In this article, we develop a numerical method to find optimal control pulses that accounts for the separation of timescales between the variation of the input control fields and the applied Hamiltonian. In traditional numerical optimization methods, these timescales are treated as being the same. While this approximation has had much success, in applications where the input controls are filtered substantially or mixed with a fast carrier, the resulting optimized pulses have little relation to the applied physical fields. Our technique remains numerically efficient in that the dimension of our search space is only dependent on the variation of the input control fields, while our simulation of the quantum evolution is accurate on the timescale of the fast variation in the applied Hamiltonian.In this article, we develop a numerical method to find optimal control pulses that accounts for the separation of timescales between the variation of the input control fields and the applied Hamiltonian. In traditional numerical optimization methods, these timescales are treated as being the same. While this approximation has had much success, in applications where the input controls are filtered substantially or mixed with a fast carrier, the resulting optimized pulses have little relation to the applied physical fields. Our technique remains numerically efficient in that the dimension of our search space is only dependent on the variation of the input control fields, while our simulation of the quantum evolution is accurate on the timescale of the fast variation in the applied Hamiltonian.

Feb 2, 20111 citation

Control optimization for parametric Hamiltonians by pulse reconstruction

Piero Luchi, Francesco Turro, Sofia Quaglioni +5 more

Abstract Optimal control techniques provide a means to tailor the control pulses required to generate customized quantum gates, which helps to improve the resilience of quantum simulations to gate errors and device noise. However, the significant amount of (classical) computation required to generate customized gates can quickly undermine the effectiveness of this approach, especially when pulse optimization needs to be iterated. We propose a method to reduce the computational time required to generate the control pulse for a Hamiltonian that is parametrically dependent on a time-varying quantity. We use simple interpolation schemes to accurately reconstruct the control pulses from a set of pulses obtained in advance for a discrete set of predetermined parameter values. We obtain a reconstruction with very high fidelity and a significant reduction in computational effort. We report the results of the application of the proposed method to device-level quantum simulations of the unitary (real) time evolution of two interacting neutrons based on superconducting qubits.Abstract Optimal control techniques provide a means to tailor the control pulses required to generate customized quantum gates, which helps to improve the resilience of quantum simulations to gate errors and device noise. However, the significant amount of (classical) computation required to generate customized gates can quickly undermine the effectiveness of this approach, especially when pulse optimization needs to be iterated. We propose a method to reduce the computational time required to generate the control pulse for a Hamiltonian that is parametrically dependent on a time-varying quantity. We use simple interpolation schemes to accurately reconstruct the control pulses from a set of pulses obtained in advance for a discrete set of predetermined parameter values. We obtain a reconstruction with very high fidelity and a significant reduction in computational effort. We report the results of the application of the proposed method to device-level quantum simulations of the unitary (real) time evolution of two interacting neutrons based on superconducting qubits.

arXiv:2102.12316The European Physical Journal ASep 1, 20236 citations

Computational techniques for optimal control of quantum systems

Evangelos A. Coutsias, G. von Winckel

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.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.

Jan 1, 20110 citations

Optimal control of time-dependent targets

I. Serban, Jan Werschnik, Eberhard K. U. Gross

In this work, we investigate how and to which extent a quantum system can be driven along a prescribed path in Hilbert space by a suitably shaped laser pulse. To calculate the optimal, i.e., the variationally best pulse, a properly defined functional is maximized. This leads to a monotonically convergent algorithm which is computationally not more expensive than the standard optimal-control techniques to push a system, without specifying the path, from a given initial to a given final state. The method is successfully applied to drive the time-dependent density along a given trajectory in real space and to control the time-dependent occupation numbers of a two-level system and of a one-dimensional model for the hydrogen atom.In this work, we investigate how and to which extent a quantum system can be driven along a prescribed path in Hilbert space by a suitably shaped laser pulse. To calculate the optimal, i.e., the variationally best pulse, a properly defined functional is maximized. This leads to a monotonically convergent algorithm which is computationally not more expensive than the standard optimal-control techniques to push a system, without specifying the path, from a given initial to a given final state. The method is successfully applied to drive the time-dependent density along a given trajectory in real space and to control the time-dependent occupation numbers of a two-level system and of a one-dimensional model for the hydrogen atom.

arXiv:quant-ph/0409124Physical Review AMay 24, 200573 citations

Efficient Quantum Algorithms for Quantum Optimal Control

Xiantao Li, Chunhao Wang

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.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.

arXiv:2304.02613Apr 5, 20231 citation

Optimal pulse design in quantum control: A unified computational method

Jr-Shin Li, Justin Ruths, Tsyr‐Yan Yu +2 more

Many key aspects of control of quantum systems involve manipulating a large quantum ensemble exhibiting variation in the value of parameters characterizing the system dynamics. Developing electromagnetic pulses to produce a desired evolution in the presence of such variation is a fundamental and challenging problem in this research area. We present such robust pulse designs as an optimal control problem of a continuum of bilinear systems with a common control function. We map this control problem of infinite dimension to a problem of polynomial approximation employing tools from geometric control theory. We then adopt this new notion and develop a unified computational method for optimal pulse design using ideas from pseudospectral approximations, by which a continuous-time optimal control problem of pulse design can be discretized to a constrained optimization problem with spectral accuracy. Furthermore, this is a highly flexible and efficient numerical method that requires low order of discretization and yields inherently smooth solutions. We demonstrate this method by designing effective broadband π/2 and π pulses with reduced rf energy and pulse duration, which show significant sensitivity enhancement at the edge of the spectrum over conventional pulses in 1D and 2D NMR spectroscopy experiments.Many key aspects of control of quantum systems involve manipulating a large quantum ensemble exhibiting variation in the value of parameters characterizing the system dynamics. Developing electromagnetic pulses to produce a desired evolution in the presence of such variation is a fundamental and challenging problem in this research area. We present such robust pulse designs as an optimal control problem of a continuum of bilinear systems with a common control function. We map this control problem of infinite dimension to a problem of polynomial approximation employing tools from geometric control theory. We then adopt this new notion and develop a unified computational method for optimal pulse design using ideas from pseudospectral approximations, by which a continuous-time optimal control problem of pulse design can be discretized to a constrained optimization problem with spectral accuracy. Furthermore, this is a highly flexible and efficient numerical method that requires low order of discretization and yields inherently smooth solutions. We demonstrate this method by designing effective broadband π/2 and π pulses with reduced rf energy and pulse duration, which show significant sensitivity enhancement at the edge of the spectrum over conventional pulses in 1D and 2D NMR spectroscopy experiments.

Proceedings of the National Academy of SciencesJan 18, 2011131 citations

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, 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.