Quantum Optimal Control Problems Research Articles

MISTER-T: An open-source software package for quantum optimal control of multi-electron systems on arbitrary geometries

We present an open-source software package, MISTER-T (Manipulating an Interacting System of Total Electrons in Real-Time), for the quantum optimal control of interacting electrons within a time-dependent Kohn-Sham formalism. In contrast to other implementations restricted to simple models on rectangular domains, our method enables quantum optimal control calculations for multi-electron systems (in the effective mass formulation) on nonuniform meshes with arbitrary two-dimensional cross-sectional geometries. Our approach is enabled by forward and backward propagator integration methods to evolve the Kohn-Sham equations with a pseudoskeleton decomposition algorithm for enhanced computational efficiency. We provide several examples of the versatility and efficiency of the MISTER-T code in handling complex geometries and quantum control mechanisms. The capabilities of the MISTER-T code provide insight into the implications of varying propagation times and local control mechanisms to understand a variety of strategies for manipulating electron dynamics in these complex systems. Program summaryProgram Title: MISTER-TCPC Library link to program files:https://doi.org/10.17632/psymy4ddnw.1Licensing provisions: GNU General Public License 3Programming language: MATLABSupplementary material: animated movies of total electron densities under the influence of optimal control fields for (1) an asymmetric double-well potential for long propagation times, (2) an asymmetric double-well potential for short propagation times, and (3) a triple-well potential with a position-dependent effective mass.Nature of problem: The MISTER-T code solves quantum optimal control problems for interacting electrons within a time-dependent Kohn-Sham formalism. It can handle two-dimensional systems with arbitrary cross-sectional geometries within the effective mass formulation. The user-friendly code uses forward and backward propagator integration methods to evolve the Kohn-Sham equations with a pseudoskeleton decomposition algorithm for enhanced computational efficiency.Solution method: iterative solution of the quantum optimal control equations using finite element methods, effective mass formulation, pseudoskeleton decomposition, sparse matrix linear algebra, and nonuniform fast Fourier transforms.

Solving quantum optimal control problems using projection-operator-based Newton steps

Solving quantum optimal control problems using projection-operator-based Newton steps

QuOCS: The quantum optimal control suite

Quantum optimal control includes a family of pulse-shaping algorithms that aim to unlock the full potential of a variety of quantum technologies. The Quantum Optimal Control Suite (QuOCS) unites experimental focus and model-based approaches in a unified framework. Easy usage and installation presented here and the availability of various combinable optimization strategies is designed to improve the performance of many quantum technology platforms, such as color defects in diamond, superconducting qubits, atom- or ion-based quantum computers. It can also be applied to the study of more general phenomena in physics. In this paper, we describe the software and the toolbox of gradient-free and gradient-based algorithms. We then show how the user can connect it to their experiment. In addition, we provide illustrative examples where our optimization suite solves typical quantum optimal control problems, in both open- and closed-loop settings. Integration into existing experimental control software is already provided for the experiment control software Qudi (Binder et al., 2017 [41]), and further extensions are investigated and highly encouraged.QuOCS is available from GitHub, under Apache License 2.0, and can be found on the PyPI repository. Program summaryProgram Title: QuOCS - Quantum Optimal Control SuiteCPC Library link to program files:https://doi.org/10.17632/wjjch757fk.1Developer's repository link:https://github.com/Quantum-OCS/QuOCSLicensing provisions: Apache-2.0Programming language: PythonExternal routines: NumPy [1], SciPy [1], JAX [2]Nature of problem: Quantum systems are typically controlled by time-dependent electromagnetic fields to perform a certain set of quantum operations. Those operations may in turn provide building blocks for various quantum information processing tasks such as quantum computation, communication, simulation, sensing, and metrology. Numerous control strategies exist to design and improve such operations [3]. While some strategies are constructed to target a rather specific problem with high efficiency, others are more general to solve a wide range of applications [4,5]. To access the different algorithms, one has to download different optimization suites with different input and output parameters, making them hard to compare and combine. To benefit from the variety of algorithms, we have devised a customizable and intuitive optimization suite that simultaneously provides access to some of the most popular quantum optimal control algorithms.Solution method: We combine, in a unified framework, some of the frequently used optimal control algorithms which are the dressed Chopped Random Basis method (dCRAB) [6], and Gradient Ascent Pulse Engineering (GRAPE) [7], with an extension to make use of Automatic Differentiation (AD) [8]. The software is able to connect to both models of quantum dynamics in simulations and real-time quantum experiments to perform open- and closed-loop optimization, respectively. With minimal knowledge of optimal control theory, the user can manage to run optimizations of quantum processes using a variety of additional features such as stopping criteria and drift compensation. Logging and data management of the optimization progress and results are also handled by the suite. Its modular structure allows for extensions that accommodate customized algorithms and can be implemented by the user straightforwardly.Additional comments including unusual features: The connection to the experiments is performed by an extension that enables a direct integration to a laboratory control software Qudi [9].

Solving quantum optimal control problems by wavelets method

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 Schrodinger 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.

Crowdsourcing human common sense for quantum control

Citizen science methodologies have over the past decade been applied with great success to help solve highly complex numerical challenges. Here, we take early steps in the quantum physics arena by introducing a citizen science game, Quantum Moves 2, and compare the performance of different optimization methods across three different quantum optimal control problems of varying difficulty. Inside the game, players can apply a gradient-based algorithm (running locally on their device) to optimize their solutions and we find that these results perform roughly on par with the best of the tested standard optimization methods performed on a computer cluster. In addition, cluster-optimized player seeds was the only method to exhibit roughly optimal performance across all three challenges. This highlights the potential for crowdsourcing the solution of future quantum research problems.

Fourier compression: A customization method for quantum control protocols

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 \textit{et al}, arXiv:1911.07105]. Originally, the technique involved the computation of the exact Hessian matrix. In this paper, we show that the navigation can be alternatively performed with a finite-difference approximation scheme, thus enabling the application of this procedure in systems where computing the exact Hessian is out of reach.

Exploiting landscape geometry to enhance quantum optimal control

The successful application of Quantum Optimal Control (QOC) over the past decades 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.

A sequential quadratic Hamiltonian scheme for solving non-smooth quantum control problems with sparsity

A sequential quadratic Hamiltonian (SQH) scheme is investigated for solving non-smooth quantum optimal control problems with control costs that promote sparsity. Quantum control problems are a representative class of bilinear control problems that are central in application in nanosciences, and in this context the use of sparsity promoting functionals are desirable to facilitate the implementation of optimal controls in the laboratory.In this framework, it is shown that the SQH scheme provides many advantages with respect to other state-of-the-art methods, since it allows to also consider non-convex and discontinuous costs, without the need of regularization techniques. A theoretical discussion and results of numerical experiments are presented that demonstrate the effectiveness of the SQH scheme in solving control problems governed by a quantum spin system.

Quantum Optimal Control Problems with a Sparsity Cost Functional

ABSTRACTIn this article, the investigation of a class of quantum optimal control problems with L1 sparsity cost functionals is presented. The focus is on quantum systems modeled by Schrödinger-type equations with a bilinear control structure as it appears in many applications in nuclear magnetic resonance spectroscopy, quantum imaging, quantum computing, and in chemical and photochemical processes. In these problems, the choice of L1 control spaces promotes sparse optimal control functions that are conveniently produced by laboratory pulse shapers. The characterization of L1 quantum optimal controls and an efficient numerical semi-smooth Newton solution procedure are discussed.

A LONE code for the sparse control of quantum systems

In many applications with quantum spin systems, control functions with a sparse and pulse-shaped structure are often required. These controls can be obtained by solving quantum optimal control problems with L1-penalized cost functionals. In this paper, the MATLAB package LONE is presented aimed to solving L1-penalized optimal control problems governed by unitary-operator quantum spin models. This package implements a new strategy that includes a globalized semi-smooth Krylov–Newton scheme and a continuation procedure. Results of numerical experiments demonstrate the ability of the LONE code in computing accurate sparse optimal control solutions. Program summaryProgram title: LONECatalogue identifier: AEYV_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEYV_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 1683No. of bytes in distributed program, including test data, etc.: 8654Distribution format: tar.gzProgramming language: MATLAB - (OCTAVE).Computer: Any capable of running MATLAB or OCTAVE.Operating system: Any capable of running MATLAB or OCTAVE.RAM: BytesClassification: 4.9.External routines: gmres MATLAB routineNature of problem: A semi-smooth Newton scheme for solving quantum spin-12 sparse optimal control problems.Solution method: Full-complex semi-smooth Newton method and continuation techniquesRunning time: 60–600 sec

SKRYN: A fast semismooth-Krylov–Newton method for controlling Ising spin systems

The modeling and control of Ising spin systems is of fundamental importance in NMR spectroscopy applications. In this paper, two computer packages, ReHaG and SKRYN, are presented. Their purpose is to set-up and solve quantum optimal control problems governed by the Liouville master equation modeling Ising spin-12 systems with pointwise control constraints. In particular, the MATLAB package ReHaG allows to compute a real matrix representation of the master equation. The MATLAB package SKRYN implements a new strategy resulting in a globalized semismooth matrix-free Krylov–Newton scheme. To discretize the real representation of the Liouville master equation, a norm-preserving modified Crank–Nicolson scheme is used. Results of numerical experiments demonstrate that the SKRYN code is able to provide fast and accurate solutions to the Ising spin quantum optimization problem. Program summaryProgram title: SKRYN, ReHaGCatalogue identifier: AEVN_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEVN_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 8452No. of bytes in distributed program, including test data, etc.: 602389Distribution format: tar.gzProgramming language: MATLAB - (OCTAVE).Computer: Any running Matlab.Operating system: Any running Matlab.RAM: bytesClassification: 4.9.External routines: gmres MATLAB routineNature of problem: Set-up and solution of quantum Ising spin- 1/2 optimal control problems, for NMR spectroscopy applications.Solution method: Krylov-matrix-free implementation of the semismooth NewtonRunning time: 60–600 s

Design of coherent quantum observers for linear quantum systems

Quantum versions of control problems are often more difficult than their classical counterparts because of the additional constraints imposed by quantum dynamics. For example, the quantum LQG and quantum optimal control problems remain open. To make further progress, new, systematic and tractable methods need to be developed. This paper gives three algorithms for designing coherent quantum observers, i.e., quantum systems that are connected to a quantum plant and their outputs provide information about the internal state of the plant. Importantly, coherent quantum observers avoid measurements of the plant outputs. We compare our coherent quantum observers with a classical (measurement-based) observer by way of an example involving an optical cavity with thermal and vacuum noises as inputs.

Information theoretical analysis of quantum optimal control.

We study the relations between classical information and the feasibility of accurate manipulation of quantum system dynamics. We show that if an efficient classical representation of the dynamics exists, optimal control problems on many-body quantum systems can be solved efficiently with finite precision. In particular, one-dimensional slightly entangled dynamics can be efficiently controlled. We provide a bound for the minimal time necessary to perform the optimal process given the bandwidth of the control pulse, which is the continuous version of the Solovay-Kitaev theorem. Finally, we quantify how noise affects the presented results.

Wei–Norman equations for a unitary evolution

The Wei–Norman technique allows one to express the solution of a system of linear non-autonomous differential equations in terms of product of exponentials with time-dependent exponents being solutions of a system of nonlinear differential equations. We show that in the unitary case, i.e. when the solution of the linear system is given by a unitary evolution operator, the nonlinear system, by an appropriate choice of ordering, can be reduced to a hierarchy of matrix Riccati equations. To this end, we consider a general linear non-autonomous dynamical system on the special linear group . The unitary case, of particular significance for quantum optimal control problems, is then obtained by restriction to anti-Hermitian generators. We also point to the connections of the obtained results with the theory of the so-called Lie systems.

QUCON: A fast Krylov–Newton code for dipole quantum control problems

A computer package (QUCON) is presented aimed at the solution of dipole quantum optimal control problems. This MATLAB package is based on a recently developed computational strategy based on a globalized reduced Hessian Krylov–Newton scheme and a discretize-before-optimize approach. To discretize the governing Schrödinger model a norm-preserving modified Crank–Nicolson scheme is used. The discretize-before-optimize criteria allows the formulation of accurate gradients and a symmetric Hessian. Robustness of the resulting Newton approach is guaranteed using a robust linesearch procedure. Results of experiments demonstrate that the QUCON code is able to provide fast and accurate controls for high-energy state transitions. Program summary Program title: QUCON Catalogue identifier: AEGZ_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEGZ_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 543 No. of bytes in distributed program, including test data, etc.: 11 419 Distribution format: tar.gz Programming language: MATLAB Computer: PCs, AMD Athlon 64 × 2 Dual, 2.21 GHz, 1.5 GB RAM Operating system: Microsoft Windows XP Word size: 32 bit Classification: 4.9 Nature of problem: Dipole quantum control Solution method: Iterative Running time: 60–600 sec

Monotonically convergent algorithms for solving quantum optimal control problems of a dynamical system nonlinearly interacting with a control

We develop a family of monotonically convergent iterative algorithms for solving a wide class of optimal control problems in which a dynamical system interacts nonlinearly with a control. The key idea is to divide a control into identical components and to introduce auxiliary steps to update each component at every iteration step. The algorithms are proved to exhibit monotonic convergence, which is also numerically confirmed through a case study of the control of molecular orientation. The numerical results show that high-quality solutions can be obtained by using the present algorithms.

A cascadic monotonic time-discretized algorithm for finite-level quantum control computation

A computer package (CNMS) is presented aimed at the solution of finite-level quantum optimal control problems. This package is based on a recently developed computational strategy known as monotonic schemes. Quantum optimal control problems arise in particular in quantum optics where the optimization of a control representing laser pulses is required. The purpose of the external control field is to channel the system's wavefunction between given states in its most efficient way. Physically motivated constraints, such as limited laser resources, are accommodated through appropriately chosen cost functionals. Program summary Program title: CNMS Catalogue identifier: ADEB_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADEB_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 770 No. of bytes in distributed program, including test data, etc.: 7098 Distribution format: tar.gz Programming language: MATLAB 6 Computer: AMD Athlon 64 × 2 Dual, 2:21 GHz, 1:5 GB RAM Operating system: Microsoft Windows XP Word size: 32 Classification: 4.9 Nature of problem: Quantum control Solution method: Iterative Running time: 60–600 sec

Formulation and numerical solution of finite-level quantum optimal control problems

Optimal control of finite-level quantum systems is investigated, and iterative solution schemes for the optimization of a control representing laser pulses are developed. The purpose of this external field is to channel the system's wavefunction between given states in its most efficient way. Physically motivated constraints, such as limited laser resources or population suppression of certain states, are accounted for through an appropriately chosen cost functional. First-order necessary optimality conditions and second-order sufficient optimality conditions are investigated. For solving the optimal control problems, a cascadic non-linear conjugate gradient scheme and a monotonic scheme are discussed. Results of numerical experiments with a representative finite-level quantum system demonstrate the effectiveness of the optimal control formulation and efficiency and robustness of the proposed approaches.

Monotonically convergent algorithms for solving quantum optimal control problems described by an integrodifferential equation of motion

A family of monotonically convergent algorithms is presented for solving a wide class of quantum optimal control problems satisfying an inhomogeneous integrodifferential equation of motion. The convergence behavior is examined using a four-level model system under the influence of non-Markovian relaxation. The results show that high quality solutions can be obtained over a wide range of parameters that characterize the algorithms, independent of the presence or absence of relaxation.

Generalized monotonically convergent algorithms for solving quantum optimal control problems.

A wide range of cost functionals that describe the criteria for designing optimal pulses can be reduced to two basic functionals by the introduction of product spaces. We extend previous monotonically convergent algorithms to solve the generalized pulse design equations derived from those basic functionals. The new algorithms are proved to exhibit monotonic convergence. Numerical tests are implemented in four-level model systems employing stationary and/or nonstationary targets in the absence and/or presence of relaxation. Trajectory plots that conveniently present the global nature of the convergence behavior show that slow convergence may often be attributed to "trapping" and that relaxation processes may remove such unfavorable behavior.