Optimal control theory with arbitrary superpositions of waveforms

Abstract

Abstract. Standard optimal control methods perform optimization in the time domain. However, many experimental settings demand the expression of the control signal as a superposition of given waveforms, a case that cannot easily be accommodated using time-local constraints. Previous approaches [1,2] have circumvented this difficulty by performing optimization in a parameter space, using the chain rule to make a connection to the time domain. In this paper, we present an extension to Optimal Control Theory which allows gradient-based optimization for superpositions of arbitrary waveforms directly in a time-domain subspace. Its key is the use of the Moore-Penrose pseudoinverse as an efficient means of transforming between a time-local and waveform-based descriptions. To illustrate this optimization technique, we study the parametrically driven harmonic oscillator as model system and reduce its energy, considering both Hamiltonian dynamics and stochastic dynamics under the influence of a thermal reservoir. We demonstrate the viability and efficiency of the method for these test cases and find significant advantages in the case of waveforms which do not form an orthogonal basis.

Training Schrödinger’s cat: quantum optimal control

“…In this case, the control only depends on a limited set of parameters which are optimized [118][119][120]. Applications of numerical optimal control are discussed below in Sections 3 to 5 but by now have grown too numerous for a complete bibliography.…”

The Moore–Penrose inverse: a hundred years on a frontline of physics research

“…Further papers demonstrating applications of the Moore-Penrose matrix inverse within the scope of theoretical physics include, among other, articles: Beylkin et al (2008), where the formulae for the inverse of modified matrices were exploited in a Green's function iteration algorithm introduced to solve the timeindependent, multiparticle Schrödinger equation, He et al (2012), which introduced phase-entanglement and phase-squeezing criteria for two bosonic fields that are robust against a number of fluctuations using the inverse to normalize the particle number operator, Huang and Li (2020), where formulae for the resistance distance and Kirchhoff index of a linear hexagonal (cylinder) chain were derived by means of the inverses of Laplacian matrices, Kametaka et al (2015), where the inverse of singular discrete Laplacian was used to solve difference equations to estimate a maximal deviation of a carbon atom from the steady state in C60 fullerene buckyball, Kirkland (2015) dealing with a quantum state transfer in a quantum walk on a graph, with the inverse used to derive expressions for the first and second partial derivatives of the fidelity of the transfer with respect to a weight of an edge, Kougioumtzoglou et al (2017), where an inverse based frequency response function was introduced to generalize frequency domain random vibration solution methodologies to account for linear and nonlinear structural systems with singular matrices, Lian et al (2019), where the inverse was exploited for calculating charge density distribution through Hartree potential to disclose the physical mechanism of electrostatic potential anomaly in 2D Janus transition metal dichalcogenides, McCartin (2009) reexpressing the Rayleigh-Schrödinger perturbation theory procedure in terms of the inverse, Meister et al (2014), where the inverse was used to formulate an optimal control algorithm with a control subspace defined by a superposition of arbitrary waveforms, Pignier et al (2017), where a model of an aeroacoustic sound source was created based on compressible flow simulations, with the inverse used to compute the sound source strengths, Ranjan and Zhang (2013) exploring the geometry of complex networks in terms of an Euclidean embedding represented by the inverse of its graph Laplacian, Yang et al (2018), where the inverse was used to solve an equilibrium equation originating in an empirical mode decomposition method combining the static and dynamic information for structural damage detection, and Yang et al (2020), where an expression for the inverse of Laplacian matrices of two connected weighted graphs was established and utilized to derive a recursion formula for the resistance distance.…”

Second order phase dispersion by optimized rotation pulses

“…Derived from Landau-Zener-Stückelberg-Majorana theory [37], the ensemble of Eq. (1) can be controlled with adiabatic evolution [44][45][46] and is described as a linear frequency sweep over the two-level systems. The phase dispersion resulting from a linear frequency sweep is quadratic and can be used to define the local targets of an optimal control method.…”

“…The phase dispersion resulting from a linear frequency sweep is quadratic and can be used to define the local targets of an optimal control method. Previous work on this theme has optimized state-to-state problems with linear phase dispersion [47][48][49] and quadratic phase dispersion [44,[50][51][52].…”