2000 J. Magn. Reson. 146 369
[56]	Elster A D 1993 Radiology 186 1
[57]	Bloembergen N, Pound R V 1954 Phys. Rev. 95 1
[58]	Szöke A, Meiboom S
[59]	Altafini C, Cappellaro P, Cory D 2010 Systems & Control Letters 59 782
[60]	Mao X A, Ye C H 1997 Concepts Magn. Reson. 9 173
[61]	Born M, Fock V A
[62]	Farhi E, Goldstone J, Gutmann S, Lapan J, Lundgren A, Preda D 2001 Science 292 472
[63]	Mizel A, Lidar D A, Mitchell M 2007 Phys. Rev. Lett. 99 070502
[64]	Amin M H S 2008 Phys. Rev. Lett. 100 130503
[65]	Roland J, Cerf N J 2002 Phys. Rev. A 65 042308
[66]	Peng X H, Liao Z Y, Xu N Y, Qin G, Zhou X Y, Suter D, Du J F 2008 Phys. Rev. Lett. 101 220405
[67]	Peng X H, Zhang J F, Du J F, Suter D 2009 Phys. Rev. Lett. 103 140501
[68]	Steffen M, van Dam W, Hogg T, Breyta G, Chuang I 2003 Phys. Rev. Lett. 90 067903
[69]	Mitra A, Mitra A, Ghosh A, Das R, Patel A, Kumar A 2005 J. Magn. Res. 177 285
[70]	Xu N Y, Zhu J, Lu D W, Zhou X Y, Peng X H, Du J F 2012 Phys. Rev. Lett. 108 130501
[71]	Garey M R, Johnson D S 1979 Computers and Intractability: A Guide to the Theory of NP-Completeness (San Francisco: Freeman)
[72]	Žnidarič M, Horvat M 2006 Phys. Rev. A 73 022329
[73]	Hogg T 2003 Phys. Rev. A 67 022314
[74]	Žnidarič M 2005 Phys. Rev. A 71 062305
[75]	Peng X H, Du J F, Dieter S 2005 Phys. Rev. A 71 012307
[76]	Li Z K, Zhou H, Ju C Y, Chen H W, Zheng W Q, Lu D W, Rong X, Duan C K, Peng X H, Du J F 2014 Phys. Rev. Lett. 112 220501
[77]	Peng X H, Luo Z H, Zheng W Q, Kou S P, Suter D, Du J F 2014 Phys. Rev. Lett. 113 080404
[78]	Gershenfeld N, Chuang I L 1997 Science 275 350
[79]	Knill E, Chuang I, Laflamme R 1998 Phys. Rev. A 57 3348
[80]	Cory D G, Price M D, Havel T F 1998 Physica D 120 82
[81]	Schulman L J, Mor T, Weinstein Y 2005 Phys. Rev. Lett. 94 120501
[82]	Ryan C A, Moussa O, Baugh J, Laflamme R 2008 Phys. Rev. Lett. 100 140501
[83]	Fortunato E M, Pravia M A, Boulant N, Teklemariam G, Havel T F, Cory D G 2002 J. Chem. Phys. 116 7599
[84]	Jeffries C D 1963 Dynamic Nuclear Orientation (New York: Wiley)
[85]	Muus L T, Atkins P W, McLauchlan K A, Pedersen J B 1977 Chemically Induced Magnetic Polarization (Dordrecht: D. Reidel)
[86]	Li J, Lu D W, Luo Z H, Laflamme R, Peng X H, Du J F 2014 arXiv:1412.4146v1
[87]	Ernst R R, Bodenhausen G, Wokaun A 1987 Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford: Oxford University Press)
[88]	Sorensen O W 1989 Progress in NMR Spectroscopy 21 503
[89]	Sorensen O W
[90]	Stoustrup J, Schedletzky O, Glaser S J, Griesinger C, Nielsen N C, Sorensen O W 1995 Phys. Rev. Lett. 74 2921
[91]	Horn R A, Johnson C R 2013 Matrix Analysis (Cambridge: Cambridge University Press)
[92]	Waugh J S, Huber L M, Haeberlen U 1968 Phys. Rev. Lett. 20 180
[93]	Mansfield P 1971 J. Phys. C: Solid State Physics 4 1444
[94]	Burum D P, Linder M, Ernst R R
[95]	Uhrig G S 2007 Phys. Rev. Lett. 98 100504
[96]	Maudsley A A
[97]	Khodjasteh K, Lidar D A 2007 Phys. Rev. A 75 062310
[98]	Khaneja N, Brockett R, Glaser S J 2001 Phys. Rev. A 63 032308
[99]	Khaneja N, Glaser S J 2001 Chem. Phys. 267 11
[100]	Li B, Yu Z H, Fei S M, Li-Jost X Q 2013 Sci. China: Phys. Mech. Astron. 56 2116
[101]	Bonnard B, Sugny D
[102]	Boulant N, Edmonds K, Yang J, Pravia M A, Cory D G 2003 Phys. Rev. A 68 032305
[103]	Khaneja N, Reiss T, Kehlet C, Herbrüggen T S, Glaser S J 2005 J. Magn. Reson. 172 296
[104]	Rabitz A H, Hsieh M M, Rosenthal C M 2004 Science 303 1998
[105]	Ho T S, Dominy J, Rabitz H 2009 Phys. Rev. A 79 013422
[106]	Hsieh M, Rabitz H 2008 Phys. Rev. A 77 042306
[107]	Hsieh M, Rabitz H, Lidar D 2010 Phys. Rev. A 81 062352
[108]	Knill E, Laflamme R, Martinez R, Tseng C H 2000 Nature 404 368
[109]	Wiseman H M
[110]	Judson R S, Rabitz H
[111]	Back T 1996 Evolutionary Algorithms in Theory and Practice (Oxford: Oxford University Press)
[112]	Belavkin V P
[113]	Wiseman H M, Milburn G J 1993 Phys. Rev. Lett. 70 548
[114]	Wiseman H M 1994 Phys. Rev. A 49 2133
[115]	Chen C L, Wang L C, Wang Y Y
[116]	Huang S Y, Witzel T, Wald L L 2008 Magnet. Reson. Med. 60 5
[117]	Huang S Y, Chung A P, Lin Y Y
[118]	Huang S Y, Yang S S, Lin Y Y
[119]	Altafini C 2007 Quantum Information Processing 6 1
[120]	Jäger G 2014 Phys. Rev. A 90 033628
[121]	Said R S, Twamley J 2009 Phys. Rev. A 80 032303
[122]	Spörl A, Schulte-Herbrüggen T, Glaser S J 2007 Phys. Rev. A 75 012302

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Vol. 64, Issue 16 - 2015-08-20

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