A closer look at quantum control landscapes and their implication for control optimization. (English) Zbl 1276.81067
Summary: The control landscape for various canonical quantum control problems is considered. For the class of pure-state transfer problems, analysis of the fidelity as a functional over the unitary group reveals no suboptimal attractive critical points (traps). For the actual optimization problem over controls in \(L^{2}(0, T)\), however, there are critical points for which the fidelity can assume any value in \((0,1)\), critical points for which the second order analysis is inconclusive, and traps. For the class of unitary operator optimization problems analysis of the fidelity over the unitary group shows that while there are no traps over \(\mathbf U(N)\), traps already emerge when the domain is restricted to the special unitary group. The traps on the group can be eliminated by modifying the performance index, corresponding to optimization over the projective unitary group. However, again, the set of critical points for the actual optimization problem for controls in \(L^{2}(0, T)\) is larger and includes traps, some of which remain traps even when the target time is allowed to vary.
References:
| [1] | DOI: 10.1088/1367-2630/11/10/105030 · doi:10.1088/1367-2630/11/10/105030 |
| [2] | DOI: 10.1016/0301-0104(93)80108-L · doi:10.1016/0301-0104(93)80108-L |
| [3] | DOI: 10.1103/PhysRevA.80.042305 · doi:10.1103/PhysRevA.80.042305 |
| [4] | DOI: 10.1088/1367-2630/11/10/105032 · doi:10.1088/1367-2630/11/10/105032 |
| [5] | DOI: 10.1080/01442350701633300 · doi:10.1080/01442350701633300 |
| [6] | DOI: 10.1016/0301-0104(93)80108-L · doi:10.1016/0301-0104(93)80108-L |
| [7] | DOI: 10.1063/1.1564043 · doi:10.1063/1.1564043 |
| [8] | DOI: 10.1103/PhysRevLett.89.188301 · doi:10.1103/PhysRevLett.89.188301 |
| [9] | DOI: 10.1016/j.jmr.2004.11.004 · doi:10.1016/j.jmr.2004.11.004 |
| [10] | DOI: 10.1103/PhysRevA.84.022305 · doi:10.1103/PhysRevA.84.022305 |
| [11] | DOI: 10.1126/science.1093649 · doi:10.1126/science.1093649 |
| [12] | DOI: 10.1080/09500340408231805 · doi:10.1080/09500340408231805 |
| [13] | DOI: 10.1103/PhysRevA.72.052337 · doi:10.1103/PhysRevA.72.052337 |
| [14] | DOI: 10.1063/1.2198837 · doi:10.1063/1.2198837 |
| [15] | Rabitz H., Phys. Rev. A 74 pp 012721–9– |
| [16] | DOI: 10.1016/j.jphotochem.2006.03.038 · doi:10.1016/j.jphotochem.2006.03.038 |
| [17] | Hsieh M., Phys. Rev. A 77 pp 042306–5– |
| [18] | DOI: 10.1088/0953-4075/41/7/074020 · doi:10.1088/0953-4075/41/7/074020 |
| [19] | DOI: 10.1088/1751-8113/41/1/015006 · Zbl 1134.82025 · doi:10.1088/1751-8113/41/1/015006 |
| [20] | Ho T., Phys. Rev. A 79 pp 013422–16– |
| [21] | DOI: 10.1103/PhysRevA.86.052117 · doi:10.1103/PhysRevA.86.052117 |
| [22] | DOI: 10.1137/S0363012993244945 · Zbl 0816.49019 · doi:10.1137/S0363012993244945 |
| [23] | Bonnard B., Non-Linear Anal. 14 pp 167– |
| [24] | Chitour Y., J. Differential Geom. 73 pp 45– · Zbl 1102.53019 · doi:10.4310/jdg/1146680512 |
| [25] | DOI: 10.1137/060663003 · Zbl 1157.49041 · doi:10.1137/060663003 |
| [26] | DOI: 10.1103/PhysRevA.86.013405 · doi:10.1103/PhysRevA.86.013405 |
| [27] | DOI: 10.1103/PhysRevLett.106.120402 · doi:10.1103/PhysRevLett.106.120402 |
| [28] | Albertini F., IEEE Decision and Control · Zbl 0988.82501 |
| [29] | DOI: 10.1088/0305-4470/35/18/309 · Zbl 1040.93007 · doi:10.1088/0305-4470/35/18/309 |
| [30] | Jurdjevic V., Geometric Control Theory (1997) · Zbl 0940.93005 |
| [31] | T. T. Frankel, Differential and Combinatorial Topology: A Symposium in Honor of Marston Morse (Princeton Univ. Press, Princeton, NJ, 1965) pp. 37–53. |
| [32] | DOI: 10.1007/b98874 · Zbl 0930.65067 · doi:10.1007/b98874 |
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