We present the quantum equation and synthesize an optimal control proce dure for this equation. We develop a theoretical method for the analysis of quantum optimal control system given by the time depending Schrödinger equation. The Legendre wavelet method is proposed for solving this problem. This can be used as an efficient and accurate computational method for obtaining numerical solutions of different quantum optimal control problems. The distinguishing feature of this paper is that it makes the method, previously used to solve non-quantum control equations based on Legendre wavelets, usable by using a change of variables for quantum control equations.
We present the quantum equation and synthesize an optimal control proce dure for this equation. We develop a theoretical method for the analysis of quantum optimal control system given by the time depending Schrödinger equation. The Legendre wavelet method is proposed for solving this problem. This can be used as an efficient and accurate computational method for obtaining numerical solutions of different quantum optimal control problems. The distinguishing feature of this paper is that it makes the method, previously used to solve non-quantum control equations based on Legendre wavelets, usable by using a change of variables for quantum control equations.
In this paper, we present efficient quantum algorithms that are exponentially faster than classical algorithms for solving the quantum optimal control problem. This problem involves finding the control variable that maximizes a physical quantity at time T, where the system is governed by a time-dependent Schrödinger equation. This type of control problem also has an intricate relation with machine learning. Our algorithms are based on a time-dependent Hamiltonian simulation method and a fast gradient-estimation algorithm. We also provide a comprehensive error analysis to quantify the total error from various steps, such as the finite-dimensional representation of the control function, the discretization of the Schrödinger equation, the numerical quadrature, and optimization. Our quantum algorithms require fault-tolerant quantum computers.
Abstract This paper presents a numerical method for solving Schrödinger equations with arbitrary potentials with the adoption of the qunatum diagonalization scheme and basis functions in the discrete wavelet transform. The quantum diagonalization scheme, developed to solve the Hubbard model and so on, is known to allow the diagonalization of a Hamiltonian matrix of a much large size because it does not require to maintain the entire matrix in memory. And, the adoption of wavelet basis functions enables representing eigenstates efficiently using a number of bases much smaller than the number of grid points to integral and reducing the computation time to perform the diagonalization process. With the help of another additional techniques, the method allows to find solutions of a Schrödinger equation much exactly and efficiently. The validity of the method and its efficiency depending on several conditions are proven and surveyed by applying it to several problems for which the exact solutions are known.
When applied quantization to classical control system, quantum mechanics opens completely new perspectives; by exploiting quantum mechanical features, such as, time evolution of quantum state and costate (generalized momenta) of the Hamiltonian of the system, one can formulate and solve the optimal control problems of the quantum mechanical system more efficiently than the classical system. The physical perspective of quantization of the optimal feedback system is outlined by exploiting the formulation of the optimization problem of the system within the context of Hamilton-Jacobi theory and Pontryagin maximum/minimum principle in abstract Hubert space. One of the most striking example is the optimal control problem of synthesizing the feedback law of quantum diffusion process. The optimal diffusion process is shown to satisfy the Schrödinger wave equation of energy with the feedback law as potential function.
In this work, time-depended Schrodinger equation described particles at matter (crystal, catalysis, metal) surface could be considered as propose of numerical control of quantum system. Accessing existing physical experimental results on the motion of particles (molecules, atoms) at surface, based on variational method of quantum control theory in Hilbert space, using density function theory (DFT), time-depended Schrodinger equation to proceed the investigation of computational approach. To do quantum calculation at surface, physically, first needs a concept as control goal: such as breaking a chemical bond as target; reducing energy of high intensity shaped laser pulse. Particles at surface is a kind of constrain control for spatial variable. Optimal control is to find and characterize the quantum optima for minimizing or maximizing the cost functional. Control methods contain selecting chemical reagent, designing chemical reaction, making control scope for a quantized system: time varying Schrodinger equation. Precisely, for general quadratic cost function, in two or three dimensional cases, a semi discrete (time continuous, spatial discrete) algorithm consisting of finite element method and conjugate gradient method, would be utilized for solving a numerical solution of state system, and obtaining quantum optimal control from a initial guess of control input. It is quite curious: what is the difference of control particles occurred at surface than control free particles? whether one can develop a suit of theory or methodology for quantum surface control? It is certainly expected to connect theoretical control, to numerical or computational control, and to experimental control as carrying out quantum system control of particles on the surface. It is desired that quantum control theory (QCT) for quantum dot at surface would be evidenced in visualization method, and attained confidential verification in the guidance of real-time computer-aided experiments in the viewpoint of chemistry and physics.
In this work, time-depended Schrodinger equation described particles at matter (crystal, catalysis, metal) surface could be considered as propose of numerical control of quantum system. Accessing existing physical experimental results on the motion of particles (molecules, atoms) at surface, based on variational method of quantum control theory in Hilbert space, using density function theory (DFT), time-depended Schrodinger equation to proceed the investigation of computational approach. To do quantum calculation at surface, physically, first needs a concept as control goal: such as breaking a chemical bond as target; reducing energy of high intensity shaped laser pulse. Particles at surface is a kind of constrain control for spatial variable. Optimal control is to find and characterize the quantum optima for minimizing or maximizing the cost functional. Control methods contain selecting chemical reagent, designing chemical reaction, making control scope for a quantized system: time varying Schrodinger equation. Precisely, for general quadratic cost function, in two or three dimensional cases, a semi discrete (time continuous, spatial discrete) algorithm consisting of finite element method and conjugate gradient method, would be utilized for solving a numerical solution of state system, and obtaining quantum optimal control from a initial guess of control input. It is quite curious: what is the difference of control particles occurred at surface than control free particles? whether one can develop a suit of theory or methodology for quantum surface control? It is certainly expected to connect theoretical control, to numerical or computational control, and to experimental control as carrying out quantum system control of particles on the surface. It is desired that quantum control theory (QCT) for quantum dot at surface would be evidenced in visualization method, and attained confidential verification in the guidance of real-time computer-aided experiments in the viewpoint of chemistry and physics.
The control of matter and energy at a fundamental level will be a cornerstone of new technologies for years to come. This idea is exemplified in a distilled form by controlling the dynamics of quantum mechanical systems via a time-dependent potential. The contributions detailed within this work focus on the computational aspects of formulating and solving quantum control problems efficiently.
The accurate numerical computation of optimal controls of infinite dimensional quantum control problems is a very difficult task that requires to take into account the features of the original infinite dimensional problem. An important issue is the choice of the functional space where the minimization process is defined. A systematic comparison of L2 versus H1 based minimization shows that the choice of the appropriate functional space matters and has many consequences in the implementation of some optimization techniques.
A matrix-free cascadic BFGS algorithm is introduced in the L 2 and H1 settings and it is demonstrated that the choice of H1 over L2 results in a substantial performance and robustness increase. A comparison between optimal control resulting from function space minimization and the control obtained by minimization over Chebyshev and POD basis function coefficients is presented.
A theoretical and computational framework is presented to obtain accurate controls for fast quantum state transitions that are needed in a host of applications such as nano electronic devices and quantum computing. This method is based on a reduced Hessian Krylov-Newton scheme applied to a norm-preserving discrete model of a dipole quantum control problem. The use of second-order numerical methods for solving the control problem is justified proving existence of optimal solutions and analyzing first- and second-order optimality conditions. Criteria for the discretization of the non-convex optimization problem and for the formulation of the Hessian are given to ensure accurate gradients and a symmetric Hessian. Robustness of the Newton approach is obtained using a globalization strategy with a robust linesearch procedure. Results of numerical experiments demonstrate that the Newton approach presented in this dissertation is able to provide fast and accurate controls for high-energy state transitions.
Control of bound-to-bound and bound-to-continuum transitions in open quantum systems and vector field control of two-dimensional systems is presented. An efficient space-time spectral discretization of the time-dependent Schrodinger equation and preconditioning strategy for a fast approximate solution with Krylov methods is outlined.