Hybrid quantum-classical control problems

Emanuel-Cristian Boghiu; Jesús Clemente-Gallardo; Jorge A. Jover-Galtier; David Martínez-Crespo; Emanuel-Cristian Boghiu; Jesús Clemente-Gallardo; Jorge A. Jover-Galtier; David Martínez-Crespo

doi:10.3934/cam.2024034

Communications in Analysis and Mechanics

2024, Volume 16, Issue 4: 786-812. doi: 10.3934/cam.2024034

Previous Article Next Article

Research article Special Issues

Hybrid quantum-classical control problems

1.
ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels (Barcelona) 08860, Spain
2.
Department of Theoretical Physics, University of Zaragoza, c/Pedro Cerbuna 12, 50009 Zaragoza, Spain
3.
Department of Applied Mathematics & IUMA, University of Zaragoza, c/María de Luna 3, 50018 Zaragoza, Spain
4.
Center for Astroparticles and High Energy Physics, University of Zaragoza, c/Pedro Cerbuna 12, 50009 Zaragoza, Spain

Received: 29 February 2024 Revised: 30 July 2024 Accepted: 03 September 2024 Published: 28 October 2024
93B05, 93B29

The notion of hybrid quantum-classical control system was introduced as a control dynamical system which combined classical and quantum degrees of freedom. Classical and quantum objects were combined within a geometrical description of both types of systems. We also considered the notion of hybrid quantum-classical controllability by means of the usual definitions of geometric control theory, and we discussed how the different concepts associated to quantum controllability are lost in the hybrid context because of the nonlinearity of the dynamics. We also considered several examples of physically relevant problems, such as the spin-boson model or the notion of hybrid spline.
- hybrid control system,
- classical controllability,
- quantum controllability,
- hybrid controllability,
- splines
Citation: Emanuel-Cristian Boghiu, Jesús Clemente-Gallardo, Jorge A. Jover-Galtier, David Martínez-Crespo. Hybrid quantum-classical control problems[J]. Communications in Analysis and Mechanics, 2024, 16(4): 786-812. doi: 10.3934/cam.2024034

Related Papers:

Abstract

The notion of hybrid quantum-classical control system was introduced as a control dynamical system which combined classical and quantum degrees of freedom. Classical and quantum objects were combined within a geometrical description of both types of systems. We also considered the notion of hybrid quantum-classical controllability by means of the usual definitions of geometric control theory, and we discussed how the different concepts associated to quantum controllability are lost in the hybrid context because of the nonlinearity of the dynamics. We also considered several examples of physically relevant problems, such as the spin-boson model or the notion of hybrid spline.

References

[1]	J. C. Maxwell, I. On govenors, Proc. R. Soc. London, 16 (1868), 270–283. https://doi.org/10.1098/rspl.1867.0055
[2]	V. Jurdjevic Geometric Control Theory, Cambridge University Press, 1996. https://doi.org/10.1017/CBO9780511530036
[3]	H. J. Sussmann, J. C. Willems, 300 years of optimal control: from the brachystochrone to the maximum principle, IEEE Control Syst Mag., 17 (1997), 32–44. https://doi.org/10.1109/37.588098 doi: 10.1109/37.588098
[4]	G. M. Huang, T. J. Tarn, J. W. Clark, On the controllability of quantum-mechanical systems, J. Math. Phys., 24 (1983), 2608–2618. https://doi.org/10.1063/1.525634 doi: 10.1063/1.525634
[5]	J. Werschnik, E. K. U. Gross, Quantum optimal control theory, J. Phys. B: At. Mol. Opt. Phys., 40 (2007), R175. https://doi.org/10.1088/0953-4075/40/18/R01 doi: 10.1088/0953-4075/40/18/R01
[6]	A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, et al., Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses, Science, 282 (1998), 919–922. https://doi.org/10.1126/science.282.5390.919 doi: 10.1126/science.282.5390.919
[7]	R. J. Levis, G. M. Menkir, H. Rabitz, Selective bond dissociation and rearrangement with optimally tailored, strong-field laser pulses, Science, 292 (2001), 709–713. https://doi.org/10.1126/science.1059133 doi: 10.1126/science.1059133
[8]	M. M. Wilde, Quantum Information Theory, Cambridge University Press, 2017. https://doi.org/10.1017/9781316809976
[9]	M. Grifoni, P. Hänggi, Driven quantum tunneling, Phys. Rep., 304 (1998), 229–354. https://doi.org/10.1016/S0370-1573(98)00022-2 doi: 10.1016/S0370-1573(98)00022-2
[10]	N. Khaneja, R. Brockett, S. J. Glaser, Time optimal control in spin systems, Phys. Rev. A, 63 (2001), 032308. https://doi.org/10.1103/PhysRevA.63.032308 doi: 10.1103/PhysRevA.63.032308
[11]	N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, S. J. Glaser, Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms, J. Magn. Reson., 172 (2005), 296–305. https://doi.org/10.1016/j.jmr.2004.11.004 doi: 10.1016/j.jmr.2004.11.004
[12]	D. J. Tannor, A. Bartana, On the interplay of control fields and spontaneous emission in laser cooling, J. Phys. Chem. A, 103 (1999), 10359–10363. https://doi.org/10.1021/jp992544x doi: 10.1021/jp992544x
[13]	N. Weaver, Time-optimal control of finite quantum systems, J. Math. Phys., 41 (2000), 5262–5269. https://doi.org/10.1063/1.533407 doi: 10.1063/1.533407
[14]	W. Zhu, J. Botina, H. Rabitz, Rapidly convergent iteration methods for quantum optimal control of population, J. Chem. Phys., 108 (1998), 1953–1963. https://doi.org/10.1063/1.475576 doi: 10.1063/1.475576
[15]	F. Albertini, D. D'Alessandro, Notions of controllability for quantum mechanical systems, Proc. 40th IEEE Conference on Decision and Control, 2 (2001), 1589–1594. https://doi.org/10.1109/CDC.2001.981126 doi: 10.1109/CDC.2001.981126
[16]	C. Altafini, Controllability properties for finite dimensional quantum Markovian master equations, J. Math. Phys., 44 (2003), 2357–2372. https://doi.org/10.1063/1.1571221 doi: 10.1063/1.1571221
[17]	C. Altafini, F. Ticozzi, Modeling and control of quantum systems: an introduction, IEEE Trans. Automat. Contr., 57 (2012), 1898–1917. https://doi.org/10.1109/TAC.2012.2195830 doi: 10.1109/TAC.2012.2195830
[18]	I. Kurniawan, G. Dirr, U. Helmke, Controllability aspects of quantum dynamics: a unified approach for closed and open systems, IEEE Trans. Automat. Contr., 57 (2012), 1984–1996. https://doi.org/10.1109/TAC.2012.2195870 doi: 10.1109/TAC.2012.2195870
[19]	L. Abrunheiro, M. Camarinha, J. Clemente-Gallardo, J. C. Cuchí, P. Santos, A general framework for quantum splines, Int. J. Geom. Methods Mod. Phys., 15 (2018), 1850147. https://doi.org/10.1142/S0219887818501475 doi: 10.1142/S0219887818501475
[20]	D. C. Brody, D. D. Holm, D. M. Meier, Quantum splines, Phys. Rev. Lett., 109 (2012), 100501. https://doi.org/10.1103/PhysRevLett.109.100501
[21]	A. Garon, S. J. Glaser, D. Sugny, Time-optimal control of $SU(2)$ quantum operations, Phys. Rev. A, 88 (2013), 043422. https://doi.org/10.1103/PhysRevA.88.043422 doi: 10.1103/PhysRevA.88.043422
[22]	F. A. Bornemann, P. Nettesheim, C. Schütte, Quantum-classical molecular dynamics as an approximation to full quantum dynamics, J. Chem. Phys., 105 (1996), 1074–1083. https://doi.org/10.1063/1.471952 doi: 10.1063/1.471952
[23]	D. C. Rapaport, The art of molecular dynamics simulation, Cambridge University Press, 2004. https://doi.org/10.1017/CBO9780511816581
[24]	J. C. Tully, Mixed quantum-classical dynamics, Faraday Discussions, 110 (1998), 407–419. https://doi.org/10.1039/a801824c doi: 10.1039/a801824c
[25]	U. Weiss, Quantum dissipative systems, World Scientific, 2008. https://doi.org/10.1142/6738
[26]	R. Abraham. J. E. Marsden, Foundations of Mechanics, 2nd ed., Addison-Wesley, 1978.
[27]	V. I. Arnold, Mathematical methods of classical mechanics, Springer New York, 1989. https://doi.org/10.1007/978-1-4757-2063-1
[28]	J. F. Cariñena, A. Ibort, G. Marmo, G. Morandi, Geometry from dynamics: classical and quantum, Springer, 2015. https://doi.org/10.1007/978-94-017-9220-2
[29]	D. C. Brody, L. P. Hughston, Geometric quantum mechanics, J. Geom. Phys., 38 (2001), 19–53. https://doi.org/10.1016/S0393-0440(00)00052-8 doi: 10.1016/S0393-0440(00)00052-8
[30]	J. Clemente-Gallardo, G. Marmo, Basics of quantum mechanics, geometrization and some applications to quantum information, Int. J. Geom. Methods Mod. Phys., 5 (2008), 989–1032. https://doi.org/10.1142/S0219887808003156 doi: 10.1142/S0219887808003156
[31]	J. Clemente-Gallardo, The geometrical formulation of quantum mechanics, Rev. Real Academia de Ciencias, 67 (2012), 51–103.
[32]	A. Heslot, Quantum mechanics as a classical theory, Phys. Rev. D, 31 (1985), 1341. https://doi.org/10.1103/PhysRevD.31.1341 doi: 10.1103/PhysRevD.31.1341
[33]	J. A. Jover-Galtier, Sistemas cuánticos abiertos: descripción geométrica, dinámica y control, (spainsh) [Open quantum systems: geometrical description, dynamics and control], PhD thesis, University of Zaragoza, 2017. Available from: https://zaguan.unizar.es/record/61849.
[34]	T. W. B. Kibble, Geometrization of quantum mechanics, Commun. Math. Phys., 65 (1979), 189–201. https://doi.org/10.1007/BF01225149 doi: 10.1007/BF01225149
[35]	J. L. Alonso, A. Castro, J. Clemente-Gallardo, J. C. Cuchí, P. Echenique-Robba, F. Falceto, Statistics and Nosé formalism for Ehrenfest dynamics, J. Phys. A Math. Theor., 44 (2011), 395004. https://doi.org/10.1088/1751-8113/44/39/395004 doi: 10.1088/1751-8113/44/39/395004
[36]	J. L. Alonso, J. Clemente-Gallardo, J. C. Cuchí, P. Echenique-Robba, F. Falceto, Ehrenfest dynamics is purity non-preserving: a necessary ingredient for decoherence, J. Chem. Phys. 137 (2012), 054106. https://doi.org/10.1063/1.4737861
[37]	E. C. Boghiu, Formulación geométrica de la dinámica y el control de sistemas híbridos clásico-cuánticos, (Spanish) [Geometrical formluation of dynamics and control of hybrid quantum-classical systems], BSc Thesis, University of Zaragoza, 2018. Available from: https://zaguan.unizar.es/record/77739.
[38]	J. Clemente-Gallardo, G. Marmo, The Ehrenfest picture and the geometry of quantum mechanics, Nuovo Cim. C, 36 (2013), 35–52. https://doi.org/10.1393/ncc/i2013-11522-6 doi: 10.1393/ncc/i2013-11522-6
[39]	J. L. Alonso, P. Bruscolini, A. Castro, J. Clemente-Gallardo, J. C. Cuchí, J. A. Jover-Galtier, Ehrenfest statistical dynamics in chemistry: study of decoherence effects, J. Chem. Theory Comput., 14 (2018) 3975–3985. https://doi.org/10.1021/acs.jctc.8b00511
[40]	C. Bouthelier-Madre, J. Clemente-Gallardo, L. González-Bravo, D. Martínez-Crespo, Hybrid Koopman C-formalism and the hybrid quantum-master equation, J. Phys. A Math. Theor.*, 56 (2023), 374001. https://doi.org/10.1088/1751-8121/aceed5 doi: 10.1088/1751-8121/aceed5
[41]	D. Martínez Crespo, Formalismo geométrico de la mecánica cuántica y sus aplicaciones a modelos moleculares, (Spanish) [Geometrical formalism of quantum mechanics and its applications to molecular models], BSc Thesis, University of Zaragoza, 2019. Available from: https://zaguan.unizar.es/record/87503.
[42]	J. L. Alonso, C. Bouthelier, A. Castro, J. Clemente-Gallardo, J. A. Jover-Galtier, Entropy and canonical ensemble of hybrid quantum classical systems, Phys. Rev. E, 102 (2020), 042118. https://doi.org/10.1103/PhysRevE.102.042118 doi: 10.1103/PhysRevE.102.042118
[43]	J. L. Alonso, C. Bouthelier-Madre, A. Castro, J. Clemente-Gallardo, J. A. Jover-Galtier, About the computation of finite temperature ensemble averages of hybrid quantum-classical systems with molecular dynamics, New J. Phys., 23 (2021), 063011. https://doi.org/10.1088/1367-2630/abf9b3 doi: 10.1088/1367-2630/abf9b3
[44]	J. L. Alonso, C. Bouthelier-Madre, J. Clemente-Gallardo, D. Martínez-Crespo, Effective nonlinear Ehrenfest hybrid quantum-classical dynamics, Eur. Phys. J. Plus, 138 (2023), 649. https://doi.org/10.1140/epjp/s13360-023-04266-w doi: 10.1140/epjp/s13360-023-04266-w
[45]	J. L. Alonso, C. Bouthelier-Madre, J. Clemente-Gallardo, D. Martínez-Crespo, Hybrid geometrodynamics: a Hamiltonian description of classical gravity coupled to quantum matter, Class. Quantum Grav., 41 (2024), 105004. https://doi.org/10.1088/1361-6382/ad3459 doi: 10.1088/1361-6382/ad3459
[46]	C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics, vol. I, Wiley, 1977.
[47]	J. F. Cariñena, J. Clemente-Gallardo, J. A. Jover-Galtier, G. Marmo, Tensorial dynamics on the space of quantum states, J. Phys. A Math. Theor., 50 (2017), 365301. https://doi.org/10.1088/1751-8121/aa8182 doi: 10.1088/1751-8121/aa8182
[48]	J. F. Cariñena, J. Clemente-Gallardo, J. A. Jover-Galtier, J. de Lucas, Application of Lie systems to quantum mechanics: superposition rules, in Classical and Quantum Physics, G. Marmo, D. Martín de Diego, M. Muñoz Lecanda, Springer, 2019, 85–119. https://doi.org/10.1007/978-3-030-24748-5_6
[49]	F. Albertini, D. D'Alessandro, Notions of controllability for bilinear multilevel quantum systems, IEEE Trans. Automat. Contr., 48 (2003), 1399–1403. https://doi.org/10.1109/TAC.2003.815027 doi: 10.1109/TAC.2003.815027
[50]	S. Zarychta, T. Sagan, M. Balcerzak, A. Dabrowski, A. Stefanski, T. Kapitaniak, A novel, Fourier series based method of control optimization and its application to a discontinuous capsule drive model, Int. J. Mech. Sci., 219 (2022), 107104. https://doi.org/10.1016/j.ijmecsci.2022.107104 doi: 10.1016/j.ijmecsci.2022.107104
[51]	T. Caneva, T. Calarco, S. Montangero, Choped random basis quantum optimization, Phys. Rev A, 84 (2011), 022326. https://doi.org/10.1103/PhysRevA.84.022326 doi: 10.1103/PhysRevA.84.022326
[52]	L. Abrunheiro, M. Camarinha, J. Clemente-Gallardo, Cubic polynomials on Lie groups: reduction of the Hamiltonian system, J. Phys. A Math. Theor., 44 (2011), 355203. https://doi.org/10.1088/1751-8113/44/35/355203 doi: 10.1088/1751-8113/44/35/355203
[53]	M. Camarinha, F. Silva Leite, P. Crouch, Splines of class $C^k$ on non-euclidean spaces, IMA J. Math. Control Inf., 12 (1995), 399-–410. https://doi.org/10.1093/imamci/12.4.399 doi: 10.1093/imamci/12.4.399
[54]	L. Noakes, G. Heinzinger, B. Paden, Cubic splines on curved spaces, IMA J. Math. Control Inf., 6 (1989), 465–-473. https://doi.org/10.1093/imamci/6.4.465 doi: 10.1093/imamci/6.4.465
[55]	M. Barbero-Liñán, M. C. Muñoz-Lecanda, Geometric approach to Pontryagin's maximum principle, Acta. Appl. Math., 108 (2009), 429–485. https://doi.org/10.1007/s10440-008-9320-5 doi: 10.1007/s10440-008-9320-5
[56]	L. S. Pontryagin, V. G. Boltyanski, R. V. Gamkrelidze, E. F. Mischenko, The Mathematical Theory of Optimal Processes. Wiley Interscience, New York, 1962.

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)