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Herschel Rabitz

Publications and source records attributed to Herschel Rabitz.

At least 109 records · Page 6Linked to original sources
arXiv2011

Exploring the top and bottom of the quantum control landscape

A controlled quantum system possesses a search landscape defined by the target physical objective as a function of the controls. This paper focuses on the landscape for the transition probability Pif between the states of a finite level quantum system. Traditionally, the controls are applied fields; here we extend the notion of control to also include the Hamiltonian structure, in the form of time independent matrix elements. Level sets of controls that produce the same transition probability value are shown to exist at the bottom Pif = 0.0 and top Pif = 1.0 of the landscape with the field and/or Hamiltonian structure as controls. We present an algorithm to continuously explore these level sets starting from an initial point residing at either extreme value of Pif . The technique can also identify control solutions that exhibit the desirable properties of (a) robustness at the top and (b) the ability to rapidly rise towards an optimal control from the bottom. Numerical simulations are presented to illustrate the varied control behavior at the top and bottom of the landscape for several simple model systems.

physics.chem-ph↗
arXiv2011

Volume Fractions of the Kinematic "Near-Critical" Sets of the Quantum Ensemble Control Landscape

An estimate is derived for the volume fraction of a subset $C_ε^{P} = \{U : ||grad J(U)|\leq ε\}\subset\mathrm{U}(N)$ in the neighborhood of the critical set $C^{P}\simeq\mathrm{U}(\mathbf{n})P\mathrm{U}(\mathbf{m})$ of the kinematic quantum ensemble control landscape J(U) = Tr(UρU' O), where $U$ represents the unitary time evolution operator, ρ is the initial density matrix of the ensemble, and O is an observable operator. This estimate is based on the Hilbert-Schmidt geometry for the unitary group and a first-order approximation of $||grad J(U)||^2$. An upper bound on these near-critical volumes is conjectured and supported by numerical simulation, leading to an asymptotic analysis as the dimension $N$ of the quantum system rises in which the volume fractions of these "near-critical" sets decrease to zero as $N$ increases. This result helps explain the apparent lack of influence exerted by the many saddles of $J$ over the gradient flow.

quant-ph↗
arXiv2011

Hamiltonian identification through enhanced observability utilizing quantum control

This paper considers Hamiltonian identification for a controllable quantum system with non-degenerate transitions and a known initial state. We assume to have at our disposal a single scalar control input and the population measure of only one state at an (arbitrarily large) final time T. We prove that the quantum dipole moment matrix is locally observable in the following sense: for any two close but distinct dipole moment matrices, we construct discriminating controls giving two different measurements. Such discriminating controls are constructed to have three well defined temporal components, as inspired by Ramsey interferometry. This result suggests that what may appear at first to be very restrictive measurements are actually rich for identification, when combined with well designed discriminating controls, to uniquely identify the complete dipole moment of such systems. The assessment supports the employment of quantum control as a promising means to achieve high quality identification of a Hamiltonian.

quant-ph↗
arXiv2011

Fidelity Between Unitary Operators and the Generation of Gates Robust Against Off-Resonance Perturbations

We perform a functional expansion of the fidelity between two unitary matrices in order to find the necessary conditions for the robust implementation of a target gate. Comparison of these conditions with those obtained from the Magnus expansion and Dyson series shows that they are equivalent in first order. By exploiting techniques from robust design optimization, we account for issues of experimental feasibility by introducing an additional criterion to the search for control pulses. This search is accomplished by exploring the competition between the multiple objectives in the implementation of the NOT gate by means of evolutionary multi-objective optimization.

quant-ph↗
arXiv2011

The role of controllability in optimizing quantum dynamics

This paper discusses the important role of controllability played on the complexity of optimizing quantum mechanical control systems. The study is based on a topology analysis of the corresponding quantum control landscape, which is referred to as the optimization objective as a functional of control fields. We find that the degree of controllability is closely relevant with the ruggedness of the landscape, which determines the search efficiency for global optima. This effect is demonstrated via the gate fidelity control landscape of a system whose controllability is restricted on a SU(2) dynamic symmetry group. We show that multiple local false traps (i.e., non-global suboptima) exist even if the target gate is realizable and that the number of these traps is increased by the loss of controllability, while the controllable systems are always devoid of false traps.

quant-ph↗
arXiv2011

All Possible Coupling Schemes in XY Spin Chains for Perfect State Transfer

We investigate quantum state transfer in XY spin chains and propose a recursive procedure to construct the nonuniform couplings of these chains with arbitrary length to achieve perfect state transfer(PST). We show that this method is capable of finding all possible coupling schemes for PST. These schemes, without external control fields, only involve preengineered couplings but not dynamical control of them, so they can be simply realized experimentally. The analytical solutions provide all information for coupling design.

quant-ph↗
arXiv2010

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 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.

quant-ph↗
arXiv2010

Unified analysis of terminal-time control in classical and quantum systems

Many phenomena in physics, chemistry, and biology involve seeking an optimal control to maximize an objective for a classical or quantum system which is open and interacting with its environment. The complexity of finding an optimal control for maximizing an objective is strongly affected by the possible existence of sub-optimal maxima. Within a unified framework under specified conditions, control objectives for maximizing at a terminal time physical observables of open classical and quantum systems are shown to be inherently free of sub-optimal maxima. This attractive feature is of central importance for enabling the discovery of controls in a seamless fashion in a wide range of phenomena transcending the quantum and classical regimes.

quant-ph↗
arXiv2010

General unifying features of controlled quantum phenomena

Many proposals have been put forth for controlling quantum phenomena, including open-loop, adaptive feedback, and real-time feedback control. Each of these approaches has been viewed as operationally, and even physically, distinct from the others. This work shows that all such scenarios inherently share the same fundamental control features residing in the topology of the landscape relating the target physical observable to the applied controls. This unified foundation may provide a basis for development of hybrid control schemes that would combine the advantages of the existing approaches to achieve the best overall performance.

quant-ph↗
arXiv2010

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. The current experimental successes would be impossible without the development of intense femtosecond laser sources and pulse shapers. The two most critical theoretical insights were (1) realizing that ultrafast atomic and molecular dynamics can be controlled via manipulation of quantum interferences and (2) 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, 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 adaptive feedback control experiments is guided by a learning algorithm, with stochastic methods proving to be especially effective. Adaptive feedback control 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 designs in experimental realizations, and real-time quantum feedback control. The paper concludes with a prospective of open research directions that are likely to attract significant attention in the future.

quant-ph↗
arXiv2009

Environment-invariant measure of distance between evolutions of an open quantum system

The problem of quantifying the difference between evolutions of an open quantum system (in particular, between the actual evolution of an open system and the ideal target operation on the corresponding closed system) is important in quantum control, especially in control of quantum information processing. Motivated by this problem, we develop a measure for evaluating the distance between unitary evolution operators of a composite quantum system that consists of a sub-system of interest (e.g., a quantum information processor) and environment. The main characteristic of this measure is the invariance with respect to the effect of the evolution operator on the environment, which follows from an equivalence relation that exists between unitary operators acting on the composite system, when the effect on only the sub-system of interest is considered. The invariance to the environment's transformation makes it possible to quantitatively compare the evolution of an open quantum system and its closed counterpart. The distance measure also determines the fidelity bounds of a general quantum channel (a completely positive and trace-preserving map acting on the sub-system of interest) with respect to a unitary target transformation. This measure is also independent of the initial state of the system and straightforward to numerically calculate. As an example, the measure is used in numerical simulations to evaluate fidelities of optimally controlled quantum gate operations (for one- and two-qubit systems), in the presence of a decohering environment. This example illustrates the utility of this measure for optimal control of quantum operations in the realistic case of open-system dynamics.

quant-ph↗
arXiv2009

The landscape of quantum transitions driven by single-qubit unitary transformations with implications for entanglement

This paper considers the control landscape of quantum transitions in multi-qubit systems driven by unitary transformations with single-qubit interaction terms. The two-qubit case is fully analyzed to reveal the features of the landscape including the nature of the absolute maximum and minimum, the saddle points and the absence of traps. The results permit calculating the Schmidt state starting from an arbitrary two-qubit state following the local gradient flow. The analysis of multi-qubit systems is more challenging, but the generalized Schmidt states may also be located by following the local gradient flow. Finally, we show the relation between the generalized Schmidt states and the entanglement measure based on the Bures distance.

quant-ph↗
arXiv2009

Singularities of Quantum Control Landscapes

A quantum control landscape is defined as the objective to be optimized as a function of the control variables. Existing empirical and theoretical studies reveal that most realistic quantum control landscapes are generally devoid of false traps. However, the impact of singular controls has yet to be investigated, which can arise due to a singularity on the mapping from the control to the final quantum state. We provide an explicit characterization of such controls that are strongly Hamiltonian-dependent and investigate their associated landscape geometry. Although in principle the singularities may correspond to local traps, we did not find any in numerical simulations. Also, as they occupy a small portion of the entire set of possible critical controls, their influence is expected to be much smaller than controls corresponding to the commonly located regular extremals. This observation supports the established ease of optimal searches to find high-quality controls in simulations and experiments.

quant-ph↗
arXiv2009

The Optimal Control Landscape for the Generation of Unitary Transformations with Constrained Dynamics

The reliable and precise generation of quantum unitary transformations is essential to the realization of a number of fundamental objectives, such as quantum control and quantum information processing. Prior work has explored the optimal control problem of generating such unitary transformations as a surface optimization problem over the quantum control landscape, defined as a metric for realizing a desired unitary transformation as a function of the control variables. It was found that under the assumption of non-dissipative and controllable dynamics, the landscape topology is trap-free, implying that any reasonable optimization heuristic should be able to identify globally optimal solutions. The present work is a control landscape analysis incorporating specific constraints in the Hamiltonian corresponding to certain dynamical symmetries in the underlying physical system. It is found that the presence of such symmetries does not destroy the trap-free topology. These findings expand the class of quantum dynamical systems on which control problems are intrinsically amenable to solution by optimal control.

quant-ph↗
arXiv2009

Strategies for optimal design for electrostatic energy storage in quantum multiwell heterostructures

The physical principles are studied for the optimal design of a quantum multiwell heterostructure working as an electrostatic energy storage device. We performed the search for an optimal multiwell trapping potential for electrons that results in the maximum static palarizability of the system. The response of the heterostructure is modeled quantum mechanically using nonlocal linear response theory. Three main design strategies are identified, which lead to the maximization of the stored energy. We found that the efficiency of each strategy crucially depends on the temperature and the broadening of electron levels. The energy density for optimized heterostructures can exceed the nonoptimized value by a factor more than $400$. These findings provide a basis for the development of new nanoscale capacitors with high energy density storage capabilities.

cond-mat.mes-hall↗