The Experts below are selected from a list of 273 Experts worldwide ranked by ideXlab platform
Uros Kalabic - One of the best experts on this subject based on the ideXlab platform.
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Application of Pontryagin's Minimum Principle to Grover's quantum search problem
Physical Review A, 2019Co-Authors: Chungwei Lin, Yebin Wang, Grigory Kolesov, Uros KalabicAbstract:Grover's algorithm is one of the most famous algorithms which explicitly demonstrates how the quantum nature can be utilized to accelerate the searching process. In this work, Grover's quantum search problem is mapped to a time-optimal control problem. Resorting to Pontryagin's Minimum Principle we find that the time-optimal solution has the bang-singular-bang structure. This structure can be derived naturally, without integrating the differential equations, using the geometric control technique where Hamiltonians in the Schr\"odinger's equation are represented as vector fields. In view of optimal control, Grover's algorithm uses the bang-bang protocol to approximate the optimal protocol with a minimized number of bang-to-bang switchings to reduce the query complexity. Our work provides a concrete example how Pontryagin's Minimum Principle is connected to quantum computation, and offers insight into how a quantum algorithm can be designed.
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application of pontryagin s Minimum Principle to grover s quantum search problem
Physical Review A, 2019Co-Authors: Chungwei Lin, Yebin Wang, Grigory Kolesov, Uros KalabicAbstract:Grover's algorithm is one of the most famous algorithms which explicitly demonstrates how the quantum nature can be utilized to accelerate the searching process. In this work, Grover's quantum search problem is mapped to a time-optimal control problem. Resorting to Pontryagin's Minimum Principle, we find that the time-optimal solution has the bang-singular-bang structure. This structure can be derived naturally, without integrating the differential equations, using the geometric control technique where Hamiltonians in Schr\"odinger's equation are represented as vector fields. In view of optimal control, Grover's algorithm uses the bang-bang protocol to approximate the optimal protocol with a minimized number of bang-to-bang switchings to reduce the query complexity. Our work provides a concrete example of how Pontryagin's Minimum Principle is connected to quantum computation, and offers insight into how a quantum algorithm can be designed.
Peter E. Caines - One of the best experts on this subject based on the ideXlab platform.
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On the Hybrid Minimum Principle
arXiv: Optimization and Control, 2017Co-Authors: Ali Pakniyat, Peter E. CainesAbstract:The Hybrid Minimum Principle (HMP) is established for the optimal control of deterministic hybrid systems with both autonomous and controlled switchings and jumps where state jumps at the switching instants are permitted to be accompanied by changes in the dimension of the state space. First order variational analysis is performed via the needle variation methodology and the necessary optimality conditions are established in the form of the HMP. A feature of special interest in this work is the explicit presentations of boundary conditions on the Hamiltonians and the adjoint processes before and after switchings and jumps. In addition to an analytic example, the HMP results are illustrated for the optimal control of an electric vehicle with transmission, where the modelling of the powertrain requires the consideration of both autonomous and controlled switchings accompanied by dimension changes.
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on the relation between the Minimum Principle and dynamic programming for classical and hybrid control systems
IEEE Transactions on Automatic Control, 2017Co-Authors: Ali Pakniyat, Peter E. CainesAbstract:Hybrid optimal control problems are studied for a general class of hybrid systems, where autonomous and controlled state jumps are allowed at the switching instants, and in addition to terminal and running costs, switching between discrete states incurs costs. The statements of the Hybrid Minimum Principle and Hybrid Dynamic Programming are presented in this framework, and it is shown that under certain assumptions, the adjoint process in the Hybrid Minimum Principle and the gradient of the value function in Hybrid Dynamic Programming are governed by the same set of differential equations and have the same boundary conditions and hence are almost everywhere identical to each other along optimal trajectories. Analytic examples are provided to illustrate the results.
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on the stochastic Minimum Principle for hybrid systems
Conference on Decision and Control, 2016Co-Authors: Ali Pakniyat, Peter E. CainesAbstract:A class of stochastic hybrid systems with both autonomous and controlled switchings and jumps is considered where autonomous and controlled state jumps at the switching instants are accompanied by changes in the dimension of the state space. Optimal control problems associated with this class of stochastic hybrid systems are studied where in addition to running and terminal costs, switching between discrete states incurs costs. Necessary optimality conditions are established in the form of the Stochastic Hybrid Minimum Principle. A feature of special importance is the effect of hard constraints imposed by switching manifolds on diffusion-driven state trajectories which influence the boundary conditions for the stochastic Hamiltonian and adjoint processes.
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CDC - On the stochastic Minimum Principle for hybrid systems
2016 IEEE 55th Conference on Decision and Control (CDC), 2016Co-Authors: Ali Pakniyat, Peter E. CainesAbstract:A class of stochastic hybrid systems with both autonomous and controlled switchings and jumps is considered where autonomous and controlled state jumps at the switching instants are accompanied by changes in the dimension of the state space. Optimal control problems associated with this class of stochastic hybrid systems are studied where in addition to running and terminal costs, switching between discrete states incurs costs. Necessary optimality conditions are established in the form of the Stochastic Hybrid Minimum Principle. A feature of special importance is the effect of hard constraints imposed by switching manifolds on diffusion-driven state trajectories which influence the boundary conditions for the stochastic Hamiltonian and adjoint processes.
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on the relation between the hybrid Minimum Principle and hybrid dynamic programming a linear quadratic example
IFAC-PapersOnLine, 2015Co-Authors: Ali Pakniyat, Peter E. CainesAbstract:Abstract Hybrid optimal control problems are studied for systems where autonomous and controlled state jumps are allowed at the switching instants and in addition to running costs, switching between discrete states incurs costs. Key aspects of the analysis are the relationship between the Hamiltonian and the adjoint process in the Hybrid Minimum Principle before and after the switching instants, the boundary conditions on the value function in Hybrid Dynamic Programming at these switching times, as well as the relationship between the adjoint process in the Hybrid Minimum Principle and the gradient process of the value function in Hybrid Dynamic Programming. The results are illustrated through an analytic example with linear dynamics and quadratic costs.
Grigory Kolesov - One of the best experts on this subject based on the ideXlab platform.
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Application of Pontryagin's Minimum Principle to Grover's quantum search problem
Physical Review A, 2019Co-Authors: Chungwei Lin, Yebin Wang, Grigory Kolesov, Uros KalabicAbstract:Grover's algorithm is one of the most famous algorithms which explicitly demonstrates how the quantum nature can be utilized to accelerate the searching process. In this work, Grover's quantum search problem is mapped to a time-optimal control problem. Resorting to Pontryagin's Minimum Principle we find that the time-optimal solution has the bang-singular-bang structure. This structure can be derived naturally, without integrating the differential equations, using the geometric control technique where Hamiltonians in the Schr\"odinger's equation are represented as vector fields. In view of optimal control, Grover's algorithm uses the bang-bang protocol to approximate the optimal protocol with a minimized number of bang-to-bang switchings to reduce the query complexity. Our work provides a concrete example how Pontryagin's Minimum Principle is connected to quantum computation, and offers insight into how a quantum algorithm can be designed.
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application of pontryagin s Minimum Principle to grover s quantum search problem
Physical Review A, 2019Co-Authors: Chungwei Lin, Yebin Wang, Grigory Kolesov, Uros KalabicAbstract:Grover's algorithm is one of the most famous algorithms which explicitly demonstrates how the quantum nature can be utilized to accelerate the searching process. In this work, Grover's quantum search problem is mapped to a time-optimal control problem. Resorting to Pontryagin's Minimum Principle, we find that the time-optimal solution has the bang-singular-bang structure. This structure can be derived naturally, without integrating the differential equations, using the geometric control technique where Hamiltonians in Schr\"odinger's equation are represented as vector fields. In view of optimal control, Grover's algorithm uses the bang-bang protocol to approximate the optimal protocol with a minimized number of bang-to-bang switchings to reduce the query complexity. Our work provides a concrete example of how Pontryagin's Minimum Principle is connected to quantum computation, and offers insight into how a quantum algorithm can be designed.
Chungwei Lin - One of the best experts on this subject based on the ideXlab platform.
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Application of Pontryagin's Minimum Principle to Grover's quantum search problem
Physical Review A, 2019Co-Authors: Chungwei Lin, Yebin Wang, Grigory Kolesov, Uros KalabicAbstract:Grover's algorithm is one of the most famous algorithms which explicitly demonstrates how the quantum nature can be utilized to accelerate the searching process. In this work, Grover's quantum search problem is mapped to a time-optimal control problem. Resorting to Pontryagin's Minimum Principle we find that the time-optimal solution has the bang-singular-bang structure. This structure can be derived naturally, without integrating the differential equations, using the geometric control technique where Hamiltonians in the Schr\"odinger's equation are represented as vector fields. In view of optimal control, Grover's algorithm uses the bang-bang protocol to approximate the optimal protocol with a minimized number of bang-to-bang switchings to reduce the query complexity. Our work provides a concrete example how Pontryagin's Minimum Principle is connected to quantum computation, and offers insight into how a quantum algorithm can be designed.
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application of pontryagin s Minimum Principle to grover s quantum search problem
Physical Review A, 2019Co-Authors: Chungwei Lin, Yebin Wang, Grigory Kolesov, Uros KalabicAbstract:Grover's algorithm is one of the most famous algorithms which explicitly demonstrates how the quantum nature can be utilized to accelerate the searching process. In this work, Grover's quantum search problem is mapped to a time-optimal control problem. Resorting to Pontryagin's Minimum Principle, we find that the time-optimal solution has the bang-singular-bang structure. This structure can be derived naturally, without integrating the differential equations, using the geometric control technique where Hamiltonians in Schr\"odinger's equation are represented as vector fields. In view of optimal control, Grover's algorithm uses the bang-bang protocol to approximate the optimal protocol with a minimized number of bang-to-bang switchings to reduce the query complexity. Our work provides a concrete example of how Pontryagin's Minimum Principle is connected to quantum computation, and offers insight into how a quantum algorithm can be designed.
David Peleg - One of the best experts on this subject based on the ideXlab platform.
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the Minimum Principle of sinr a useful discretization tool for wireless communication
Foundations of Computer Science, 2015Co-Authors: Erez Kantor, Zvi Lotker, Merav Parter, David PelegAbstract:Theoretical study of optimization problems in wireless communication often deals with zero-dimensional tasks. For example, the power control problem requires computing a power assignment guaranteeing that each transmitting station is successfully received at a single receiver point. This paper aims at addressing communication applications that require handling 2-dimensional tasks (e.g., Guaranteeing successful transmission in entire regions rather than in specific points). A natural approach to such tasks is to discretize the 2-dimensional optimization domain, e.g., By sampling points within the domain. This approach, however, might incur high time and memory requirements, and moreover, it cannot guarantee exact solutions. Towards this goal, we establish the Minimum Principle for the SINR function with free-space path loss (i.e., When the signal decays in proportion to the square of the distance between the transmitter and receiver). We then utilize it as a discretization technique for solving two-dimensional problems in the SINR model. This approach is shown to be useful for handling optimization problems over two dimensions (e.g., Power control, energy minimization), in providing tight bounds on the number of null-cells in the reception map, and in approximating geometrical and topological properties of the wireless reception map (e.g., Maximum inscribed sphere). Essentially, the Minimum Principle allows us to reduce the dimension of the optimization domain without losing anything in the accuracy or quality of the solution. More specifically, when the two dimensional optimization domain is bounded and free from any interfering station, the Minimum Principle implies that it is sufficient to optimize over the boundary of the domain, as the "hardest" points to be satisfied reside on boundary and not in the interior. We believe that the Minimum Principle, as well as the interplay between continuous and discrete analysis presented in this paper, may pave the way to future study of algorithmic SINR in higher dimensions.
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FOCS - The Minimum Principle of SINR: A Useful Discretization Tool for Wireless Communication
2015 IEEE 56th Annual Symposium on Foundations of Computer Science, 2015Co-Authors: Erez Kantor, Zvi Lotker, Merav Parter, David PelegAbstract:Theoretical study of optimization problems in wireless communication often deals with zero-dimensional tasks. For example, the power control problem requires computing a power assignment guaranteeing that each transmitting station is successfully received at a single receiver point. This paper aims at addressing communication applications that require handling 2-dimensional tasks (e.g., Guaranteeing successful transmission in entire regions rather than in specific points). A natural approach to such tasks is to discretize the 2-dimensional optimization domain, e.g., By sampling points within the domain. This approach, however, might incur high time and memory requirements, and moreover, it cannot guarantee exact solutions. Towards this goal, we establish the Minimum Principle for the SINR function with free-space path loss (i.e., When the signal decays in proportion to the square of the distance between the transmitter and receiver). We then utilize it as a discretization technique for solving two-dimensional problems in the SINR model. This approach is shown to be useful for handling optimization problems over two dimensions (e.g., Power control, energy minimization), in providing tight bounds on the number of null-cells in the reception map, and in approximating geometrical and topological properties of the wireless reception map (e.g., Maximum inscribed sphere). Essentially, the Minimum Principle allows us to reduce the dimension of the optimization domain without losing anything in the accuracy or quality of the solution. More specifically, when the two dimensional optimization domain is bounded and free from any interfering station, the Minimum Principle implies that it is sufficient to optimize over the boundary of the domain, as the "hardest" points to be satisfied reside on boundary and not in the interior. We believe that the Minimum Principle, as well as the interplay between continuous and discrete analysis presented in this paper, may pave the way to future study of algorithmic SINR in higher dimensions.
