Quantum Optimal Control Research Articles

Dressing the chopped-random-basis optimization: A bandwidth-limited access to the trap-free landscape

In quantum optimal control theory the success of an optimization algorithm is highly influenced by how the figure of merit to be optimized behaves as a function of the control field, i.e. by the control landscape. Constraints on the control field introduce local minima in the landscape --false traps-- which might prevent an efficient solution of the optimal control problem. Rabitz et al. [Science 303, 1998 (2004)] showed that local minima occur only rarely for unconstrained optimization. Here, we extend this result to the case of bandwidth-limited control pulses showing that in this case one can eliminate the false traps arising from the constraint. Based on this theoretical understanding, we modify the Chopped Random Basis (CRAB) optimal control algorithm and show that this development exploits the advantages of both (unconstrained) gradient algorithms and of truncated basis methods, allowing to always follow the gradient of the unconstrained landscape by bandwidth-limited control functions. We study the effects of additional constraints and show that for reasonable constraints the convergence properties are still maintained. Finally, we numerically show that this approach saturates the theoretical bound on the minimal bandwidth of the control needed to optimally drive the system.

Quantum speed limit and optimal control of many-boson dynamics

We extend the concept of quantum speed limit -- the minimal time needed to perform a driven evolution -- to complex interacting many-body systems. We investigate a prototypical many-body system, a bosonic Josephson junction, at increasing levels of complexity: (a) within the two-mode approximation {corresponding to} a nonlinear two-level system, (b) at the mean-field level by solving the nonlinear Gross-Pitaevskii equation in a double well potential, and (c) at an exact many-body level by solving the time-dependent many-body Schr\"odinger equation. We propose a control protocol to transfer atoms from the ground state of a well to the ground state of the neighbouring well. Furthermore, we show that the detrimental effects of the inter-particle repulsion can be eliminated by means of a compensating control pulse, yielding, quite surprisingly, an enhancement of the transfer speed because of the particle interaction -- in contrast to the self-trapping scenario. Finally, we perform numerical optimisations of both the nonlinear and the (exact) many-body quantum dynamics in order to further enhance the transfer efficiency close to the quantum speed limit.

Training Schrödinger’s cat: quantum optimal control

It is control that turns scientific knowledge into useful technology: in physics and engineering it provides a systematic way for driving a system from a given initial state into a desired target state with minimized expenditure of energy and resources -- as famously applied in the Apollo programme. As one of the cornerstones for enabling quantum technologies, optimal quantum control keeps evolving and expanding into areas as diverse as quantum-enhanced sensing, manipulation of single spins, photons, or atoms, optical spectroscopy, photochemistry, magnetic resonance (spectroscopy as well as medical imaging), quantum information processing and quantum simulation. --- Here state-of-the-art quantum control techniques are reviewed and put into perspective by a consortium uniting expertise in optimal control theory and applications to spectroscopy, imaging, quantum dynamics of closed and open systems. We address key challenges and sketch a roadmap to future developments.

Reduced-dynamics approach for optimally designing unitary transformations

Quantum optimal control is a powerful tool for designing a pulse that executes a specified gate, i.e., a unitary transformation in quantum computation. If we consider qubits characterized by different spectroscopic properties, we may design a pulse that executes a specified gate on a specified qubit ($S$ qubit) with the inclusion of neighboring qubits ($B$ qubits) but does not explicitly include the other qubits. In this situation, first we introduce sets of reduced time-evolution operators associated with the $S$ and $B$ qubits. Then, we apply quantum optimal control to these sets to design optimal pulses that could selectively execute the specified gate on the $S$ qubit while actively removing the effects of the pulse on the $B$ qubits, which would lead to a scalable pulse-design procedure. The present pulse design is applied to the quantum computation proposed by DeMille with the aim of executing the Hadamard and controlled-not gates on the $S$ qubits while actively suppressing the pulse-induced population transitions of the $B$ qubit states. The numerical results illustrate high fidelity even after introducing the extra qubits that are not included in the pulse design.

Role of control constraints in quantum optimal control

The problems of optimizing the value of an arbitrary observable of the two-level system at both a fixed time and the shortest possible time is theoretically explored. Complete identification and classification along with comprehensive analysis of globally optimal control policies and traps (i.e. policies which are locally but not globally optimal) is presented. The central question addressed is whether the control landscape remains trap-free if control constraints of the inequality type are imposed. The answer is astonishingly controversial, namely, although formally it is always negative, in practice it is positive provided that the control time is fixed and chosen long enough.

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

Smooth Optimal Quantum Control for Robust Solid-State Spin Magnetometry.

We experimentally demonstrate a simple yet versatile optimal quantum control technique that achieves tailored robustness against qubit inhomogeneities and control errors while requiring minimal bandwidth. We apply the technique to nitrogen-vacancy (NV) centers in diamond and verify its performance using quantum process tomography. In a wide-field NV center magnetometry scenario, we achieve a homogeneous sensitivity across a 33% drop in control amplitude, and we improve the sensitivity by up to 2 orders of magnitude for a normalized detuning as large as 40%, achieving a value of 20 nT Hz(-1/2) μm(3/2) in sensitivity times square root volume.

Efficient photo-dissociation of CH4 and H2CO molecules with optimized ultra-short laser pulses

The fragmentation dynamics of CH4 and H2CO molecules have been studied with ultra-short pulses at laser intensityof up to 1015Wcm−2. Three dimensional molecular dynamics calculations for finding the optimized laser pulses are presented based on time-dependent density functional theory and quantum optimal control theory. A comparison of the results for orientation dependence in the ionization process shows that the electron distribution for CH4 is more isotropic than H2CO molecule. Total conversion yields of up to 70% at an orientation angle of 30o for CH4 and 65% at 900 for H2CO are achieved which lead to enhancement of dissociation probability.

Analytical optimal pulse shapes obtained with the aid of genetic algorithms.

We propose a methodology to design optimal pulses for achieving quantum optimal control on molecular systems. Our approach constrains pulse shapes to linear combinations of a fixed number of experimentally relevant pulse functions. Quantum optimal control is obtained by maximizing a multi-target fitness function using genetic algorithms. As a first application of the methodology, we generated an optimal pulse that successfully maximized the yield on a selected dissociation channel of a diatomic molecule. Our pulse is obtained as a linear combination of linearly chirped pulse functions. Data recorded along the evolution of the genetic algorithm contained important information regarding the interplay between radiative and diabatic processes. We performed a principal component analysis on these data to retrieve the most relevant processes along the optimal path. Our proposed methodology could be useful for performing quantum optimal control on more complex systems by employing a wider variety of pulse shape functions.

Quantum optimal control within the rotating-wave approximation

We study the interplay between rotating wave approximation and optimal control. In particular, we show that for a wide class of optimal control problems one can choose the control field such that the Hamiltonian becomes time-independent under the rotating wave approximation. Thus, we show how to recast the functional minimization defined by the optimal control problem into a simpler multi-variable function minimization. We provide the analytic solution to the state-to-state transfer of the paradigmatic two-level system and to the more general star configuration of an $N$-level system. We demonstrate numerically the usefulness of this approach in the more general class of connected acyclic $N$-level systems with random spectra. Finally, we use it to design a protocol to entangle Rydberg via constant laser pulses atoms in an experimentally relevant range of parameters.

Auxiliary matrix formalism for interaction representation transformations, optimal control, and spin relaxation theories.

Auxiliary matrix exponential method is used to derive simple and numerically efficient general expressions for the following, historically rather cumbersome, and hard to compute, theoretical methods: (1) average Hamiltonian theory following interaction representation transformations; (2) Bloch-Redfield-Wangsness theory of nuclear and electron relaxation; (3) gradient ascent pulse engineering version of quantum optimal control theory. In the context of spin dynamics, the auxiliary matrix exponential method is more efficient than methods based on matrix factorizations and also exhibits more favourable complexity scaling with the dimension of the Hamiltonian matrix.

Enhancement of quantum speed limit time due to cooperative effects in multilevel systems

Deriving minimum evolution times is of paramount importance in quantum mechanics. Bounds on the speed of evolution are given by the so called quantum speed limit (QSL). In this work we use quantum optimal control methods to study the QSL for driven many level systems which exhibit local two-level interactions in the form of avoided crossings (ACs). Remarkably, we find that optimal evolution times are proportionally smaller than those predicted by the well-known two-level case, even when the ACs are isolated. We show that the physical mechanism for such enhancement is due to non-trivial cooperative effects between the AC and other levels, which are dynamically induced by the shape of the optimized control field.

Laser pulses for coherent xuv Raman excitation

We combine multi-channel electronic structure theory with quantum optimal control to derive Raman pulse sequences that coherently populate a valence excited state. For a neon atom, Raman target populations of up to 13% are obtained. Superpositions of the ground and valence Raman states with a controllable relative phase are found to be reachable with up to 4.5% population and phase control facilitated by the pump pulse carrier envelope phase. Our results open a route to creating core-hole excitations in molecules and aggregates that locally address specific atoms and represent the first step towards realization of multidimensional spectroscopy in the xuv and x-ray regimes.

Optimal State Estimation for Cavity Optomechanical Systems.

We demonstrate optimal state estimation for a cavity optomechanical system through Kalman filtering. By taking into account nontrivial experimental noise sources, such as colored laser noise and spurious mechanical modes, we implement a realistic state-space model. This allows us to obtain the conditional system state, i.e., conditioned on previous measurements, with a minimal least-squares estimation error. We apply this method to estimate the mechanical state, as well as optomechanical correlations both in the weak and strong coupling regime. The application of the Kalman filter is an important next step for achieving real-time optimal (classical and quantum) control of cavity optomechanical systems.

Optimal control of fast and high-fidelity quantum gates with electron and nuclear spins of a nitrogen-vacancy center in diamond

A negatively charged nitrogen vacancy (NV) center in diamond has been recognized as a good solid-state qubit. A system consisting of the electronic spin of the NV center and hyperfine-coupled nitrogen and additionally nearby carbon nuclear spins can form a quantum register of several qubits for quantum information processing or as a node in a quantum repeater. Several impressive experiments on the hybrid electron and nuclear spin register have been reported, but fidelities achieved so far are not yet at or below the thresholds required for fault-tolerant quantum computation (FTQC). Using quantum optimal control theory based on the Krotov method, we show here that fast and high-fidelity single-qubit and two-qubit gates in the universal quantum gate set for FTQC, taking into account the effects of the leakage state, nearby noise qubits and distant bath spins, can be achieved with errors less than those required by the threshold theorem of FTQC.

Limits of optimal control yields achievable with quantum controllers

In quantum optimal control theory, kinematic bounds are the minimum and maximum values of the control objective achievable for any physically realizable system dynamics. For a given initial state of the system, these bounds depend on the nature and state of the controller. We consider a general situation where the controlled quantum system is coupled to both an external classical field (referred to as a classical controller) and an auxiliary quantum system (referred to as a quantum controller). In this general situation, the kinematic bound is between the classical kinematic bound (CKB), corresponding to the case when only the classical controller is available, and the quantum kinematic bound (QKB), corresponding to the ultimate physical limit of the objective's value. Specifically, when the control objective is the expectation value of a quantum observable (a Hermitian operator on the system's Hilbert space), the QKBs are the minimum and maximum eigenvalues of this operator. We present, both qualitatively and quantitatively, the necessary and sufficient conditions for surpassing the CKB and reaching the QKB, through the use of a quantum controller. The general conditions are illustrated by examples in which the system and controller are initially in thermal states. The obtained results provide a basis for the design of quantum controllers capable of maximizing the control yield and reaching the ultimate physical limit.

Searching for quantum optimal controls under severe constraints

The success of quantum optimal control for both experimental and theoretical objectives is connected to the topology of the corresponding control landscapes, which are free from local traps if three conditions are met: (1) the quantum system is controllable, (2) the Jacobian of the map from the control field to the evolution operator is of full rank, and (3) there are no constraints on the control field. This paper investigates how the violation of assumption (3) affects gradient searches for globally optimal control fields. The satisfaction of assumptions (1) and (2) ensures that the control landscape lacks fundamental traps, but certain control constraints can still introduce artificial traps. Proper management of these constraints is an issue of great practical importance for numerical simulations as well as optimization in the laboratory. Using optimal control simulations, we show that constraints on quantities such as the number of control variables, the control duration, and the field strength are potentially severe enough to prevent successful optimization of the objective. For each such constraint, we show that exceeding quantifiable limits can prevent gradient searches from reaching a globally optimal solution. These results demonstrate that careful choice of relevant control parameters helps to eliminate artificial traps and facilitate successful optimization.

Completely positive approximate solutions of driven open quantum systems.

We define a perturbative approximation for the solution of Lindblad master equations with time-dependent generators that satisfies the fundamental property of complete positivity, as essential for quantum simulations and optimal control. With explicit examples we show that ensuring this property substantially improves the accuracy of the perturbative approximation.

Suppression of strong-field ionization by optimal pulse shaping: Application to hydrogen and the hydrogen molecular ion

We investigate the ability of quantum optimal control theory to shape pulses suppressing strong-field ionization of a hydrogen atom and a ${\mathrm{H}}_{2}{}^{+}$ molecule. We show that considerable suppression of the ionization yield can be achieved for both H and ${\mathrm{H}}_{2}{}^{+}$ with optimal pulse shaping for a fixed fluence and pulse length. The mechanisms responsible for ionization suppression and the shape of the optimized pulse are different for infrared and ultraviolet laser fields. In the low-frequency regime the optimized pulse reduces the ionization yield by suppressing the highest peaks of the laser field. For the higher laser frequencies considered the ionization yield of H can be decreased by exciting low-lying resonances.

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