×

Control landscapes are almost always trap free: a geometric assessment. (English) Zbl 1367.81074

Summary: A proof is presented that almost all closed, finite dimensional quantum systems have trap free (i.e. free from local optima) landscapes for a large and physically general class of circumstances, which includes qubit evolutions in quantum computing. This result offers an explanation for why gradient-based methods succeed so frequently in quantum control. The role of singular controls is analyzed using geometric tools in the case of the control of the propagator, and thus in the case of observables as well. Singular controls have been implicated as a source of landscape traps. The conditions under which singular controls can introduce traps, and thus interrupt the progress of a control optimization, are discussed and a geometrical characterization of the issue is presented. It is shown that a control being singular is not sufficient to cause control optimization progress to halt, and sufficient conditions for a trap free landscape are presented. It is further shown that the local surjectivity (full rank) assumption of landscape analysis can be refined to the condition that the end-point map is transverse to each of the level sets of the fidelity function. This mild condition is shown to be sufficient for a quantum system’s landscape to be trap free. The control landscape is shown to be trap free for all but a null set of Hamiltonians using a geometric technique based on the parametric transversality theorem. Numerical evidence confirming this analysis is also presented. This new result is the analogue of the work of Altifini, wherein it was shown that controllability holds for all but a null set of quantum systems in the dipole approximation. These collective results indicate that the availability of adequate control resources remains the most physically relevant issue for achieving high fidelity control performance while also avoiding landscape traps.

MSC:

81Q93 Quantum control
81P68 Quantum computation
93C70 Time-scale analysis and singular perturbations in control/observation systems
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Russell B and Stepney S 2014 Zermelo navigation and a speed limit to quantum information processing Phys. Rev. A 90 012303 · doi:10.1103/PhysRevA.90.012303
[2] Russell B and Stepney S 2015 Zermelo navigation in the quantum brachistochrone J. Phys. A: Math. Theor.48 115303 · Zbl 1348.37053 · doi:10.1088/1751-8113/48/11/115303
[3] Wang X, Allegra M, Jacobs K, Lloyd S, Lupo C and Mohseni M 2015 Quantum brachistochrone curves as geodesics: obtaining accurate minimum-time protocols for the control of quantum systems Phys. Rev. Lett.114 170501 · doi:10.1103/PhysRevLett.114.170501
[4] Brody D C and Meier D M 2015 Elementary solution to the time-independent quantum navigation problem J. Phys. A: Math. Theor.48 055302 · Zbl 1307.81043 · doi:10.1088/1751-8113/48/5/055302
[5] Brody D C, Gibbons G W and Meier D M 2015 Time-optimal navigation through quantum wind New J. Phys.17 033048 · Zbl 1452.81122 · doi:10.1088/1367-2630/17/3/033048
[6] Brody D C and Meier D M 2015 Solution to the quantum zermelo navigation problem Phys. Rev. Lett.114 100502 · doi:10.1103/PhysRevLett.114.100502
[7] Carlini A, Hosoya A, Koike T and Okudaira Y 2006 Time-optimal quantum evolution Phys. Rev. Lett.96 060503 · doi:10.1103/PhysRevLett.96.060503
[8] Brody D C and Hughston L P 2001 Geometric quantum mechanics J. Geom. Phys.38 19-53 · Zbl 1067.81081 · doi:10.1016/S0393-0440(00)00052-8
[9] Nielsen M A 2005 A geometric approach to quantum circuit lower bounds (arXiv:quant-ph/0502070)
[10] Altafini C 2002 Controllability of quantum mechanical systems by root space decomposition of su(N) J. Math. Phys.43 2051-62 · Zbl 1059.93016 · doi:10.1063/1.1467611
[11] Sard A 1942 The measure of the critical values of differentiable maps Bull. Amer. Math. Soc.48 883-90 · Zbl 0063.06720 · doi:10.1090/S0002-9904-1942-07811-6
[12] Arnold V I, Varchenko A and Gusein-Zade S M 1985 Singularities of Differentiable Maps(Volume 1: The Classification of Critical Points Caustics and Wave Fronts) (Birkhäuser: Springer) · Zbl 0554.58001 · doi:10.1007/978-1-4612-5154-5
[13] Soare A, Ball H, Hayes D, Sastrawan J, Jarratt M C, McLoughlin J J, Zhen X, Green T J and Biercuk M J 2014 Experimental noise filtering by quantum control Nat. Phys.10 825-9 · doi:10.1038/nphys3115
[14] Roslund J and Rabitz H 2009 Gradient algorithm applied to laboratory quantum control Phys. Rev. A 79 053417 · doi:10.1103/PhysRevA.79.053417
[15] Roslund J and Rabitz H 2009 Experimental quantum control landscapes: inherent monotonicity and artificial structure Phys. Rev. A 80 013408 · doi:10.1103/PhysRevA.80.013408
[16] Riviello G, Brif C, Long R, Wu R-B, Tibbetts K M, Ho T-S and Rabitz H 2014 Searching for quantum optimal control fields in the presence of singular critical points Phys. Rev. A 90 013404 · doi:10.1103/PhysRevA.90.013404
[17] Riviello G, Tibbetts K M, Brif C, Long R, Wu R-B, Ho T-S and Rabitz H 2015 Searching for quantum optimal controls under severe constraints Phys. Rev. A 91 043401 · doi:10.1103/PhysRevA.91.043401
[18] Moore K W and Rabitz H 2011 Exploring quantum control landscapes: topology, features, and optimization scaling Phys. Rev. A 84 012109 · doi:10.1103/PhysRevA.84.012109
[19] Moore K W, Chakrabarti R, Riviello G and Rabitz H 2011 Search complexity and resource scaling for the quantum optimal control of unitary transformations Phys. Rev. A 83 012326 · doi:10.1103/PhysRevA.83.012326
[20] De Fouquieres P and Schirmer S G 2013 A closer look at quantum control landscapes and their implication for control optimization Infinite Dimens. Anal. Quantum Probab. Relat. Top.16 1350021 · Zbl 1276.81067 · doi:10.1142/S0219025713500215
[21] Pechen A N and Tannor D J 2011 Are there traps in quantum control landscapes? Phys. Rev. Lett.106 120402 · doi:10.1103/PhysRevLett.106.120402
[22] Rabitz H, Ho T-S, Long R, Wu R and Brif C 2012 Comment on ‘Are there traps in quantum control landscapes?’ Phys. Rev. Lett.108 198901 · doi:10.1103/PhysRevLett.108.198901
[23] Wu R-B, Long R, Dominy J, Ho T-S and Rabitz H 2012 Singularities of quantum control landscapes Phys. Rev. A 86 013405 · doi:10.1103/PhysRevA.86.013405
[24] Long R, Riviello G and Rabitz H 2013 The gradient flow for control of closed quantum systems IEEE Trans. Autom. Control58 2665-9 · Zbl 1369.49056 · doi:10.1109/TAC.2013.2256677
[25] Moore K, Hsieh M and Rabitz H 2008 On the relationship between quantum control landscape structure and optimization complexity J. Chem. Phys.128 154117 · doi:10.1063/1.2907740
[26] Franks J M 2009 A (terse) Introduction to Lebesgue Integration(Student Mathematical Library) (Providence, RI: American Mathematical Society) · Zbl 1177.28001 · doi:10.1090/stml/048
[27] Hirsch M W 1997 Differential Topology(Graduate Texts in Mathematics) (New York: Springer)
[28] Joe-Wong C, Ho T S and Rabitz H J 2016 On choosing the form of the objective functional for optimal control of molecules J. Math. Chem.54 · Zbl 1349.81106 · doi:10.1007/s10910-015-0558-7
[29] Judson R S and Rabitz H 1992 Teaching lasers to control molecules Phys. Rev. Lett.68 1500-3 · doi:10.1103/PhysRevLett.68.1500
[30] Joe-Wong C, Ho T S and Rabitz H J 2009 Landscape of unitary transformations in controlled quantum dynamics Phys. Rev. A 79 013422 · doi:10.1103/PhysRevA.79.013422
[31] Jurdjevic V 1997 Geometric Control Theory(Cambridge Studies in Advanced Mathematics) (Cambridge: Cambridge University Press) · Zbl 0940.93005
[32] Brif C, Chakrabarti R and Rabitz H 2010 Control of quantum phenomena: past, present and future New J. Phys.12 075008 · Zbl 1445.81029 · doi:10.1088/1367-2630/12/7/075008
[33] Russell B, Rabitz H and Wu R 2016 Quantum control landscapes beyond the dipole approximation (arXiv: 1602.06250)
[34] Arora R K 2015 Optimization: Algorithms and Applications (Boca Raton, FL: CRC Press) · Zbl 1314.90001 · doi:10.1201/b18469
[35] Sun Q, Pelczer I, Riviello G, Wu R-B and Rabitz H 2015 Experimental observation of saddle points over the quantum control landscape of a two-spin system Phys. Rev. A 91 043412 · doi:10.1103/PhysRevA.91.043412
[36] Sun Q, Pelczer I, Riviello G, Wu R-B and Rabitz H 2014 Experimental exploration over a quantum control landscape through nuclear magnetic resonance Phys. Rev. A 89 033413 · doi:10.1103/PhysRevA.89.033413
[37] Arenz C, Gualdi G and Burgarth D 2014 Control of open quantum systems: case study of the central spin model New J. Phys.16 065023 · Zbl 1451.81254 · doi:10.1088/1367-2630/16/6/065023
[38] Kriegl A and Michor P W 1997 The Convenient Setting of Global Analysis(Mathematical Surveys) (Providence, RI: American Mathematical Society) · Zbl 0889.58001 · doi:10.1090/surv/053
[39] Maruyama K and Burgarth D 2017 Gateway schemes of quantum control for spin networks Electron Spin Resonance (ESR) Based Quantum Computing pp 167-92 · doi:10.1007/978-1-4939-3658-8_6
[40] Chakrabarti R and Rabitz H 2007 Quantum control landscapes Int. Rev. Phys. Chem.26 671-735 · doi:10.1080/01442350701633300
[41] Fulton W and Harris J 1991 Representation Theory: a First Course(Graduate Texts in Mathematics) (New York: Springer) · Zbl 0744.22001
[42] Hall B 2003 Lie Groups, Lie Algebras, and Representations: an Elementary Introduction(Graduate Texts in Mathematics) (Berlin: Springer) · Zbl 1026.22001 · doi:10.1007/978-0-387-21554-9
[43] Russell B and Stepney S 2014 Applications of finsler geometry to speed limits to quantum information processing Int. J. Found. Comput. Sci.25 489-505 · Zbl 1360.68474 · doi:10.1142/S0129054114400073
[44] Tiglay F and Vizman C 2011 Generalized euler-poincaré equations on lie groups and homogeneous spaces, orbit invariants and applications Lett. Math. Phys.97 45-60 · Zbl 1219.35198 · doi:10.1007/s11005-011-0464-2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.