New formulations of monotonically convergent quantum control algorithms

Controlling NMR spin systems for quantum computation

Nuclear magnetic resonance is arguably both the best available quantum technology for implementing simple quantum computing experiments and the worst technology for building large scale quantum computers that has ever been seriously put forward. After a few years of rapid growth, leading to an implementation of Shor's quantum factoring algorithm in a seven-spin system, the field started to reach its natural limits and further progress became challenging. Rather than pursuing more complex algorithms on larger systems, interest has now largely moved into developing techniques for the precise and efficient manipulation of spin states with the aim of developing methods that can be applied in other more scalable technologies and within conventional NMR. However, the user friendliness of NMR implementations means that they remain popular for proof-of-principle demonstrations of simple quantum information protocols.

Quantum optimal control of multiple weakly interacting molecular rotors in the time-dependent Hartree approximation

We perform quantum optimal control simulations, based on the Time-Dependent Hartree (TDH) approximation, for systems of three to five dipole-dipole coupled OCS rotors. A control electric field is used to steer all of the individual rotors, arranged in chains and regular polygons in a plane, toward either identical or unique objectives. The goal is to explore the utility of the TDH approximation to model the field-induced dynamics of multiple interacting rotors in the weak dipole-dipole coupling regime. A stochastic hill climbing approach is employed to seek an optimal control field that achieves the desired objectives at a specified target time. We first show that multiple rotors in chain and polygon geometries can be identically oriented in the same direction; these cases do not significantly depend on the presence of the dipole-dipole interaction. Additionally, in particular geometrical arrangements, we demonstrate that individual rotors can be uniquely manipulated toward different objectives with the same field. Specifically, it is shown that for a three rotor chain, the two end rotors can be identically oriented in a specific direction while keeping the middle rotor in its ground state, and for an equilateral triangle, two rotors can be identically oriented in a specific direction while the third rotor is oriented in the opposite direction. These multirotor unique objective cases exploit the shape of the field in coordination with dipole-dipole coupling between the rotors. Comparisons to numerically exact calculations, utilizing the TDH-determined fields, are given for all optimal control studies involving systems of three rotors.

Optimal control of orientation and entanglement for two dipole–dipole coupled quantum planar rotors

Optimal control simulations are performed for orientation and entanglement of two dipole–dipole coupled identical quantum rotors.

Real-time 2D spatially selective MRI experiments: Comparative analysis of optimal control design methods

There is an increasing need for development of advanced radio-frequency (RF) pulse techniques in modern magnetic resonance imaging (MRI) systems driven by recent advancements in ultra-high magnetic field systems, new parallel transmit/receive coil designs, and accessible powerful computational facilities. 2D spatially selective RF pulses are an example of advanced pulses that have many applications of clinical relevance, e.g., reduced field of view imaging, and MR spectroscopy. The 2D spatially selective RF pulses are mostly generated and optimised with numerical methods that can handle vast controls and multiple constraints. With this study we aim at demonstrating that numerical, optimal control (OC) algorithms are efficient for the design of 2D spatially selective MRI experiments, when robustness towards e.g. field inhomogeneity is in focus. We have chosen three popular OC algorithms; two which are gradient-based, concurrent methods using first- and second-order derivatives, respectively; and a third that belongs to the sequential, monotonically convergent family. We used two experimental models: a water phantom, and an in vivo human head. Taking into consideration the challenging experimental setup, our analysis suggests the use of the sequential, monotonic approach and the second-order gradient-based approach as computational speed, experimental robustness, and image quality is key. All algorithms used in this work were implemented in the MATLAB environment and are freely available to the MRI community.

Optimized multiple-quantum filter for robust selective excitation of metabolite signals

The selective excitation of metabolite signals in vivo requires the use of specially adapted pulse techniques, in particular when the signals are weak and the resonances overlap with those of unwanted molecules. Several pulse sequences have been proposed for this spectral editing task. However, their performance is strongly degraded by unavoidable experimental imperfections. Here, we show that optimal control theory can be used to generate pulses and sequences that perform almost ideally over a range of rf field strengths and frequency offsets that can be chosen according to the specifics of the spectrometer or scanner being used. We demonstrate this scheme by applying it to lactate editing. In addition to the robust excitation, we also have designed the pulses to minimize the signal of unwanted molecular species.

Spin system trajectory analysis under optimal control pulses

Several methods are proposed for the analysis, visualization and interpretation of high-dimensional spin system trajectories produced by quantum mechanical simulations. It is noted that expectation values of specific observables in large spin systems often feature fast, complicated and hard-to-interpret time dynamics and suggested that populations of carefully selected subspaces of states are much easier to analyze and interpret. As an illustration of the utility of the proposed methods, it is demonstrated that the apparent "noisy" appearance of many optimal control pulses in NMR and EPR spectroscopy is an illusion - the underlying spin dynamics is shown to be smooth, orderly and very tightly controlled.

Dynamic nuclear polarization and optimal control spatial-selective 13C MRI and MRS

Aimed at (13)C metabolic magnetic resonance imaging (MRI) and spectroscopy (MRS) applications, we demonstrate that dynamic nuclear polarization (DNP) may be combined with optimal control 2D spatial selection to simultaneously obtain high sensitivity and well-defined spatial restriction. This is achieved through the development of spatial-selective single-shot spiral-readout MRI and MRS experiments combined with dynamic nuclear polarization hyperpolarized [1-(13)C]pyruvate on a 4.7 T pre-clinical MR scanner. The method stands out from related techniques by facilitating anatomic shaped region-of-interest (ROI) single metabolite signals available for higher image resolution or single-peak spectra. The 2D spatial-selective rf pulses were designed using a novel Krotov-based optimal control approach capable of iteratively fast providing successful pulse sequences in the absence of qualified initial guesses. The technique may be important for early detection of abnormal metabolism, monitoring disease progression, and drug research.

Fast numerical design of spatial-selective rf pulses in MRI using Krotov and quasi-Newton based optimal control methods

The use of increasingly strong magnetic fields in magnetic resonance imaging (MRI) improves sensitivity, susceptibility contrast, and spatial or spectral resolution for functional and localized spectroscopic imaging applications. However, along with these benefits come the challenges of increasing static field (B(0)) and rf field (B(1)) inhomogeneities induced by radial field susceptibility differences and poorer dielectric properties of objects in the scanner. Increasing fields also impose the need for rf irradiation at higher frequencies which may lead to elevated patient energy absorption, eventually posing a safety risk. These reasons have motivated the use of multidimensional rf pulses and parallel rf transmission, and their combination with tailoring of rf pulses for fast and low-power rf performance. For the latter application, analytical and approximate solutions are well-established in linear regimes, however, with increasing nonlinearities and constraints on the rf pulses, numerical iterative methods become attractive. Among such procedures, optimal control methods have recently demonstrated great potential. Here, we present a Krotov-based optimal control approach which as compared to earlier approaches provides very fast, monotonic convergence even without educated initial guesses. This is essential for in vivo MRI applications. The method is compared to a second-order gradient ascent method relying on the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method, and a hybrid scheme Krotov-BFGS is also introduced in this study. These optimal control approaches are demonstrated by the design of a 2D spatial selective rf pulse exciting the letters "JCP" in a water phantom.

Recent advances in nuclear magnetic resonance quantum information processing

Quantum information processors have the potential to drastically change the way we communicate and process information. Nuclear magnetic resonance (NMR) has been one of the first experimental implementations of quantum information processing (QIP) and continues to be an excellent testbed to develop new QIP techniques. We review the recent progress made in NMR QIP, focusing on decoupling, pulse engineering and indirect nuclear control. These advances have enhanced the capabilities of NMR QIP, and have useful applications in both traditional NMR and other QIP architectures.

Implementing quantum gates by optimal control with doubly exponential convergence

We introduce a novel algorithm for the task of coherently controlling a quantum mechanical system to implement any chosen unitary dynamics. It performs faster than existing state of the art methods by 1 to 3 orders of magnitude (depending on which one we compare to), particularly for quantum information processing purposes. This substantially enhances the ability to both study the control capabilities of physical systems within their coherence times, and constrain solutions for control tasks to lie within experimentally feasible regions. Natural extensions of the algorithm are also discussed.

Accelerated monotonic convergence of optimal control over quantum dynamics

The control of quantum dynamics is often concerned with finding time-dependent optimal control fields that can take a system from an initial state to a final state to attain the desired value of an observable. This paper presents a general method for formulating monotonically convergent algorithms to iteratively improve control fields. The formulation is based on a two-point boundary-value quantum control paradigm (TBQCP) expressed as a nonlinear integral equation of the first kind arising from dynamical invariant tracking control. TBQCP is shown to be related to various existing techniques, including local control theory, the Krotov method, and optimal control theory. Several accelerated monotonic convergence schemes for iteratively computing control fields are derived based on TBQCP. Numerical simulations are compared with the Krotov method showing that the new TBQCP schemes are efficient and remain monotonically convergent over a wide range of the iteration step parameters and the control pulse lengths, which is attributable to the trap-free character of the transition probability quantum dynamics control landscape.

Acceleration of quantum optimal control theory algorithms with mixing strategies

We propose the use of mixing strategies to accelerate the convergence of the common iterative algorithms utilized in quantum optimal control theory (QOCT). We show how the nonlinear equations of QOCT can be viewed as a "fixed-point" nonlinear problem. The iterative algorithms for this class of problems may benefit from mixing strategies, as it happens, e.g., in the quest for the ground-state density in Kohn-Sham density-functional theory. We demonstrate, with some numerical examples, how the same mixing schemes utilized in this latter nonlinear problem may significantly accelerate the QOCT iterative procedures.

Vibrationally selective optimal control of alignment and orientation using infrared laser pulses: application to carbon monoxide

Optimal control methods are used to study molecular alignment and orientation using infrared laser pulses. High order molecule-field interactions are taken into account through the use of the electric-nuclear Born-Oppenheimer approximation [G. G. Balint-Kurti et al., J. Chem. Phys. 122, 084110 (2005)]. High degrees of alignment and orientation are achieved by optimized infrared laser pulses of duration on the order of one rotational period of the molecule. It is shown that, through the incorporation of a vibrational projection operator into the optimization procedure, it is possible not only to maximize the alignment and orientation but also to bring the whole system into a single prescribed vibrational manifold. Numerical calculations are performed for carbon monoxide using ab initio potential energies computed in the presence of external electric fields.

Femtosecond laser pulse control of multidimensional vibrational dynamics: Computational studies on the pyrazine molecule

The multiconfiguration time-dependent Hartree (MCTDH) method is combined with the optimal control theory (OCT) to study femtosecond laser pulse control of multidimensional vibrational dynamics. Simulations are presented for the widely discussed three-electronic-level vibronic coupling model of pyrazine either in a three or four vibrational coordinate version. Thus, for the first time OCT is applied to a four-coordinate system. Different control tasks are investigated and also some general aspects of the OCT-MCTDH method combination are analyzed.

Pulse shaping for optimal control of molecular processes

In this paper, a new method is proposed to design optimized control fields with desired temporal and/or spectral properties. The method is based on penalizing the difference between an optimized field obtained from an iterative scheme and a reference field with desired temporal and/or spectral properties. Compared with the standard optimal control theory, the current method allows a simple, experimentally accessible field be found on the fly; while compared with parameter space searching optimization, the iterative nature of this method allows automatic exploration of the intrinsic mechanism of the population transfer. The method is illustrated by examing the optimal control of vibrational excitation of the Cl-O bond with both temporally and spectrally restricted pulses.

On the relationship between the local tracking procedures and monotonic schemes in quantum optimal control

Numerical simulations of (bilinear) quantum control often rely on either monotonically convergent algorithms or tracking schemes. However, despite their mathematical simplicity, very limited intuitive understanding exists at this time to explain the former type of algorithms. Departing from the usual mathematical formalization, we present in this paper an interpretation of the monotonic algorithms as finite horizon, local in time, tracking schemes. Our purpose is not to present a new class of procedures but rather to introduce the necessary rigorous framework that supports this interpretation. As a by-product we show that at each instant, estimates of the future quality of the current control field are available and used in the optimization. When the target is expressed as reaching a prescribed final state, we also present an intuitive geometrical interpretation as the minimization of the distance between two correlated trajectories: one starting from the given initial state and the other backward in time from the target state. As an illustration, a stochastic monotonic algorithm is introduced. Numerical discretizations of the two procedures are also presented.

Optimal molecular alignment and orientation through rotational ladder climbing

We study the control by electromagnetic fields of molecular alignment and orientation in a linear, rigid-rotor model. With the help of a monotonically convergent algorithm, we find that the optimal field is in the microwave part of the spectrum and acts by resonantly exciting the rotation of the molecule progressively from the ground state, i.e., by rotational ladder climbing. This mechanism is present not only when maximizing orientation or alignment, but also when using prescribed target states that simultaneously optimize the efficiency of orientation/alignment and its duration. The extension of the optimization method to consider a finite rotational temperature is also presented.

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.