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Li X. Optimal control of quantum state preparation and entanglement creation in two-qubit quantum system with bounded amplitude. Sci Rep 2023; 13:14734. [PMID: 37679384 PMCID: PMC10484962 DOI: 10.1038/s41598-023-41688-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/29/2023] [Accepted: 08/30/2023] [Indexed: 09/09/2023] Open
Abstract
We consider the optimal control problem in a two-qubit system with bounded amplitude. Two cases are studied: quantum state preparation and entanglement creation. Cost functions, fidelity and concurrence, are optimized over bang-off controls for various values of the total duration, respectively. For quantum state preparation problem, three critical time points are determined accurately, and optimal controls are estimated. A better estimation of the quantum speed limit is obtained, so is the time-optimal control. For entanglement creation problem, two critical time points are determined, one of them is the minimal time to achieve maximal entanglement (unit concurrence) starting from the product state. In addition, the comparisons between bang-off and chopped random basis (CRAB) are made.
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Affiliation(s)
- Xikun Li
- School of Physics and Optoelectronic Engineering, Anhui University, Hefei, 230601, China.
- Max-Planck-Institut für Physik komplexer Systeme, 01187, Dresden, Germany.
- Department of Physics and Astronomy, Aarhus University, 8000, Aarhus, Denmark.
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2
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Rabitz H, Russell B, Ho TS. The Surprising Ease of Finding Optimal Solutions for Controlling Nonlinear Phenomena in Quantum and Classical Complex Systems. J Phys Chem A 2023; 127:4224-4236. [PMID: 37142303 DOI: 10.1021/acs.jpca.3c01896] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 05/06/2023]
Abstract
This Perspective addresses the often observed surprising ease of achieving optimal control of nonlinear phenomena in quantum and classical complex systems. The circumstances involved are wide-ranging, with scenarios including manipulation of atomic scale processes, maximization of chemical and material properties or synthesis yields, Nature's optimization of species' populations by natural selection, and directed evolution. Natural evolution will mainly be discussed in terms of laboratory experiments with microorganisms, and the field is also distinct from the other domains where a scientist specifies the goal(s) and oversees the control process. We use the word "control" in reference to all of the available variables, regardless of the circumstance. The empirical observations on the ease of achieving at least good, if not excellent, control in diverse domains of science raise the question of why this occurs despite the generally inherent complexity of the systems in each scenario. The key to addressing the question lies in examining the associated control landscape, which is defined as the optimization objective as a function of the control variables that can be as diverse as the phenomena under consideration. Controls may range from laser pulses, chemical reagents, chemical processing conditions, out to nucleic acids in the genome and more. This Perspective presents a conjecture, based on present findings, that the systematics of readily finding good outcomes from controlled phenomena may be unified through consideration of control landscapes with the same common set of three underlying assumptions─the existence of an optimal solution, the ability for local movement on the landscape, and the availability of sufficient control resources─whose validity needs assessment in each scenario. In practice, many cases permit using myopic gradient-like algorithms while other circumstances utilize algorithms having some elements of stochasticity or introduced noise, depending on whether the landscape is locally smooth or rough. The overarching observation is that only relatively short searches are required despite the common high dimensionality of the available controls in typical scenarios.
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Affiliation(s)
- Herschel Rabitz
- Department of Chemistry, Princeton University, Princeton, New Jersey 08544, United States
| | - Benjamin Russell
- Department of Chemistry, Princeton University, Princeton, New Jersey 08544, United States
| | - Tak-San Ho
- Department of Chemistry, Princeton University, Princeton, New Jersey 08544, United States
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3
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Friedland S, Eckstein M, Cole S, Życzkowski K. Quantum Monge-Kantorovich Problem and Transport Distance between Density Matrices. PHYSICAL REVIEW LETTERS 2022; 129:110402. [PMID: 36154415 DOI: 10.1103/physrevlett.129.110402] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/23/2021] [Revised: 08/01/2022] [Accepted: 08/02/2022] [Indexed: 06/16/2023]
Abstract
A quantum version of the Monge-Kantorovich optimal transport problem is analyzed. The transport cost is minimized over the set of all bipartite coupling states ρ^{AB} such that both of its reduced density matrices ρ^{A} and ρ^{B} of dimension N are fixed. We show that, selecting the quantum cost matrix to be proportional to the projector on the antisymmetric subspace, the minimal transport cost leads to a semidistance between ρ^{A} and ρ^{B}, which is bounded from below by the rescaled Bures distance and from above by the root infidelity. In the single-qubit case, we provide a semianalytic expression for the optimal transport cost between any two states and prove that its square root satisfies the triangle inequality and yields an analog of the Wasserstein distance of the order of 2 on the set of density matrices. We introduce an associated measure of proximity of quantum states, called swap fidelity, and discuss its properties and applications in quantum machine learning.
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Affiliation(s)
- Shmuel Friedland
- Department of Mathematics, Statistics and Computer Science, University of Illinois, Chicago, Illinois 60607-7045, USA
| | - Michał Eckstein
- Institute of Theoretical Physics, Jagiellonian University and Mark Kac Center, ul. Łojasiewicza 11, 30-348 Kraków, Poland
| | - Sam Cole
- Department of Mathematics, University of Missouri, Columbia, Missouri 65211, USA
| | - Karol Życzkowski
- Institute of Theoretical Physics, Jagiellonian University and Mark Kac Center, ul. Łojasiewicza 11, 30-348 Kraków, Poland
- Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warszawa, Poland
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4
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Li JF, Hu JR, Wan F, He DS. Optimization two-qubit quantum gate by two optical control methods in molecular pendular states. Sci Rep 2022; 12:14918. [PMID: 36050511 PMCID: PMC9437090 DOI: 10.1038/s41598-022-18967-2] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/26/2022] [Accepted: 08/23/2022] [Indexed: 11/09/2022] Open
Abstract
Implementation of quantum gates are important for quantum computations in physical system made of polar molecules. We investigate the feasibility of implementing gates based on pendular states of the molecular system by two different quantum optical control methods. Firstly, the Multi-Target optimal control theory and the Multi-Constraint optimal control theory are described for optimizing control fields and accomplish the optimization of quantum gates. Numerical results show that the controlled NOT gate (CNOT) can be realized under the control of above methods with high fidelities (0.975 and 0.999) respectively. In addition, in order to examine the dependence of the fidelity on energy difference in the same molecular system, the SWAP gate in the molecular system is also optimized with high fidelity (0.999) by the Multi-Constraint optimal control theory with the zero-area and constant-fluence constraints.
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Affiliation(s)
- Jin-Fang Li
- Department of Physics and Electronic Engineering, Xianyang Normal University, Shaanxi, 712000, China. .,State Key Laboratory of Precision Spectroscopy, Department of Physics, East China Normal University, Shanghai, 200062, China.
| | - Jie-Ru Hu
- State Key Laboratory of Precision Spectroscopy, Department of Physics, East China Normal University, Shanghai, 200062, China
| | - Feng Wan
- Department of Physics and Electronic Engineering, Xianyang Normal University, Shaanxi, 712000, China
| | - Dong-Shan He
- Department of Physics and Electronic Engineering, Xianyang Normal University, Shaanxi, 712000, China
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5
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Li J, Yang X, Peng X, Sun CP. Hybrid Quantum-Classical Approach to Quantum Optimal Control. PHYSICAL REVIEW LETTERS 2017; 118:150503. [PMID: 28452527 DOI: 10.1103/physrevlett.118.150503] [Citation(s) in RCA: 22] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/10/2016] [Indexed: 05/25/2023]
Abstract
A central challenge in quantum computing is to identify more computational problems for which utilization of quantum resources can offer significant speedup. Here, we propose a hybrid quantum-classical scheme to tackle the quantum optimal control problem. We show that the most computationally demanding part of gradient-based algorithms, namely, computing the fitness function and its gradient for a control input, can be accomplished by the process of evolution and measurement on a quantum simulator. By posing queries to and receiving answers from the quantum simulator, classical computing devices update the control parameters until an optimal control solution is found. To demonstrate the quantum-classical scheme in experiment, we use a seven-qubit nuclear magnetic resonance system, on which we have succeeded in optimizing state preparation without involving classical computation of the large Hilbert space evolution.
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Affiliation(s)
- Jun Li
- Beijing Computational Science Research Center, Beijing 100193, China
| | - Xiaodong Yang
- Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Xinhua Peng
- Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
- Synergetic Innovation Centre of Quantum Information & Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Chang-Pu Sun
- Beijing Computational Science Research Center, Beijing 100193, China
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Russell B, Rabitz H. Common foundations of optimal control across the sciences: evidence of a free lunch. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2017; 375:rsta.2016.0210. [PMID: 28115607 PMCID: PMC5311431 DOI: 10.1098/rsta.2016.0210] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 11/20/2016] [Indexed: 05/22/2023]
Abstract
A common goal in the sciences is optimization of an objective function by selecting control variables such that a desired outcome is achieved. This scenario can be expressed in terms of a control landscape of an objective considered as a function of the control variables. At the most basic level, it is known that the vast majority of quantum control landscapes possess no traps, whose presence would hinder reaching the objective. This paper reviews and extends the quantum control landscape assessment, presenting evidence that the same highly favourable landscape features exist in many other domains of science. The implications of this broader evidence are discussed. Specifically, control landscape examples from quantum mechanics, chemistry and evolutionary biology are presented. Despite the obvious differences, commonalities between these areas are highlighted within a unified mathematical framework. This mathematical framework is driven by the wide-ranging experimental evidence on the ease of finding optimal controls (in terms of the required algorithmic search effort beyond the laboratory set-up overhead). The full scope and implications of this observed common control behaviour pose an open question for assessment in further work.This article is part of the themed issue 'Horizons of cybernetical physics'.
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Affiliation(s)
- Benjamin Russell
- Department of Chemistry, Princeton University, Princeton, NJ 08540, USA
| | - Herschel Rabitz
- Department of Chemistry, Princeton University, Princeton, NJ 08540, USA
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7
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Wigley PB, Everitt PJ, van den Hengel A, Bastian JW, Sooriyabandara MA, McDonald GD, Hardman KS, Quinlivan CD, Manju P, Kuhn CCN, Petersen IR, Luiten AN, Hope JJ, Robins NP, Hush MR. Fast machine-learning online optimization of ultra-cold-atom experiments. Sci Rep 2016; 6:25890. [PMID: 27180805 PMCID: PMC4867626 DOI: 10.1038/srep25890] [Citation(s) in RCA: 124] [Impact Index Per Article: 15.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2015] [Accepted: 04/21/2016] [Indexed: 12/04/2022] Open
Abstract
We apply an online optimization process based on machine learning to the production of Bose-Einstein condensates (BEC). BEC is typically created with an exponential evaporation ramp that is optimal for ergodic dynamics with two-body s-wave interactions and no other loss rates, but likely sub-optimal for real experiments. Through repeated machine-controlled scientific experimentation and observations our ‘learner’ discovers an optimal evaporation ramp for BEC production. In contrast to previous work, our learner uses a Gaussian process to develop a statistical model of the relationship between the parameters it controls and the quality of the BEC produced. We demonstrate that the Gaussian process machine learner is able to discover a ramp that produces high quality BECs in 10 times fewer iterations than a previously used online optimization technique. Furthermore, we show the internal model developed can be used to determine which parameters are essential in BEC creation and which are unimportant, providing insight into the optimization process of the system.
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Affiliation(s)
- P B Wigley
- Quantum Sensors and Atomlaser Lab, Department of Quantum Science, Research School of Physics and Engineering, The Australian National University, Acton, 2601, Australia
| | - P J Everitt
- Quantum Sensors and Atomlaser Lab, Department of Quantum Science, Research School of Physics and Engineering, The Australian National University, Acton, 2601, Australia
| | - A van den Hengel
- Australian Centre for Visual Technologies, University of Adelaide, Adelaide, 5005, Australia
| | - J W Bastian
- School of Computer Science, University of Adelaide, Adelaide, 5005, Australia
| | - M A Sooriyabandara
- Quantum Sensors and Atomlaser Lab, Department of Quantum Science, Research School of Physics and Engineering, The Australian National University, Acton, 2601, Australia
| | - G D McDonald
- Quantum Sensors and Atomlaser Lab, Department of Quantum Science, Research School of Physics and Engineering, The Australian National University, Acton, 2601, Australia
| | - K S Hardman
- Quantum Sensors and Atomlaser Lab, Department of Quantum Science, Research School of Physics and Engineering, The Australian National University, Acton, 2601, Australia
| | - C D Quinlivan
- Quantum Sensors and Atomlaser Lab, Department of Quantum Science, Research School of Physics and Engineering, The Australian National University, Acton, 2601, Australia
| | - P Manju
- Quantum Sensors and Atomlaser Lab, Department of Quantum Science, Research School of Physics and Engineering, The Australian National University, Acton, 2601, Australia
| | - C C N Kuhn
- Quantum Sensors and Atomlaser Lab, Department of Quantum Science, Research School of Physics and Engineering, The Australian National University, Acton, 2601, Australia
| | - I R Petersen
- School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra, 2600, Australia
| | - A N Luiten
- Institute for Photonics &Advanced Sensing, School of Physical Sciences,The University of Adelaide, Adelaide, 5005, Australia
| | - J J Hope
- Department of Quantum Science, Australian National University, Canberra, 2601, Australia
| | - N P Robins
- Quantum Sensors and Atomlaser Lab, Department of Quantum Science, Research School of Physics and Engineering, The Australian National University, Acton, 2601, Australia
| | - M R Hush
- School of Engineering and Information Technology, University of New South Wales at the Australian Defence Force Academy, Canberra, 2600, Australia
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Sun Q, Pelczer I, Riviello G, Wu RB, Rabitz H. Identifying and avoiding singularity-induced local traps over control landscapes of spin chain systems. Phys Chem Chem Phys 2015; 17:29714-22. [PMID: 26478216 DOI: 10.1039/c5cp05418d] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
The wide success of quantum optimal control in experiments and simulations is attributed to the properties of the control landscape, defined by the objective value as a functional of the controls. Prior analysis has shown that on satisfaction of some underlying assumptions, the landscapes are free of suboptimal traps that could halt the search for a global optimum with gradient-based algorithms. However, violation of one particular assumption can give rise to a so-called singular control, possibly bringing about local traps on the corresponding landscapes in some particular situations. This paper theoretically and experimentally demonstrates the existence of singular traps on the landscape in linear spin-1/2 chains with Ising couplings between nearest neighbors and with certain field components set to zero. The results in a two-spin example show how a trap influences the search trajectories passing by it, and how to avoid encountering such traps in practice by choosing sufficiently strong initial control fields. The findings are also discussed in the context of the generally observed success of quantum control.
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Affiliation(s)
- Qiuyang Sun
- Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA.
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Chenel A, Meier C, Dive G, Desouter-Lecomte M. Optimal control of a Cope rearrangement by coupling the reaction path to a dissipative bath or a second active mode. J Chem Phys 2015; 142:024307. [DOI: 10.1063/1.4905200] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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10
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Moore Tibbetts K, Rabitz H. Constrained control landscape for population transfer in a two-level system. Phys Chem Chem Phys 2015; 17:3164-78. [DOI: 10.1039/c4cp04792c] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Controlling population transfer in a two-level quantum system reveals a landscape with a rich structure containing highly connected optimal regions.
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Abstract
We consider manipulation of the transmission coefficient for a quantum particle moving in one dimension where the shape of the potential is taken as the control. We show that the control landscape, the transmission as a functional of the potential, has no traps, i.e., any maxima correspond to full transmission.
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Affiliation(s)
- Alexander N. Pechen
- Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel
- Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina str. 8, Moscow 119991, Russia
| | - David J. Tannor
- Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel
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12
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Moore Tibbetts K, Xing X, Rabitz H. Exploring control landscapes for laser-driven molecular fragmentation. J Chem Phys 2013; 139:144201. [DOI: 10.1063/1.4824153] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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13
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Chen C, Wang LC, Wang Y. Closed-loop and robust control of quantum systems. ScientificWorldJournal 2013; 2013:869285. [PMID: 23997680 PMCID: PMC3749599 DOI: 10.1155/2013/869285] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/11/2013] [Accepted: 07/16/2013] [Indexed: 11/27/2022] Open
Abstract
For most practical quantum control systems, it is important and difficult to attain robustness and reliability due to unavoidable uncertainties in the system dynamics or models. Three kinds of typical approaches (e.g., closed-loop learning control, feedback control, and robust control) have been proved to be effective to solve these problems. This work presents a self-contained survey on the closed-loop and robust control of quantum systems, as well as a brief introduction to a selection of basic theories and methods in this research area, to provide interested readers with a general idea for further studies. In the area of closed-loop learning control of quantum systems, we survey and introduce such learning control methods as gradient-based methods, genetic algorithms (GA), and reinforcement learning (RL) methods from a unified point of view of exploring the quantum control landscapes. For the feedback control approach, the paper surveys three control strategies including Lyapunov control, measurement-based control, and coherent-feedback control. Then such topics in the field of quantum robust control as H(∞) control, sliding mode control, quantum risk-sensitive control, and quantum ensemble control are reviewed. The paper concludes with a perspective of future research directions that are likely to attract more attention.
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Affiliation(s)
- Chunlin Chen
- Department of Control and System Engineering, Nanjing University, Nanjing 210093, China
| | - Lin-Cheng Wang
- School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024, China
| | - Yuanlong Wang
- Institute of Cyber-Systems and Control, State Key Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou 310027, China
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14
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Abstract
The broad success of optimally controlling quantum systems with external fields has been attributed to the favorable topology of the underlying control landscape, where the landscape is the physical observable as a function of the controls. The control landscape can be shown to contain no suboptimal trapping extrema upon satisfaction of reasonable physical assumptions, but this topological analysis does not hold when significant constraints are placed on the control resources. This work employs simulations to explore the topology and features of the control landscape for pure-state population transfer with a constrained class of control fields. The fields are parameterized in terms of a set of uniformly spaced spectral frequencies, with the associated phases acting as the controls. This restricted family of fields provides a simple illustration for assessing the impact of constraints upon seeking optimal control. Optimization results reveal that the minimum number of phase controls necessary to assure a high yield in the target state has a special dependence on the number of accessible energy levels in the quantum system, revealed from an analysis of the first- and second-order variation of the yield with respect to the controls. When an insufficient number of controls and/or a weak control fluence are employed, trapping extrema and saddle points are observed on the landscape. When the control resources are sufficiently flexible, solutions producing the globally maximal yield are found to form connected "level sets" of continuously variable control fields that preserve the yield. These optimal yield level sets are found to shrink to isolated points on the top of the landscape as the control field fluence is decreased, and further reduction of the fluence turns these points into suboptimal trapping extrema on the landscape. Although constrained control fields can come in many forms beyond the cases explored here, the behavior found in this paper is illustrative of the impacts that constraints can introduce.
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Affiliation(s)
- Katharine W Moore
- Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA
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Rabitz H, Ho TS, Long R, Wu R, Brif C. Comment on "Are there traps in quantum control landscapes?". PHYSICAL REVIEW LETTERS 2012; 108:198901-198902. [PMID: 23003098 DOI: 10.1103/physrevlett.108.198901] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/11/2011] [Indexed: 06/01/2023]
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16
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Pechen AN, Tannor DJ. Quantum Control Landscape for a Λ-atom in the Vicinity of Second-Order Traps. Isr J Chem 2012. [DOI: 10.1002/ijch.201100165] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
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