Raj Chakrabarti

Control of quantum phenomena: past, present and future

Quantum control is concerned with active manipulation of physical and chemical processes on the atomic and molecular scale. This work presents a perspective of progress in the field of control over quantum phenomena, tracing the evolution of theoretical concepts and experimental methods from early developments to the most recent advances. Among numerous theoretical insights and technological improvements that produced the present state-of- the-art in quantum control, there have been several breakthroughs of foremost importance. On the technology side, the current experimental successes would be impossible without the development of intense femtosecond laser sources and pulse shapers. On the theory side, the two most critical insights were (i) realizing that ultrafast atomic and molecular dynamics can be controlled via manipulation of quantum interferences and (ii) understanding that optimally shaped ultrafast laser pulses are the most effective means for producing the desired quantum interference patterns in the controlled system. Finally, these theoretical and experimental advances were brought together by the crucial concept of adaptive feedback control (AFC), which is a laboratory procedure employing measurement-driven, closed-loop optimization to identify the best shapes of femtosecond laser control pulses for steering quantum dynamics towards the desired objective. Optimization in AFC experiments is guided by a learning algorithm, with stochastic methods proving to be especially effective. AFC of quantum phenomena has found numerous applications in many areas of the physical and chemical sciences, and this paper reviews the extensive experiments. Other subjects discussed include quantum optimal control theory, quantum control landscapes, the role of theoretical control

The enhancement of PCR amplification by low molecular-weight sulfones.

DNA amplification by polymerase chain reaction (PCR) is frequently complicated by the problems of low yield and specificity, especially when the GC content of the target sequence is high. A common approach to the optimization of such reactions is the addition of small quantities of certain organic chemicals, such as dimethylsulfoxide (DMSO), betaine, polyethylene glycol and formamide, to the reaction mixture. Even in the presence of such additives, however, the amplification of GC-rich templates is often ineffective. In this paper, we introduce a novel class of PCR-enhancing compounds, the low molecular-weight sulfones, that are effective in the optimization of high GC template amplification. We describe here the results of an extensive structure-activity investigation in which we studied the effects of a series of six different sulfones on PCR amplification. We identify two sulfones, sulfolane and methyl sulfone, that are especially potent enhancers of high GC template amplification, and show that these compounds often outperform DMSO and betaine, two of the most effective PCR enhancers currently used. We conclude with a brief discussion of the role that the sulfone functional group may play in such enhancement.

Quantum control landscapes

Numerous lines of experimental, numerical and analytical evidence indicate that it is surprisingly easy to locate optimal controls steering quantum dynamical systems to desired objectives. This has enabled the control of complex quantum systems despite the expense of solving the Schrödinger equation in simulations and the complicating effects of environmental decoherence in the laboratory. Recent work indicates that this simplicity originates in universal properties of the solution sets to quantum control problems that are fundamentally different from their classical counterparts. Here, we review studies that aim to systematically characterize these properties, enabling the classification of quantum control mechanisms and the design of globally efficient quantum control algorithms.

Search complexity and resource scaling for the quantum optimal control of unitary transformations

The optimal control of unitary transformations is a fundamental problem in quantum control theory and quantum information processing. The feasibility of performing such optimizations is determined by the computational and control resources required, particularly for systems with large Hilbert spaces. Prior work on unitary transformation control indicates that (i) for controllable systems, local extrema in the search landscape for optimal control of quantum gates have null measure, facilitating the convergence of local search algorithms, but (ii) the required time for convergence to optimal controls can scale exponentially with the Hilbert space dimension. Depending on the control-system Hamiltonian, the landscape structure and scaling may vary. This work introduces methods for quantifying Hamiltonian-dependent and kinematic effects on control optimization dynamics in order to classify quantum systems according to the search effort and control resources required to implement arbitrary unitary transformations.

Quantum Pareto optimal control

We describe algorithms and experimental strategies for the Pareto optimal control problem of simultaneously driving an arbitrary number of quantum observable expectation values to their respective extrema. Conven- tional quantum optimal control strategies are less effective at sampling points on the Pareto frontier of multi- observable control landscapes than they are at locating optimal solutions to single observable control problems. The present algorithms facilitate multiobservable optimization by following direct paths to the Pareto front, and are capable of continuously tracing the front once it is found to explore families of viable solutions. The numerical and experimental methodologies introduced are also applicable to other problems that require the simultaneous control of large numbers of observables, such as quantum optimal mixed state preparation. The methodology of quantum optimal control has been applied extensively to problems requiring the maximization of the expectation values of single quantum observables. Re- cently, an important new class of quantum control problems has surfaced, wherein the aim is the simultaneous maximiza- tion of the expectation values of multiple quantum observ- ables or, more generally, control over the full quantum state as encoded in the density matrix. Whereas various algo- rithms for the estimation of the density matrix have been reported in the literature 1, comparatively little attention has been devoted to the optimal control of the density matrix or multiobservables 2-4. Such methodologies are impor- tant in diverse applications including the control of product selectivity in coherently driven chemical reactions 3, the dynamical discrimination of like molecules 5,6, and the precise preparation of tailored mixed states. In a recent work 4, we reported an experimentally implementable control methodology—quantum multiobserv- able tracking control MOTC—that can be used to simulta- neously drive multiple observables to desired target expecta- tion values, based on control landscape gradient information. It was shown that, due to the fact that the critical manifolds of quantum control landscapes 7 have measure zero in the search domain, tracking control algorithms are typically un- obstructed when following paths to arbitrary targets in mul- tiobservable space. Therefore, these algorithms can facilitate multiobservable control by following direct paths to the tar- get expectation values, and are typically more efficient than algorithms based on optimization of a control cost func- tional. The MOTC method falls into the general category of continuation algorithms for multiobjective optimization 8. These were introduced several years ago as alternatives to stochastic multiobjective optimization algorithms, which do not exploit landscape gradient information. In this paper, we extend the theory of MOTC to more difficult multiobjective optimization problems such as multi- observable maximization, and propose experimental tech- niques for the implementation of MOTC in such multiob- servable control scenarios. Multiobjective maximization typically seeks to identify nondominated rather than com- pletely optimal solutions, which lie on the so-called Pareto frontier 9. Conventional multiobjective control algorithms require substantial search effort in order to adequately sample the Pareto frontier and locate solutions that strike the desired balance between the various objectives. Direct appli- cation of optimal control OC10 or MOTC algorithms is also inefficient in other applications, such as quantum state preparation, that require control of a large number m of ob- servables. When naively applied, both OC and MOTC fail to exploit the simple underlying geometry of quantum control land- scapes 7, by convoluting a the dynamic Schrodinger map tUT between time-dependent control fields and asso- ciated unitary propagators at the prescribed final time T with b the kinematic map UT(UT) between unitary propagators and associated observable expectation values. In multiobservable Pareto optimal control, the Pareto frontier on the domain UN has a simple geometric structure, which can be exploited to efficiently sample the corresponding frontier on the domain t. Here, we develop strategies that combine the application of MOTC with quantum state esti- mation and kinematic optimization on UN for the purpose of solving Pareto optimal control and other large m observ- able control problems. The paper is organized as follows. In Sec. II, we provide necessary preliminaries on Pareto optimal control and quan- tum multiobservable control cost functionals. Section III pre- sents analytical results on the distribution of Pareto optima in quantum multiobservable maximization problems. In Sec. IV, we describe how MOTC can be used to locate corre- sponding points on the dynamic Pareto front and to subse- quently explore families of fields within the front that mini- mize auxiliary, experimentally relevant costs. Section V proposes measurement strategies aimed at improving the ex- perimental efficiency of MOTC. In Sec. VI, we numerically implement Pareto optimal tracking control and illustrate the use of efficient measurement strategies for difficult problems with large numbers of observables m. We also examine the impact of prior state preparation and the advantages of using MOTC versus scalar cost function optimization for Pareto

Optimal control theory for continuous-variable quantum gates

The methodology of optimal control theory is applied to the problem of implementing quantum gates in continuous-variable (CV) systems with quadratic Hamiltonians. We demonstrate that it is possible to define a fidelity measure for CV gate optimization that is devoid of traps, such that the search for optimal control fields using local algorithms will not be hindered. The optimal control of several quantum computing gates, as well as that of algorithms composed of these primitives, is investigated using several typical physical models and compared for discrete-variable and continuous-variable quantum systems. Numerical simulations indicate that the optimization of generic CV quantum gates is inherently more expensive than that of generic discrete variable quantum gates, but can be routinely achieved for all the major classes of computing primitives. The exact-time controllability of CV systems, hitherto largely ignored in the design of information processing models, is shown to play an important role in determining the maximal achievable gate fidelity. Moreover, the ability to control interactions between qunits can be exploited to delimit the total control fluence. Future experimental model systems should carefully tune these parameters so as to enable the implementation of CV quantum information processing with optimal fidelity.

Developing Globally Compatible Institutional Infrastructures for Indian Higher Education.

The authors profile developments in the globalization of Indian higher education, with an emphasis on emerging globally compatible institutional infrastructures. In recent decades, there has been an enormous amount of brain drain: the exodus of the brightest professionals and students to other countries. The article argues that the implementation of robust student support and quality-assessment methodologies in Indian universities is essential if India is to become an attractor as well as a source for international student and faculty circulation. On-site research conducted with senior administrators and student body members in India indicates that the mounting pressure to develop such infrastructures is increasing the divergence between privately funded and traditional public schools in the Indian higher education landscape, as progressive private institutions position themselves to become competitive in burgeoning international partnerships. In particular, many new innovative private university programs and infrastructures have developed that address the demands of cross-circulation with Western academic programs.

Critical Landscape Topology for Optimization on the Symplectic Group

Optimization problems over compact Lie groups have been studied extensively due to their broad applications in linear programming and optimal control. This paper analyzes an optimization problem over a noncompact symplectic Lie group Sp(2N,ℝ), i.e., minimizing the Frobenius distance from a target symplectic transformation, which can be used to assess the fidelity function over dynamical transformations in classical mechanics and quantum optics. The topology of the set of critical points is proven to have a unique local minimum and a number of saddlepoint submanifolds, exhibiting the absence of local suboptima that may hinder the search for ultimate optimal solutions. Compared with those of previously studied problems on compact Lie groups, such as the orthogonal and unitary groups, the topology is more complicated due to the significant nonlinearity brought by the incompatibility of the Frobenius norm with the pseudo-Riemannian structure on the symplectic group.

Quantum multiobservable control

We present deterministic algorithms for the simultaneous control of an arbitrary number of quantum observables. Unlike optimal control approaches based on cost function optimization, quantum multiobservable tracking control (MOTC) is capable of tracking predetermined homotopic trajectories to target expectation values in the space of multiobservables. The convergence of these algorithms is facilitated by the favorable critical topology of quantum control landscapes. Fundamental properties of quantum multiobservable control landscapes, including the MOTC Gramian matrix, are introduced. The effects of multiple control objectives on the structure and complexity of optimal fields are examined. With minor modifications, the techniques described herein can be applied to general quantum multiobjective control problems.

Robustness of controlled quantum dynamics

Control of multi-level quantum systems is sensitive to implementation errors in the control field and uncertainties associated with system Hamiltonian parameters. A small variation in the control field spectrum or the system Hamiltonian can cause an otherwise optimal field to deviate from controlling desired quantum state transitions and reaching a particular objective. An accurate analysis of robustness is thus essential in understanding and achieving model-based quantum control, such as in control of chemical reactions based on ab initio or experimental estimates of the molecular Hamiltonian. In this paper, theoretical foundations for quantum control robustness analysis are presented from both a distributional perspective - in terms of moments of the transition amplitude, interferences, and transition probability - and a worst-case perspective. Based on this theory, analytical expressions and a computationally efficient method for determining the robustness of coherently controlled quantum dynamics are derived. The robustness analysis reveals that there generally exists a set of control pathways that are more resistant to destructive interferences in the presence of control field and system parameter uncertainty. These robust pathways interfere and combine to yield a relatively accurate transition amplitude and high transition probability when uncertainty is present.