Final Report Summary - ACOLA (Analysis of controllability and control landscapes for quantum systems)
The project was performed in the field of quantum control. This is a vibrant, actively-growing field of broad interest in the physics and chemistry communities. According to Science Citation Index, more than 1300 theoretical and experimental papers are published each year in this general area, including regular publications in Science and Nature, and many researchers are actively involved in the research in Europe and worldwide. The basic reason for this growing activity stems from the simple fact that anyone probing a quantum system would also desire at some juncture to control its dynamics. Quantum control is also a highly interdisciplinary branch of science with multiple existing and prospective applications in physics and chemistry which range from the production of selective atomic or molecular excitations, where the objective is the desired atomic or molecular state and the control is the tailored laser field applied to the system, to laser control of chemical reactions and selective chemical bond breaking.
Generally control problems are formulated as the maximization of a certain objective function which describes the desired property of the controlled physical system. The objective for selective excitation of atomic or molecular states is the population of the desired atomic or molecular state and the control goal is to find a shaped laser pulse maximizing this population. The objective for selective control of chemical reactions is the concentration of the desired reaction product and the control goal is to find a shaped laser pulse to produce optimal concentration of the product. The objective for selective bond breaking is the population of a molecular state with broken desired bond; the control goal is to tailor laser pulse to break the bond while preserving the rest of the molecular structure unaffected.
The project was devoted to the analysis of two topics within this field. First topic is controllability of open, that is, interacting with the environment, quantum systems controlled by tailored laser pulse and state of the environment. Controllability is a fundamental property of any control system which determines the ability to steer any initial system state into any final state with the available controls. The common state space for a quantum system is either the set of all pure states or the set of all density matrices. The latter is the biggest and therefore controllability in this set is the strongest amongst all degrees of quantum state control.
Second topic is the analysis of traps in quantum control problems. Trap is a local maximum of the objective. Knowing traps for a given control problem is crucial for determining proper algorithms for finding optimal control fields. In the absence of traps, local search algorithms should generally be able to find globally optimal controls. If the objective has traps or perhaps even a single trap with large attracting domain then local search may converge to local maxima instead of attaining a desired globally optimal control and more sophisticated global search methods should be exploited for successful optimization. We obtained several results on these topics.
First result of the project is the construction of a control scheme to implement approximate controllability of finite-level open quantum systems in the set of all density matrices. We give a theoretical proof and suggest a physical implementation using combination of coherent and incoherent photons. The discovery of controllability of open quantum systems in the set of all density matrices provides the strongest possible degree of quantum state control. This result has a potential impact and use for making robust quantum memories, preparing many-body states and nonequilibrium quantum phases, inducing multiparticle entanglement dynamics, and building quantum computers with mixed states and non-unitary gates.
Second result is the discovery of second-order traps for a general class of quantum systems. Such traps are important as being traps for first-order and second-order local search algorithms. We found the conditions on the system parameters under which second-order traps exist. The simplest example of a system which satisfies these conditions is the Lambda-atom relevant for many applications in physics. We studied this system in details and rigorously shown that its second-ordertrap is not a local maximum. The numerical analysis was performed to find the intensity at which the second-order trap begins to significantly decrease the search for the optimal control; for some realistic system parameters the intensity was shown to be of about 10^12-10^13 W/cm^2 that is generally considered as a strong field.
Third result is the discovery of quantum control problems which are completely free of traps.
Whereas the analysis of traps for controlled quantum systems has attracted high attention of researchers, no examples of trap-free quantum systems were known before the project. In the project we proved the absence of traps for a wide class of control problems for two-level systems, including state preparation and unitary gate generation. We also analyzed how decoherence, fluctuations of the actual control due to noise or imperfections of the laboratory setup, and laboratory restrictions on the available controls influence on traps. Only very severe restrictions on the controls were found as being able to produce traps.Fourth result is the proof of absence of traps for manipulating the transmission coefficient for a particle in one-dimensional potential whose shape is used as control. Such systems represent a class of problems with infinite number of energy states. They were not at all analyzed before in the context of quantum control landscapes, where all previous effort was directed towards the study of control landscapes for simple finite-level systems. In the project, we considered systems with an infinite number of energy states and proved the absence of traps for control of transmission of a quantum particle moving through a potential barrier with adjusted shape. All maxima of the transmission coefficient were shown to be corresponding to full transmission. Second, third, and fourth result have impact and use for determining proper algorithms for the search of optimal controls for particular systems.
Fifth result is the proof of absence of traps at regular points of fitness landscapes for biological open systems. This result was formulated as Optimal landscapes in Evolution (OptiEvo) theory. The theory views a particular biological system as interacting with the environment, utilizing nucleotides as variables and optimizing a measure of fitness as the objective. OptiEvo can be based on either classical or quantum-mechanical description. The basic conclusion is that fitness landscapes in a constant homogeneous environment should not contain isolated local peaks (i.e. traps) under the assumption that biologically realizable gene changes can provide sufficient flexibility to explore the local landscape structure in the process of evolution. Violation of this assumption through restrictions on the gene changes can lead to the presence of trapping points. A comparison of the biological predictions of OptiEvo with laboratory results in the literature was performed along with a discussion of recently analyzed fitness landscapes in chemical and physical sciences. The implications of OptiEvo for the design strategy and outcome of directed evolution experiments were found. This result is important for understanding the structure of fitness landscape in biology.
In the project, several fundamental results which advance the field of quantum control were obtained that contributed significantly to the European excellence and competitiveness in the field of quantum control. These results were widely disseminated between the scientific community by publishing papers, including in the world leading journals, giving talks at numerous international conferences and seminars, and discussing the results on personal meetings with European colleagues. The project contributed to transfer of knowledge from the US to Europe by bringing skills, knowledge, and research experience of the researcher from the US to the European host institution that culminated in the obtained scientific results and in the numerous talks given by the researcher. The project created a strong persistent long-term collaboration between the researcher and the host, as well as established new collaborations between the researcher and other European scientists. The results could be relevant for physicists and chemists working on quantum technologies and on control of atomic and molecular systems.
Generally control problems are formulated as the maximization of a certain objective function which describes the desired property of the controlled physical system. The objective for selective excitation of atomic or molecular states is the population of the desired atomic or molecular state and the control goal is to find a shaped laser pulse maximizing this population. The objective for selective control of chemical reactions is the concentration of the desired reaction product and the control goal is to find a shaped laser pulse to produce optimal concentration of the product. The objective for selective bond breaking is the population of a molecular state with broken desired bond; the control goal is to tailor laser pulse to break the bond while preserving the rest of the molecular structure unaffected.
The project was devoted to the analysis of two topics within this field. First topic is controllability of open, that is, interacting with the environment, quantum systems controlled by tailored laser pulse and state of the environment. Controllability is a fundamental property of any control system which determines the ability to steer any initial system state into any final state with the available controls. The common state space for a quantum system is either the set of all pure states or the set of all density matrices. The latter is the biggest and therefore controllability in this set is the strongest amongst all degrees of quantum state control.
Second topic is the analysis of traps in quantum control problems. Trap is a local maximum of the objective. Knowing traps for a given control problem is crucial for determining proper algorithms for finding optimal control fields. In the absence of traps, local search algorithms should generally be able to find globally optimal controls. If the objective has traps or perhaps even a single trap with large attracting domain then local search may converge to local maxima instead of attaining a desired globally optimal control and more sophisticated global search methods should be exploited for successful optimization. We obtained several results on these topics.
First result of the project is the construction of a control scheme to implement approximate controllability of finite-level open quantum systems in the set of all density matrices. We give a theoretical proof and suggest a physical implementation using combination of coherent and incoherent photons. The discovery of controllability of open quantum systems in the set of all density matrices provides the strongest possible degree of quantum state control. This result has a potential impact and use for making robust quantum memories, preparing many-body states and nonequilibrium quantum phases, inducing multiparticle entanglement dynamics, and building quantum computers with mixed states and non-unitary gates.
Second result is the discovery of second-order traps for a general class of quantum systems. Such traps are important as being traps for first-order and second-order local search algorithms. We found the conditions on the system parameters under which second-order traps exist. The simplest example of a system which satisfies these conditions is the Lambda-atom relevant for many applications in physics. We studied this system in details and rigorously shown that its second-ordertrap is not a local maximum. The numerical analysis was performed to find the intensity at which the second-order trap begins to significantly decrease the search for the optimal control; for some realistic system parameters the intensity was shown to be of about 10^12-10^13 W/cm^2 that is generally considered as a strong field.
Third result is the discovery of quantum control problems which are completely free of traps.
Whereas the analysis of traps for controlled quantum systems has attracted high attention of researchers, no examples of trap-free quantum systems were known before the project. In the project we proved the absence of traps for a wide class of control problems for two-level systems, including state preparation and unitary gate generation. We also analyzed how decoherence, fluctuations of the actual control due to noise or imperfections of the laboratory setup, and laboratory restrictions on the available controls influence on traps. Only very severe restrictions on the controls were found as being able to produce traps.Fourth result is the proof of absence of traps for manipulating the transmission coefficient for a particle in one-dimensional potential whose shape is used as control. Such systems represent a class of problems with infinite number of energy states. They were not at all analyzed before in the context of quantum control landscapes, where all previous effort was directed towards the study of control landscapes for simple finite-level systems. In the project, we considered systems with an infinite number of energy states and proved the absence of traps for control of transmission of a quantum particle moving through a potential barrier with adjusted shape. All maxima of the transmission coefficient were shown to be corresponding to full transmission. Second, third, and fourth result have impact and use for determining proper algorithms for the search of optimal controls for particular systems.
Fifth result is the proof of absence of traps at regular points of fitness landscapes for biological open systems. This result was formulated as Optimal landscapes in Evolution (OptiEvo) theory. The theory views a particular biological system as interacting with the environment, utilizing nucleotides as variables and optimizing a measure of fitness as the objective. OptiEvo can be based on either classical or quantum-mechanical description. The basic conclusion is that fitness landscapes in a constant homogeneous environment should not contain isolated local peaks (i.e. traps) under the assumption that biologically realizable gene changes can provide sufficient flexibility to explore the local landscape structure in the process of evolution. Violation of this assumption through restrictions on the gene changes can lead to the presence of trapping points. A comparison of the biological predictions of OptiEvo with laboratory results in the literature was performed along with a discussion of recently analyzed fitness landscapes in chemical and physical sciences. The implications of OptiEvo for the design strategy and outcome of directed evolution experiments were found. This result is important for understanding the structure of fitness landscape in biology.
In the project, several fundamental results which advance the field of quantum control were obtained that contributed significantly to the European excellence and competitiveness in the field of quantum control. These results were widely disseminated between the scientific community by publishing papers, including in the world leading journals, giving talks at numerous international conferences and seminars, and discussing the results on personal meetings with European colleagues. The project contributed to transfer of knowledge from the US to Europe by bringing skills, knowledge, and research experience of the researcher from the US to the European host institution that culminated in the obtained scientific results and in the numerous talks given by the researcher. The project created a strong persistent long-term collaboration between the researcher and the host, as well as established new collaborations between the researcher and other European scientists. The results could be relevant for physicists and chemists working on quantum technologies and on control of atomic and molecular systems.