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arXiv:2001.05941 · Physical Review A

Fourier compression: A customization method for quantum control protocols

Martín Larocca, Esteban Calzetta, Diego A. Wisniacki

Published Sep 16, 2020• 7 citations
arXivHTMLPDF

Abstract

Quantum optimal control (QOC) is the field devoted to the production of external control protocols that actively guide quantum dynamics. Solutions to QOC problems were shown to constitute continuous submanifolds of control space. A solution navigation method exploiting this property to achieve secondary features in the control protocols was proposed [Larocca, Calzetta, and Wisniacki, Phys. Rev. A 101, 023410 (2020)]. Here, we present a navigation-powered protocol postprocessing mechanism allowing naturally inflexible optimization algorithms to produce user-customized solutions, for example, in terms of a Fourier decomposition. We test the performance of the method in two inherently different models: the two-level Landau-Zener system and the quantum harmonic oscillator.

DOI
10.1103/physreva.102.033108

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Quantum optimal control (QOC) is the field devoted to the production of external control protocols that actively guide quantum dynamics. Solutions to QOC problems were shown to constitute continuous submanifolds of control space. A solution navigation method exploiting this property to achieve secondary features in the control protocols was proposed [Larocca, Calzetta, and Wisniacki, Phys. Rev. A 101, 023410 (2020)]. Here, we present a navigation-powered protocol postprocessing mechanism allowing naturally inflexible optimization algorithms to produce user-customized solutions, for example, in terms of a Fourier decomposition. We test the performance of the method in two inherently different models: the two-level Landau-Zener system and the quantum harmonic oscillator.
Abstract Quantum optimal control (QOC) is a set of methods for designing time-varying electromagnetic fields to perform operations in quantum technologies. This tutorial paper introduces the basic elements of this theory based on the Pontryagin maximum principle, in a physicist-friendly way. An analogy with classical Lagrangian and Hamiltonian mechanics is proposed to present the main results used in this field. Emphasis is placed on the different numerical algorithms to solve a QOC problem. Several examples ranging from the control of two-level quantum systems to that of Bose–Einstein condensates (BECs) in a one-dimensional optical lattice are studied in detail, using both analytical and numerical methods. Codes based on shooting method and gradient-based algorithms are provided. The connection between optimal processes and the quantum speed limit is also discussed in two-level quantum systems. In the case of BEC, the experimental implementation of optimal control protocols is described, both for two-level and many-level cases, with the current constraints and limitations of such platforms. This presentation is illustrated by the corresponding experimental results.
The successful application of quantum optimal control (QOC) over the past decades has unlocked the possibility of directing the dynamics of quantum systems. Nevertheless, solutions obtained from QOC algorithms are usually highly irregular, making them unsuitable for direct experimental implementation. In this paper, we propose a method to reshape those unattractive optimal controls. The approach is based on the fact that solutions to QOC problems are not isolated policies but constitute multidimensional submanifolds of control space. This was originally shown for finite-dimensional systems. Here, we analytically prove that this property is still valid in a continuous variable system. The degenerate subspace can be effectively traversed by moving in the null subspace of the Hessian of the cost function, allowing for the pursuit of secondary objectives. To demonstrate the usefulness of this procedure, we apply the method to smooth and compress optimal protocols in order to meet laboratory demands.
The river-crossing problem, also known as Zermelo's navigation problem, is a classic example of an optimization problem with practical relevance and a scalable degree of complexity. It asks for the optimal trajectory of a vessel moving through a water flow field and provides a setting in which physics, variational methods, and optimization are naturally intertwined. We state a version of Zermelo's problem and then solve it as formulate it as an adiabatic quantum-computing problem using quantum trits, or qutrits for short. The construction includes a penalty term that enforces the prescribed boundary conditions and an exploration term that allows the system to move through intermediate configurations toward the optimal feasible path. In the adiabatic description, the evolution proceeds through a sequence of avoided crossings, so that the resulting fidelity can be estimated using the Landau-Zener formula. Remarkably, the regime in which this approximation is valid also provides a deterministic way to identify the correct solution with computational effort that scales only quadratically with the problem size. Thus, a quantum formulation initially motivated by the apparent exponential complexity of the problem reveals an underlying classical structure that can be exploited efficiently. Our approach also provides a pedagogical illustration of how a real-world optimization problem can be cast into a quantum-annealing framework and then analyzed using the Schrödinger equation, avoided crossings, and Landau-Zener theory.
Quantum optimal control (QOC) supports the advance of quantum technologies by tackling its problems at the pulse level: numerical approaches iteratively work toward a given target by parametrizing the applied time-dependent fields with a finite set of variables. The effectiveness of the resulting optimization depends on the complexity of the problem and the number of variables. We consider different parametrizations in terms of basis functions, asking whether the choice of the applied basis affects the quality of the optimization. Furthermore, we consider strategies to choose the most suitable basis. For comparison, we test three different randomizable bases---introducing the sinc and sigmoid bases as alternatives to the Fourier basis---on QOC problems of varying complexity. For each problem, the basis-specific convergence rates result in a unique ranking. Especially for expensive evaluations, e.g., in closed loop, a potential speedup by a factor of up to 10 may be crucial for the optimization's feasibility. We conclude that a problem-dependent basis choice is an influential factor for QOC efficiency and provide advice for its approach.
Quantum optimal control (QOC) enables the realization of accurate operations, such as quantum gates, and supports the development of quantum technologies. To date, many QOC frameworks have been developed, but those remain only naturally suited to optimize a single targeted operation at a time. We extend this concept to optimal control with a continuous family of targets, and demonstrate that an optimization based on neural networks can find families of time-dependent Hamiltonians realizing desired classes of quantum gates in minimal time.
Abstract The core problem in optimal control theory applied to quantum systems is to determine the temporal shape of an applied field in order to maximize the expected value of some physical observable. The complexity of this procedure is given by the structural and topological features of the quantum control landscape (QCL)—i.e. the functional which maps the control field into a given value of the observable. In this work, we analyze the rich structure of the QCL in the paradigmatic Landau–Zener two-level model, and focus in particular on characterizing the QCL when the total evolution time is severely constrained. By studying several features of the optimized solutions, such as their abundance, spatial distribution and fidelities, we are able to rationalize several geometrical and topological aspects of the QCL of this simple model and identify the effects produced by different types of constraint.